Pi

Pi

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Something's going on. It has to do with that number. There's an answer in that number.

You see the simplicity of the circle. You see the maddening complexity of the endless string of numbers, 3.14 off into infinity.

I'm trying to understand our world. I don't deal with petty materialists like you.

12:53-Restate my assumptions: Mathematics is the language of nature. Everything around us can be represented by numbers. If one graphs the system of any numbers, patterns emerge, therefore there are patterns everywhere in nature.

12:50, press return...

Failed treatments to date: Beta blockers, calcium channel blockers, adrenalin injections, high dose ibuprofen, steroids, Trager Mentastics, violent exercise, cafergot suppositories, caffeine, acupuncture, marijuana, Percodan, Midrine, Tenormin, Sansert, homeopathics. No results. No results...

10:15, personal note: It's fair to say I'm stepping out on a limb, but I am on the edge and that's where it happens.

My new hypothesis: If we're built from Spirals while living in a giant Spiral, then is it possible that everything we put our hands to is infused with the Spiral?

Encyclopedia


is a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 that is the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of any circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. is approximately equal to 3.14. Many formulae in mathematics, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 involve , which makes it one of the most important mathematical constants. For instance, the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...

 is equal to times the square of the radius of the circle.

is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, which means that its value cannot be expressed exactly as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 having integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

. is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Probably because of the simplicity of its definition, has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter was first adopted for the number as an abbreviation of the Greek word for perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 who provided an approximation of the number during the 3rd century BC
3rd century BC
The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period.-Overview:...

, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

, who computed a 35-digit approximation around the year 1600.

The Greek letter


The Latin name of the Greek letter
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

 is pi. When referring to the constant, the symbol is pronounced like the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "" because "" is the first letter of the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 word περιφέρεια "periphery" (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1737. William Jones wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to  ...  3.14159, etc. =  ...


When used as a symbol for the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

, the Greek letter is not capitalized at the beginning of a sentence. The capital letter (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition


In Euclidean plane geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, is defined as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

  to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

  :


The ratio is constant, regardless of a circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference , preserving the ratio .

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, which can be a problem when occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define without reference to geometry, instead selecting one of its analytic
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 properties as a definition. A common choice is to define as twice the smallest positive for which the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 cos equals zero.


Irrationality and transcendence


is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, meaning that it cannot be written as the ratio of two integers
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, meaning that there is no polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 coefficients for which is a root. An important consequence of the transcendence of is the fact that it is not constructible
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.

Decimal representation


The decimal representation of truncated
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

 to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 is:

.


Various online web sites provide to many more digits. While the decimal representation of has been computed to more than a trillion (1012) digits, elementary applications
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

 of less than one millimetre, and the decimal representation of truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 with precision comparable to the radius of a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

.

Because is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, its decimal representation does not repeat
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating 's properties. Despite much analytical work, and supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

 calculations that have determined over 1 trillion digits of the decimal representation of , no simple base-10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 pattern in the digits has ever been found. Digits of the decimal representation of are available on many web pages, and there is software for calculating the decimal representation of  to billions of digits on any personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...

.

Estimating the value


Numeral system Approximation of
Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

Hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

Sexagesimal (used by ancients, including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

)
= 377/120
Rational approximations 3, , , , , , ...
(listed in order of increasing accuracy)
Continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...


(This fraction is not periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. Shown in linear notation)
Generalized continued fraction expression

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999
Feynman point
The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

The earliest numerical approximation of is almost certainly the value .

In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of an inscribed regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 hexagon to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, is to calculate the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, , of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 with sides circumscribed around a circle with diameter . Then compute the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 as increases to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

:


This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:


can also be calculated using purely mathematical methods. Due to the transcendental nature of , there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to . The more terms included in a calculation, the closer to the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. However, some are quite simple, such as this form of the Gregory–Leibniz series:


While that series is easy to write and calculate, it is not immediately obvious why it yields . In addition, this series converges so slowly that nearly 300 terms are needed to calculate correctly to two decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...




and then define


then computing will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

For many purposes, 3.14 or 227 is close enough, although engineers often use 3.1416 (5 significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...

) or 3.14159 (6 significant figures) for more precision. The approximations 227 and 355113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 expansion of . The approximation 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

; the next good approximation 5216316604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of . For extremely accurate approximations, either Ramanujan's approximation of (3.14159265258...) or 10399333102 (3.14159265301...) are used, which are both accurate to 10 significant figures.

History



The Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...

 at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2. The same apotropaic proportions were used earlier at the Pyramid of Meidum
Meidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...

 c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of , in practice they used it". Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design". Others have argued that the Ancient Egyptians had no concept of and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).

The early history of from textual sources roughly parallels the development of mathematics as a whole.

Antiquity


The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.

The Indian text Shatapatha Brahmana
Shatapatha Brahmana
The Shatapatha Brahmana is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina and Kanva , with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17...

 (composed between the 8th to 6th centuries BCE, Iron Age India
Iron Age India
Iron Age India, the Iron Age in the Indian subcontinent, succeeds the Late Harappan culture, also known as the last phase of the Indus Valley Tradition...

) gives as 339/108 ≈ 3.139. It has been suggested that passages in the and discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.

Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 (287–212 BC) was the first to estimate rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s and calculating the outer and inner polygons' respective perimeters: By using the equivalent of 96-sided polygons, he proved that The average of these values is about 3.14185.

Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, in his Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, gives a value of 3.1416, which he may have obtained from Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

.

Around AD 265, the Wei Kingdom mathematician Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

 provided a simple and rigorous iterative algorithm to calculate to any degree of accuracy. He himself carried through the calculation to a 3072-gon (i.e. a 3072-sided polygon) and obtained an approximate value for of 3.1416. Later, Liu Hui invented a quick method of calculating and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 demonstrated that  ≈ 355/113 (≈ 3.1415929
), and showed that 3.1415926 <  < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of available for the next 900 years.

Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

 mentions with certainty the irrationality of in the 12th century. This was proved in 1768 by Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 calculus. One of those, due to Ivan Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...

, is widely known. A somewhat earlier similar proof is by Mary Cartwright
Mary Cartwright
Dame Mary Lucy Cartwright DBE FRS was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England...

.

Second millennium AD


Until the second millennium AD
2nd millennium
File:2nd millennium montage.png|From left, clockwise: In 1492, Christopher Columbus; The American Revolution; The French Revolution; The Atomic Bomb from World War II; An alternate source of light, the Light Bulb; For the first time, a human being sets foot on the moon in 1969 during the Apollo 11...

, estimations of were accurate to fewer than 10 decimal digits. The next major advances in the study of came with the development of infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and subsequently with the discovery of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, which permit the estimation of to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 found the first known such series:


This is now known as the Madhava–Leibniz series or Gregory–Leibniz series since it was rediscovered by James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into


Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 was able to estimate as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, who gave an estimate that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

 (1540–1610), who used a geometric method to give an estimate of that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

 in 1593. Another famous result is Wallis' product,


by John Wallis in 1655. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 derived the arcsin series for in 1665–66 and calculated 15 digits:


although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin
John Machin
John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.Machin's formula is:...

 was the first to compute 100 decimals of , using the arctan series in the formula


with


Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase
Zacharias Dase
Johann Martin Zacharias Dase was a German mental calculator.He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on him. He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing...

, who in 1844 employed a Machin-like formula to calculate 200 decimals of in his head at the behest of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

. The best value at the end of the 19th century was due to William Shanks
William Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...

, who took 15 years to calculate with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about 's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

 proved the irrationality of in 1761, and Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 also proved in 1794 2 to be irrational. When Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1735 solved the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

, finding the exact value of the Riemann zeta function of 2,


which is 2/6, he established a deep connection between and the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Both Legendre and Euler speculated that might be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which was finally proved in 1882 by Ferdinand von Lindemann
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

.

Computation in the computer age


Practically, one needs only 39 digits of to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

The advent of digital computers in the 20th century led to an increased rate of new calculation records. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 et al. used ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 to compute 2037 digits of in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 found many new formulas for , some remarkable for their elegance, mathematical depth and rapid convergence. One of his formulas is the series,


where ! is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 of .

A collection of some others are in the table below:


where


is the Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 for the falling factorial.

The related one found by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....

 in 1987 is


which delivers 14 digits per term. The Chudnovskys used this formula to set several computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for calculating software that runs on personal computers, as opposed to the supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s used to set modern records.
On August 6, 2010, PhysOrg.com reported that Japanese and American computer experts Shigeru Kondo and Alexander Yee said they've calculated the value of to 5 trillion decimal places on a personal computer, double the previous record.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist, born in 1946. He holds the position of Distinguished Professor of Mathematics and Computer Science with a joint appointment in the Mathematical Sciences Institute and the College of Engineering and Computer Science at...

 and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting


and iterating


until and are close enough. Then the estimate for is given by


Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...

 and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

. The methods have been used by Yasumasa Kanada
Yasumasa Kanada
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of π. He has set the record 11 of the past 21 times....

 and team to set most of the calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record was almost 2.7 trillion digits. This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

Another method for fast calculation of the constant is the method for fast summing series of special form FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

. To calculate the it's possible to use the Euler formula
and apply the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 to sum the Taylor series for
One can apply the same procedure also to the other special series approximating the constant . Besides the formulas representing the via arctangents, the new formulas for derived in the 1990s by S. Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, F. Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 and some other computer scientists, are good for fast summing via the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 and fast computation of the constant .

An important recent development was the Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 (BBP formula), discovered by Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 and named after the authors of the paper in which the formula was first published, David H. Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

, Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

, and Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

. The formula,


is remarkable because it allows extracting any individual hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

 or binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 digit of without calculating all the preceding ones. Between 1998 and 2000, the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

 project PiHex
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of Pi, the greatest calculation of Pi ever successfully attempted. 1,246 contributors used idle time slices on almost two thousand computers to make its calculations...

 used a modification of the BBP formula due to Fabrice Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 to compute the quadrillionth (1,000,000,000,000,000:th) bit of , which turned out to be 0.

If a formula of the form


were found where and are positive integers and and are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of at any position in base without computing all the preceding digits in that base, in a time just depending on the size of the integer and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer , and requiring modest computing resources. The previous formula (found by Plouffe for with  = 2 and  = 4, but also found for log(9/10) and for a few other irrational constants), implies that is a SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

 number.

In September 2010, Yahoo!
Yahoo!
Yahoo! Inc. is an American multinational internet corporation headquartered in Sunnyvale, California, United States. The company is perhaps best known for its web portal, search engine , Yahoo! Directory, Yahoo! Mail, Yahoo! News, Yahoo! Groups, Yahoo! Answers, advertising, online mapping ,...

 employee Nicholas Sze used the company's Hadoop production application to compute 256 bits of starting at a position a little before the two-quadrillionth (2,000,000,000,000,000th) bit, doubling the previous record by PiHex. The record was broken on 1,000 of Yahoo!'s computers over a 23-day period. The formula is used to compute a single bit of in a small set of mathematical steps.

In 2006, Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, using the integer relation algorithm
Integer relation algorithm
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such thata_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,...

 PSLQ, found a series of formulas. Let =
{{pp-semi|small=yes}}
{{Two other uses|the number|the Greek letter|Pi (letter)}}


{{Pi box}}
{{pi}} (sometimes written pi) is a
mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 that is the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of any circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. {{pi}} is approximately equal to 3.14. Many formulae in mathematics, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 involve {{pi}}, which makes it one of the most important mathematical constants. For instance, the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...

 is equal to {{pi}} times the square of the radius of the circle.

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, which means that its value cannot be expressed exactly as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 having integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine {{pi}} more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Probably because of the simplicity of its definition, {{pi}} has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of {{pi}} are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter {{pi}} was first adopted for the number as an abbreviation of the Greek word for perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 who provided an approximation of the number during the 3rd century BC
3rd century BC
The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period.-Overview:...

, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

, who computed a 35-digit approximation around the year 1600.

The Greek letter


{{Main|Pi (letter)}}
The Latin name of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

 is pi. When referring to the constant, the symbol {{pi}} is pronounced like the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "{{pi}}" because "{{pi}}" is the first letter of the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 word περιφέρεια "periphery"{{Citation needed|date=May 2011}} (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1737. William Jones wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to  ...  3.14159, etc. = {{pi}} ...


When used as a symbol for the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

, the Greek letter ({{pi}}) is not capitalized at the beginning of a sentence. The capital letter {{PI}} (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition


In Euclidean plane geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, {{pi}} is defined as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

  {{math|C}} to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

  {{math|d}}:


The ratio {{math|C/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{math|d}} of another circle it will also have twice the circumference {{math|C}}, preserving the ratio {{math|C/d}}.

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, which can be a problem when {{pi}} occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define {{pi}} without reference to geometry, instead selecting one of its analytic
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 properties as a definition. A common choice is to define {{pi}} as twice the smallest positive {{math|x}} for which the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 cos({{math|x}}) equals zero.


Irrationality and transcendence


{{Main|Proof that π is irrational}}

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, meaning that it cannot be written as the ratio of two integers
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, meaning that there is no polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 coefficients for which {{pi}} is a root. An important consequence of the transcendence of {{pi}} is the fact that it is not constructible
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.{{cite book
| author=Schlager, Neil; Lauer, Josh | year=2001 | page=185
| title=Science and Its Times: Understanding the Social Significance of Scientific Discovery
| volume=1 | series=Science and Its Times
| publisher=Gale Group | isbn=0787639338 }}

Decimal representation


{{See also|Approximations of π}}

The decimal representation of {{pi}} truncated
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

 to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 is:

{{gaps|lhs={{pi}}|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510|...}}.


Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (1012) digits, elementary applications
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of {{pi}} truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

 of less than one millimetre, and the decimal representation of {{pi}} truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 with precision comparable to the radius of a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

.

Because {{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, its decimal representation does not repeat
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating {{pi}}'s properties. Despite much analytical work, and supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

 calculations that have determined over 1 trillion digits of the decimal representation of {{pi}}, no simple base-10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 pattern in the digits has ever been found. Digits of the decimal representation of {{pi}} are available on many web pages, and there is software for calculating the decimal representation of {{pi}} to billions of digits on any personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...

.

Estimating the value


{{Main|Approximations of π}}
Numeral system Approximation of {{pi}}
Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

{{gaps 26535|89793|23846|26433|83279|50288...}}
Hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

{{gaps A8885|A308D|31319...}}
Sexagesimal (used by ancients, including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

)
{{gaps 3 ; 8′ 30″}} = 377/120
Rational approximations 3, {{frac|22|7}}, {{frac|333|106}}, {{frac|355|113}}, {{frac|52163|16604}}, {{frac|103993|33102}}, ...
(listed in order of increasing accuracy)
Continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

{{nowrap|[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...]}}
(This fraction is not periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. Shown in linear notation)
Generalized continued fraction expression

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999
Feynman point
The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

The earliest numerical approximation of {{pi}} is almost certainly the value {{num|3}}.{{verification failed|date=November 2011}}

In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of an inscribed regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 hexagon to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

{{pi}} can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, is to calculate the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, {{math|P}}{{math|n}}, of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 with {{math|n}} sides circumscribed around a circle with diameter {{math|d}}. Then compute the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 as {{math|n}} increases to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

:


This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:


{{pi}} can also be calculated using purely mathematical methods. Due to the transcendental nature of {{pi}}, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to {{pi}}. The more terms included in a calculation, the closer to {{pi}} the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. However, some are quite simple, such as this form of the Gregory–Leibniz series:


While that series is easy to write and calculate, it is not immediately obvious why it yields {{pi}}. In addition, this series converges so slowly that nearly 300 terms are needed to calculate {{pi}} correctly to two decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...




and then define


then computing will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

For many purposes, 3.14 or 227 is close enough, although engineers often use 3.1416 (5 significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...

) or 3.14159 (6 significant figures) for more precision.{{Citation needed|date=March 2011}} The approximations 227 and 355113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 expansion of {{pi}}. The approximation 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

; the next good approximation 5216316604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of {{pi}}. For extremely accurate approximations, either Ramanujan's approximation of (3.14159265258...) or 10399333102 (3.14159265301...) are used, which are both accurate to 10 significant figures.

History


{{See also|Chronology of computation of π|Approximations of π}}
The Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...

 at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2{{pi}}. The same apotropaic proportions were used earlier at the Pyramid of Meidum
Meidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...

 c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of {{pi}}, in practice they used it". Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design". Others have argued that the Ancient Egyptians had no concept of {{pi}} and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).

The early history of {{pi}} from textual sources roughly parallels the development of mathematics as a whole.

Antiquity


The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.

The Indian text Shatapatha Brahmana
Shatapatha Brahmana
The Shatapatha Brahmana is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina and Kanva , with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17...

 (composed between the 8th to 6th centuries BCE, Iron Age India
Iron Age India
Iron Age India, the Iron Age in the Indian subcontinent, succeeds the Late Harappan culture, also known as the last phase of the Indus Valley Tradition...

) gives {{pi}} as 339/108 ≈ 3.139. It has been suggested that passages in the {{bibleverse|1|Kings|7:23|NKJV}} and {{bibleverse|2|Chronicles|4:2|NKJV}} discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered {{pi}} to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.

{{multiple image
| direction = vertical
| width = 200
| image1 = Cutcircle2.svg
| caption1 = Estimating {{pi}} with inscribed polygons
| image2 = Archimedes pi.svg
| caption2 = Estimating {{pi}} with circumscribed and inscribed polygons
}}Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 (287–212 BC) was the first to estimate {{pi}} rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s and calculating the outer and inner polygons' respective perimeters: By using the equivalent of 96-sided polygons, he proved that The average of these values is about 3.14185.

Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, in his Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, gives a value of 3.1416, which he may have obtained from Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

.

Around AD 265, the Wei Kingdom mathematician Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

 provided a simple and rigorous iterative algorithm to calculate {{pi}} to any degree of accuracy. He himself carried through the calculation to a 3072-gon (i.e. a 3072-sided polygon) and obtained an approximate value for {{pi}} of 3.1416. Later, Liu Hui invented a quick method of calculating {{pi}} and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 demonstrated that {{pi}} ≈ 355/113 (≈ 3.1415929
), and showed that 3.1415926 < {{pi}} < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.

Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

 mentions with certainty the irrationality of {{pi}} in the 12th century. This was proved in 1768 by Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 calculus. One of those, due to Ivan Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...

, is widely known. A somewhat earlier similar proof is by Mary Cartwright
Mary Cartwright
Dame Mary Lucy Cartwright DBE FRS was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England...

.

Second millennium AD


Until the second millennium AD
2nd millennium
File:2nd millennium montage.png|From left, clockwise: In 1492, Christopher Columbus; The American Revolution; The French Revolution; The Atomic Bomb from World War II; An alternate source of light, the Light Bulb; For the first time, a human being sets foot on the moon in 1969 during the Apollo 11...

, estimations of {{pi}} were accurate to fewer than 10 decimal digits. The next major advances in the study of {{pi}} came with the development of infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and subsequently with the discovery of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, which permit the estimation of {{pi}} to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 found the first known such series:


This is now known as the Madhava–Leibniz series or Gregory–Leibniz series since it was rediscovered by James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into


Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 was able to estimate {{pi}} as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, who gave an estimate {{pi}} that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

 (1540–1610), who used a geometric method to give an estimate of {{pi}} that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. {{pi}} is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

 in 1593. Another famous result is Wallis' product,


by John Wallis in 1655. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 derived the arcsin series for {{pi}} in 1665–66 and calculated 15 digits:


although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin
John Machin
John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.Machin's formula is:...

 was the first to compute 100 decimals of {{pi}}, using the arctan series in the formula


with


Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating {{pi}} well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase
Zacharias Dase
Johann Martin Zacharias Dase was a German mental calculator.He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on him. He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing...

, who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

. The best value at the end of the 19th century was due to William Shanks
William Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...

, who took 15 years to calculate {{pi}} with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about {{pi}}'s nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

 proved the irrationality of {{pi}} in 1761, and Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 also proved in 1794 {{pi}}2 to be irrational. When Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1735 solved the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

, finding the exact value of the Riemann zeta function of 2,


which is {{pi}}2/6, he established a deep connection between {{pi}} and the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Both Legendre and Euler speculated that {{pi}} might be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which was finally proved in 1882 by Ferdinand von Lindemann
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

.

Computation in the computer age


Practically, one needs only 39 digits of {{pi}} to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

The advent of digital computers in the 20th century led to an increased rate of new {{pi}} calculation records. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 et al. used ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 to compute 2037 digits of {{pi}} in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 found many new formulas for {{pi}}, some remarkable for their elegance, mathematical depth and rapid convergence. One of his formulas is the series,


where {{math|k}}! is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 of {{math|k}}.

A collection of some others are in the table below:


where


is the Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 for the falling factorial.

The related one found by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....

 in 1987 is


which delivers 14 digits per term. The Chudnovskys used this formula to set several {{pi}} computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for {{pi}} calculating software that runs on personal computers, as opposed to the supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s used to set modern records.
On August 6, 2010, PhysOrg.com reported that Japanese and American computer experts Shigeru Kondo and Alexander Yee said they've calculated the value of {{pi}} to 5 trillion decimal places on a personal computer, double the previous record.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist, born in 1946. He holds the position of Distinguished Professor of Mathematics and Computer Science with a joint appointment in the Mathematical Sciences Institute and the College of Engineering and Computer Science at...

 and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting


and iterating


until {{math|a}}{{math|n}} and {{math|b}}{{math|n}} are close enough. Then the estimate for {{pi}} is given by


Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...

 and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

. The methods have been used by Yasumasa Kanada
Yasumasa Kanada
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of π. He has set the record 11 of the past 21 times....

 and team to set most of the {{pi}} calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record was almost 2.7 trillion digits. This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

Another method for fast calculation of the constant {{pi}} is the method for fast summing series of special form FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

. To calculate the {{pi}} it's possible to use the Euler formula
and apply the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 to sum the Taylor series for
One can apply the same procedure also to the other special series approximating the constant {{pi}}. Besides the formulas representing the {{pi}} via arctangents, the new formulas for {{pi}} derived in the 1990s by S. Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, F. Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 and some other computer scientists, are good for fast summing via the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 and fast computation of the constant {{pi}}.

An important recent development was the Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 (BBP formula), discovered by Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 and named after the authors of the paper in which the formula was first published, David H. Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

, Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

, and Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

. The formula,


is remarkable because it allows extracting any individual hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

 or binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 digit of {{pi}} without calculating all the preceding ones. Between 1998 and 2000, the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

 project PiHex
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of Pi, the greatest calculation of Pi ever successfully attempted. 1,246 contributors used idle time slices on almost two thousand computers to make its calculations...

 used a modification of the BBP formula due to Fabrice Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 to compute the quadrillionth (1,000,000,000,000,000:th) bit of {{pi}}, which turned out to be 0.

If a formula of the form


were found where {{math|b}} and {{math|c}} are positive integers and {{math|p}} and {{math|p}} are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of {{pi}} at any position in base {{math|b}}{{math|c}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{math|k}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer {{math|k}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{math|b}} = 2 and {{math|c}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

 number.{{Citation needed|date=March 2011}}

In September 2010, Yahoo!
Yahoo!
Yahoo! Inc. is an American multinational internet corporation headquartered in Sunnyvale, California, United States. The company is perhaps best known for its web portal, search engine , Yahoo! Directory, Yahoo! Mail, Yahoo! News, Yahoo! Groups, Yahoo! Answers, advertising, online mapping ,...

 employee Nicholas Sze used the company's Hadoop production application to compute 256 bits of {{pi}} starting at a position a little before the two-quadrillionth (2,000,000,000,000,000th) bit, doubling the previous record by PiHex. The record was broken on 1,000 of Yahoo!'s computers over a 23-day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.

In 2006, Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, using the integer relation algorithm
Integer relation algorithm
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such thata_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,...

 PSLQ, found a series of formulas. Let {{math|q}} =
{{pp-semi|small=yes}}
{{Two other uses|the number|the Greek letter|Pi (letter)}}


{{Pi box}}
{{pi}} (sometimes written pi) is a
mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 that is the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of any circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. {{pi}} is approximately equal to 3.14. Many formulae in mathematics, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 involve {{pi}}, which makes it one of the most important mathematical constants. For instance, the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...

 is equal to {{pi}} times the square of the radius of the circle.

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, which means that its value cannot be expressed exactly as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 having integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine {{pi}} more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Probably because of the simplicity of its definition, {{pi}} has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of {{pi}} are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter {{pi}} was first adopted for the number as an abbreviation of the Greek word for perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 who provided an approximation of the number during the 3rd century BC
3rd century BC
The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period.-Overview:...

, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

, who computed a 35-digit approximation around the year 1600.

The Greek letter


{{Main|Pi (letter)}}
The Latin name of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

 is pi. When referring to the constant, the symbol {{pi}} is pronounced like the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "{{pi}}" because "{{pi}}" is the first letter of the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 word περιφέρεια "periphery"{{Citation needed|date=May 2011}} (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1737. William Jones wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to  ...  3.14159, etc. = {{pi}} ...


When used as a symbol for the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

, the Greek letter ({{pi}}) is not capitalized at the beginning of a sentence. The capital letter {{PI}} (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition


In Euclidean plane geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, {{pi}} is defined as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

  {{math|C}} to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

  {{math|d}}:


The ratio {{math|C/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{math|d}} of another circle it will also have twice the circumference {{math|C}}, preserving the ratio {{math|C/d}}.

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, which can be a problem when {{pi}} occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define {{pi}} without reference to geometry, instead selecting one of its analytic
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 properties as a definition. A common choice is to define {{pi}} as twice the smallest positive {{math|x}} for which the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 cos({{math|x}}) equals zero.


Irrationality and transcendence


{{Main|Proof that π is irrational}}

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, meaning that it cannot be written as the ratio of two integers
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, meaning that there is no polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 coefficients for which {{pi}} is a root. An important consequence of the transcendence of {{pi}} is the fact that it is not constructible
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.{{cite book
| author=Schlager, Neil; Lauer, Josh | year=2001 | page=185
| title=Science and Its Times: Understanding the Social Significance of Scientific Discovery
| volume=1 | series=Science and Its Times
| publisher=Gale Group | isbn=0787639338 }}

Decimal representation


{{See also|Approximations of π}}

The decimal representation of {{pi}} truncated
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

 to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 is:

{{gaps|lhs={{pi}}|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510|...}}.


Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (1012) digits, elementary applications
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of {{pi}} truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

 of less than one millimetre, and the decimal representation of {{pi}} truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 with precision comparable to the radius of a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

.

Because {{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, its decimal representation does not repeat
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating {{pi}}'s properties. Despite much analytical work, and supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

 calculations that have determined over 1 trillion digits of the decimal representation of {{pi}}, no simple base-10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 pattern in the digits has ever been found. Digits of the decimal representation of {{pi}} are available on many web pages, and there is software for calculating the decimal representation of {{pi}} to billions of digits on any personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...

.

Estimating the value


{{Main|Approximations of π}}
Numeral system Approximation of {{pi}}
Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

{{gaps 26535|89793|23846|26433|83279|50288...}}
Hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

{{gaps A8885|A308D|31319...}}
Sexagesimal (used by ancients, including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

)
{{gaps 3 ; 8′ 30″}} = 377/120
Rational approximations 3, {{frac|22|7}}, {{frac|333|106}}, {{frac|355|113}}, {{frac|52163|16604}}, {{frac|103993|33102}}, ...
(listed in order of increasing accuracy)
Continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

{{nowrap|[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...]}}
(This fraction is not periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. Shown in linear notation)
Generalized continued fraction expression

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999
Feynman point
The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

The earliest numerical approximation of {{pi}} is almost certainly the value {{num|3}}.{{verification failed|date=November 2011}}

In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of an inscribed regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 hexagon to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

{{pi}} can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, is to calculate the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, {{math|P}}{{math|n}}, of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 with {{math|n}} sides circumscribed around a circle with diameter {{math|d}}. Then compute the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 as {{math|n}} increases to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

:


This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:


{{pi}} can also be calculated using purely mathematical methods. Due to the transcendental nature of {{pi}}, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to {{pi}}. The more terms included in a calculation, the closer to {{pi}} the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. However, some are quite simple, such as this form of the Gregory–Leibniz series:


While that series is easy to write and calculate, it is not immediately obvious why it yields {{pi}}. In addition, this series converges so slowly that nearly 300 terms are needed to calculate {{pi}} correctly to two decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...




and then define


then computing will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

For many purposes, 3.14 or 227 is close enough, although engineers often use 3.1416 (5 significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...

) or 3.14159 (6 significant figures) for more precision.{{Citation needed|date=March 2011}} The approximations 227 and 355113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 expansion of {{pi}}. The approximation 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

; the next good approximation 5216316604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of {{pi}}. For extremely accurate approximations, either Ramanujan's approximation of (3.14159265258...) or 10399333102 (3.14159265301...) are used, which are both accurate to 10 significant figures.

History


{{See also|Chronology of computation of π|Approximations of π}}
The Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...

 at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2{{pi}}. The same apotropaic proportions were used earlier at the Pyramid of Meidum
Meidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...

 c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of {{pi}}, in practice they used it". Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design". Others have argued that the Ancient Egyptians had no concept of {{pi}} and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).

The early history of {{pi}} from textual sources roughly parallels the development of mathematics as a whole.

Antiquity


The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.

The Indian text Shatapatha Brahmana
Shatapatha Brahmana
The Shatapatha Brahmana is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina and Kanva , with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17...

 (composed between the 8th to 6th centuries BCE, Iron Age India
Iron Age India
Iron Age India, the Iron Age in the Indian subcontinent, succeeds the Late Harappan culture, also known as the last phase of the Indus Valley Tradition...

) gives {{pi}} as 339/108 ≈ 3.139. It has been suggested that passages in the {{bibleverse|1|Kings|7:23|NKJV}} and {{bibleverse|2|Chronicles|4:2|NKJV}} discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered {{pi}} to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.

{{multiple image
| direction = vertical
| width = 200
| image1 = Cutcircle2.svg
| caption1 = Estimating {{pi}} with inscribed polygons
| image2 = Archimedes pi.svg
| caption2 = Estimating {{pi}} with circumscribed and inscribed polygons
}}Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 (287–212 BC) was the first to estimate {{pi}} rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s and calculating the outer and inner polygons' respective perimeters: By using the equivalent of 96-sided polygons, he proved that The average of these values is about 3.14185.

Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, in his Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, gives a value of 3.1416, which he may have obtained from Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

.

Around AD 265, the Wei Kingdom mathematician Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

 provided a simple and rigorous iterative algorithm to calculate {{pi}} to any degree of accuracy. He himself carried through the calculation to a 3072-gon (i.e. a 3072-sided polygon) and obtained an approximate value for {{pi}} of 3.1416. Later, Liu Hui invented a quick method of calculating {{pi}} and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 demonstrated that {{pi}} ≈ 355/113 (≈ 3.1415929
), and showed that 3.1415926 < {{pi}} < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.

Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

 mentions with certainty the irrationality of {{pi}} in the 12th century. This was proved in 1768 by Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 calculus. One of those, due to Ivan Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...

, is widely known. A somewhat earlier similar proof is by Mary Cartwright
Mary Cartwright
Dame Mary Lucy Cartwright DBE FRS was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England...

.

Second millennium AD


Until the second millennium AD
2nd millennium
File:2nd millennium montage.png|From left, clockwise: In 1492, Christopher Columbus; The American Revolution; The French Revolution; The Atomic Bomb from World War II; An alternate source of light, the Light Bulb; For the first time, a human being sets foot on the moon in 1969 during the Apollo 11...

, estimations of {{pi}} were accurate to fewer than 10 decimal digits. The next major advances in the study of {{pi}} came with the development of infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and subsequently with the discovery of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, which permit the estimation of {{pi}} to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 found the first known such series:


This is now known as the Madhava–Leibniz series or Gregory–Leibniz series since it was rediscovered by James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into


Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 was able to estimate {{pi}} as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, who gave an estimate {{pi}} that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

 (1540–1610), who used a geometric method to give an estimate of {{pi}} that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. {{pi}} is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

 in 1593. Another famous result is Wallis' product,


by John Wallis in 1655. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 derived the arcsin series for {{pi}} in 1665–66 and calculated 15 digits:


although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin
John Machin
John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.Machin's formula is:...

 was the first to compute 100 decimals of {{pi}}, using the arctan series in the formula


with


Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating {{pi}} well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase
Zacharias Dase
Johann Martin Zacharias Dase was a German mental calculator.He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on him. He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing...

, who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

. The best value at the end of the 19th century was due to William Shanks
William Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...

, who took 15 years to calculate {{pi}} with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about {{pi}}'s nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

 proved the irrationality of {{pi}} in 1761, and Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 also proved in 1794 {{pi}}2 to be irrational. When Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1735 solved the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

, finding the exact value of the Riemann zeta function of 2,


which is {{pi}}2/6, he established a deep connection between {{pi}} and the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Both Legendre and Euler speculated that {{pi}} might be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which was finally proved in 1882 by Ferdinand von Lindemann
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

.

Computation in the computer age


Practically, one needs only 39 digits of {{pi}} to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

The advent of digital computers in the 20th century led to an increased rate of new {{pi}} calculation records. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 et al. used ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 to compute 2037 digits of {{pi}} in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 found many new formulas for {{pi}}, some remarkable for their elegance, mathematical depth and rapid convergence. One of his formulas is the series,


where {{math|k}}! is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 of {{math|k}}.

A collection of some others are in the table below:


where


is the Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 for the falling factorial.

The related one found by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....

 in 1987 is


which delivers 14 digits per term. The Chudnovskys used this formula to set several {{pi}} computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for {{pi}} calculating software that runs on personal computers, as opposed to the supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s used to set modern records.
On August 6, 2010, PhysOrg.com reported that Japanese and American computer experts Shigeru Kondo and Alexander Yee said they've calculated the value of {{pi}} to 5 trillion decimal places on a personal computer, double the previous record.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist, born in 1946. He holds the position of Distinguished Professor of Mathematics and Computer Science with a joint appointment in the Mathematical Sciences Institute and the College of Engineering and Computer Science at...

 and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting


and iterating


until {{math|a}}{{math|n}} and {{math|b}}{{math|n}} are close enough. Then the estimate for {{pi}} is given by


Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...

 and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

. The methods have been used by Yasumasa Kanada
Yasumasa Kanada
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of π. He has set the record 11 of the past 21 times....

 and team to set most of the {{pi}} calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record was almost 2.7 trillion digits. This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

Another method for fast calculation of the constant {{pi}} is the method for fast summing series of special form FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

. To calculate the {{pi}} it's possible to use the Euler formula
and apply the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 to sum the Taylor series for
One can apply the same procedure also to the other special series approximating the constant {{pi}}. Besides the formulas representing the {{pi}} via arctangents, the new formulas for {{pi}} derived in the 1990s by S. Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, F. Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 and some other computer scientists, are good for fast summing via the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 and fast computation of the constant {{pi}}.

An important recent development was the Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 (BBP formula), discovered by Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 and named after the authors of the paper in which the formula was first published, David H. Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

, Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

, and Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

. The formula,


is remarkable because it allows extracting any individual hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

 or binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 digit of {{pi}} without calculating all the preceding ones. Between 1998 and 2000, the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

 project PiHex
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of Pi, the greatest calculation of Pi ever successfully attempted. 1,246 contributors used idle time slices on almost two thousand computers to make its calculations...

 used a modification of the BBP formula due to Fabrice Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 to compute the quadrillionth (1,000,000,000,000,000:th) bit of {{pi}}, which turned out to be 0.

If a formula of the form


were found where {{math|b}} and {{math|c}} are positive integers and {{math|p}} and {{math|p}} are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of {{pi}} at any position in base {{math|b}}{{math|c}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{math|k}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer {{math|k}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{math|b}} = 2 and {{math|c}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

 number.{{Citation needed|date=March 2011}}

In September 2010, Yahoo!
Yahoo!
Yahoo! Inc. is an American multinational internet corporation headquartered in Sunnyvale, California, United States. The company is perhaps best known for its web portal, search engine , Yahoo! Directory, Yahoo! Mail, Yahoo! News, Yahoo! Groups, Yahoo! Answers, advertising, online mapping ,...

 employee Nicholas Sze used the company's Hadoop production application to compute 256 bits of {{pi}} starting at a position a little before the two-quadrillionth (2,000,000,000,000,000th) bit, doubling the previous record by PiHex. The record was broken on 1,000 of Yahoo!'s computers over a 23-day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.

In 2006, Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, using the integer relation algorithm
Integer relation algorithm
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such thata_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,...

 PSLQ, found a series of formulas. Let {{math|q}} = {{math
Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...

 (Gelfond's constant), then



and others of form,


where {{math|k}} is an odd number, and {{math|a}}, {{math|b}}, {{math|c}} are rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

In the previous formula, if {{math|k}} is of the form 4{{math|m}} + 3, then the formula has the particularly simple form,


for some rational number {{math|p}} where the denominator is a highly factorable number. General expressions for these kinds of sums are known.

Representation as a continued fraction


The sequence of partial denominators of the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 of {{pi}} does not show any obvious pattern:
or

However, there are generalized continued fraction
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....

s for {{pi}} with a perfectly regular structure, such as:

Combining the last continued fraction with Machin's arctangent formula provides an even more rapidly-converging expression:

Memorizing digits


{{Main|Piphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira Haraguchi
Akira Haraguchi
Akira Haraguchi , a retired Japanese engineer, currently working as a mental health counsellor and business consultant in Mobara City, is known for memorizing and reciting digits of Pi....

, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records
Guinness World Records
Guinness World Records, known until 2000 as The Guinness Book of Records , is a reference book published annually, containing a collection of world records, both human achievements and the extremes of the natural world...

. The Guinness-recognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu Chao
Lu Chao
Lu Chao from China is the recognized Guinness record holder for reciting digits of Pi. He successfully recited 67,890 digits of pi in 24 hours and 4 minutes with an error at the 67,891st digit, saying it was a "5", when it was actually a "0". He stated he had 100,000 memorized, and was going to...

, a 24-year-old graduate student from China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of {{pi}} without an error.

There are many ways to memorize {{pi}}, including the use of "piems", which are poems that represent {{pi}} in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem, originally devised by Sir James Jeans
James Hopwood Jeans
Sir James Hopwood Jeans OM FRS MA DSc ScD LLD was an English physicist, astronomer and mathematician.-Background:...

: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. The Cadaeic Cadenza
Cadaeic Cadenza
Cadaeic Cadenza is a 1996 short story by Mike Keith. It is an example of constrained writing, a book with restrictions on how it can be written. It is also one of the most prodigious examples of piphilology, being written in "pilish"....

 contains the first 3835 digits of {{pi}} in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 to remember the digits of {{pi}}, known as piphilology
Piphilology
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant . The word is a play on Pi itself and the linguistic field of philology....

. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of {{pi}}. Other methods include remembering patterns in the numbers and the method of loci
Method of loci
The method of loci , also called the memory palace, is a mnemonic device introduced in ancient Roman rhetorical treatises . It relies on memorized spatial relationships to establish, order and recollect memorial content...

.

Open questions


One open question about {{pi}} is whether it is a normal number
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...

—whether any digit block occurs in the expansion of {{pi}} just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every integer base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of {{pi}}, although it is clear that at least two such digits must occur infinitely often, since otherwise {{pi}} would be rational, which it is not.

Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

 and Crandall
Richard Crandall
Richard E. Crandall is an American Physicist and computer scientist who has made contributions to computational number theory.He is most notable for the development of the irrational base discrete weighted transform, an important method of finding very large primes. He has, at various times, been...

 showed in 2000 that the existence of the above mentioned Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 and similar formulas imply that the normality in base 2 of {{pi}} and various other constants can be reduced to a plausible conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 of chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

.

It is also unknown whether {{pi}} and {{math
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

 are algebraically independent
Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K...

, although Yuri Nesterenko
Yuri Valentinovich Nesterenko
Yuri Valentinovich Nesterenko is a mathematician who has written papers in algebraic independence theory and transcendental number theory.In 1997 he was awarded the Ostrowski Prize for his proof that the numbers π and eπ are algebraically independent...

 proved the algebraic independence of
{{pp-semi|small=yes}}
{{Two other uses|the number|the Greek letter|Pi (letter)}}


{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 that is the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of any circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. {{pi}} is approximately equal to 3.14. Many formulae in mathematics, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 involve {{pi}}, which makes it one of the most important mathematical constants. For instance, the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...

 is equal to {{pi}} times the square of the radius of the circle.

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, which means that its value cannot be expressed exactly as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 having integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine {{pi}} more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Probably because of the simplicity of its definition, {{pi}} has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of {{pi}} are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter {{pi}} was first adopted for the number as an abbreviation of the Greek word for perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 who provided an approximation of the number during the 3rd century BC
3rd century BC
The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period.-Overview:...

, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

, who computed a 35-digit approximation around the year 1600.

The Greek letter


{{Main|Pi (letter)}}
The Latin name of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

 is pi. When referring to the constant, the symbol {{pi}} is pronounced like the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "{{pi}}" because "{{pi}}" is the first letter of the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 word περιφέρεια "periphery"{{Citation needed|date=May 2011}} (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1737. William Jones wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to  ...  3.14159, etc. = {{pi}} ...


When used as a symbol for the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

, the Greek letter ({{pi}}) is not capitalized at the beginning of a sentence. The capital letter {{PI}} (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition


In Euclidean plane geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, {{pi}} is defined as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

  {{math|C}} to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

  {{math|d}}:


The ratio {{math|C/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{math|d}} of another circle it will also have twice the circumference {{math|C}}, preserving the ratio {{math|C/d}}.

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, which can be a problem when {{pi}} occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define {{pi}} without reference to geometry, instead selecting one of its analytic
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 properties as a definition. A common choice is to define {{pi}} as twice the smallest positive {{math|x}} for which the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 cos({{math|x}}) equals zero.


Irrationality and transcendence


{{Main|Proof that π is irrational}}

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, meaning that it cannot be written as the ratio of two integers
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, meaning that there is no polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 coefficients for which {{pi}} is a root. An important consequence of the transcendence of {{pi}} is the fact that it is not constructible
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.{{cite book
| author=Schlager, Neil; Lauer, Josh | year=2001 | page=185
| title=Science and Its Times: Understanding the Social Significance of Scientific Discovery
| volume=1 | series=Science and Its Times
| publisher=Gale Group | isbn=0787639338 }}

Decimal representation


{{See also|Approximations of π}}

The decimal representation of {{pi}} truncated
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

 to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 is:

{{gaps|lhs={{pi}}|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510|...}}.


Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (1012) digits, elementary applications
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of {{pi}} truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

 of less than one millimetre, and the decimal representation of {{pi}} truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 with precision comparable to the radius of a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

.

Because {{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, its decimal representation does not repeat
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating {{pi}}'s properties. Despite much analytical work, and supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

 calculations that have determined over 1 trillion digits of the decimal representation of {{pi}}, no simple base-10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 pattern in the digits has ever been found. Digits of the decimal representation of {{pi}} are available on many web pages, and there is software for calculating the decimal representation of {{pi}} to billions of digits on any personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...

.

Estimating the value


{{Main|Approximations of π}}
Numeral system Approximation of {{pi}}
Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

{{gaps 26535|89793|23846|26433|83279|50288...}}
Hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

{{gaps A8885|A308D|31319...}}
Sexagesimal (used by ancients, including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

)
{{gaps 3 ; 8′ 30″}} = 377/120
Rational approximations 3, {{frac|22|7}}, {{frac|333|106}}, {{frac|355|113}}, {{frac|52163|16604}}, {{frac|103993|33102}}, ...
(listed in order of increasing accuracy)
Continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

{{nowrap|[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...]}}
(This fraction is not periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. Shown in linear notation)
Generalized continued fraction expression

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999
Feynman point
The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

The earliest numerical approximation of {{pi}} is almost certainly the value {{num|3}}.{{verification failed|date=November 2011}}

In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of an inscribed regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 hexagon to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

{{pi}} can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, is to calculate the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, {{math|P}}{{math|n}}, of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 with {{math|n}} sides circumscribed around a circle with diameter {{math|d}}. Then compute the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 as {{math|n}} increases to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

:


This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:


{{pi}} can also be calculated using purely mathematical methods. Due to the transcendental nature of {{pi}}, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to {{pi}}. The more terms included in a calculation, the closer to {{pi}} the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. However, some are quite simple, such as this form of the Gregory–Leibniz series:


While that series is easy to write and calculate, it is not immediately obvious why it yields {{pi}}. In addition, this series converges so slowly that nearly 300 terms are needed to calculate {{pi}} correctly to two decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...




and then define


then computing will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

For many purposes, 3.14 or 227 is close enough, although engineers often use 3.1416 (5 significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...

) or 3.14159 (6 significant figures) for more precision.{{Citation needed|date=March 2011}} The approximations 227 and 355113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 expansion of {{pi}}. The approximation 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

; the next good approximation 5216316604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of {{pi}}. For extremely accurate approximations, either Ramanujan's approximation of (3.14159265258...) or 10399333102 (3.14159265301...) are used, which are both accurate to 10 significant figures.

History


{{See also|Chronology of computation of π|Approximations of π}}
The Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...

 at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2{{pi}}. The same apotropaic proportions were used earlier at the Pyramid of Meidum
Meidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...

 c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of {{pi}}, in practice they used it". Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design". Others have argued that the Ancient Egyptians had no concept of {{pi}} and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).

The early history of {{pi}} from textual sources roughly parallels the development of mathematics as a whole.

Antiquity


The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.

The Indian text Shatapatha Brahmana
Shatapatha Brahmana
The Shatapatha Brahmana is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina and Kanva , with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17...

 (composed between the 8th to 6th centuries BCE, Iron Age India
Iron Age India
Iron Age India, the Iron Age in the Indian subcontinent, succeeds the Late Harappan culture, also known as the last phase of the Indus Valley Tradition...

) gives {{pi}} as 339/108 ≈ 3.139. It has been suggested that passages in the {{bibleverse|1|Kings|7:23|NKJV}} and {{bibleverse|2|Chronicles|4:2|NKJV}} discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered {{pi}} to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.

{{multiple image
| direction = vertical
| width = 200
| image1 = Cutcircle2.svg
| caption1 = Estimating {{pi}} with inscribed polygons
| image2 = Archimedes pi.svg
| caption2 = Estimating {{pi}} with circumscribed and inscribed polygons
}}Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 (287–212 BC) was the first to estimate {{pi}} rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s and calculating the outer and inner polygons' respective perimeters: By using the equivalent of 96-sided polygons, he proved that The average of these values is about 3.14185.

Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, in his Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, gives a value of 3.1416, which he may have obtained from Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

.

Around AD 265, the Wei Kingdom mathematician Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

 provided a simple and rigorous iterative algorithm to calculate {{pi}} to any degree of accuracy. He himself carried through the calculation to a 3072-gon (i.e. a 3072-sided polygon) and obtained an approximate value for {{pi}} of 3.1416. Later, Liu Hui invented a quick method of calculating {{pi}} and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 demonstrated that {{pi}} ≈ 355/113 (≈ 3.1415929
), and showed that 3.1415926 < {{pi}} < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.

Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

 mentions with certainty the irrationality of {{pi}} in the 12th century. This was proved in 1768 by Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 calculus. One of those, due to Ivan Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...

, is widely known. A somewhat earlier similar proof is by Mary Cartwright
Mary Cartwright
Dame Mary Lucy Cartwright DBE FRS was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England...

.

Second millennium AD


Until the second millennium AD
2nd millennium
File:2nd millennium montage.png|From left, clockwise: In 1492, Christopher Columbus; The American Revolution; The French Revolution; The Atomic Bomb from World War II; An alternate source of light, the Light Bulb; For the first time, a human being sets foot on the moon in 1969 during the Apollo 11...

, estimations of {{pi}} were accurate to fewer than 10 decimal digits. The next major advances in the study of {{pi}} came with the development of infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and subsequently with the discovery of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, which permit the estimation of {{pi}} to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 found the first known such series:


This is now known as the Madhava–Leibniz series or Gregory–Leibniz series since it was rediscovered by James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into


Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 was able to estimate {{pi}} as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, who gave an estimate {{pi}} that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

 (1540–1610), who used a geometric method to give an estimate of {{pi}} that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. {{pi}} is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

 in 1593. Another famous result is Wallis' product,


by John Wallis in 1655. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 derived the arcsin series for {{pi}} in 1665–66 and calculated 15 digits:


although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin
John Machin
John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.Machin's formula is:...

 was the first to compute 100 decimals of {{pi}}, using the arctan series in the formula


with


Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating {{pi}} well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase
Zacharias Dase
Johann Martin Zacharias Dase was a German mental calculator.He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on him. He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing...

, who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

. The best value at the end of the 19th century was due to William Shanks
William Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...

, who took 15 years to calculate {{pi}} with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about {{pi}}'s nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

 proved the irrationality of {{pi}} in 1761, and Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 also proved in 1794 {{pi}}2 to be irrational. When Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1735 solved the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

, finding the exact value of the Riemann zeta function of 2,


which is {{pi}}2/6, he established a deep connection between {{pi}} and the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Both Legendre and Euler speculated that {{pi}} might be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which was finally proved in 1882 by Ferdinand von Lindemann
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

.

Computation in the computer age


Practically, one needs only 39 digits of {{pi}} to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

The advent of digital computers in the 20th century led to an increased rate of new {{pi}} calculation records. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 et al. used ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 to compute 2037 digits of {{pi}} in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 found many new formulas for {{pi}}, some remarkable for their elegance, mathematical depth and rapid convergence. One of his formulas is the series,


where {{math|k}}! is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 of {{math|k}}.

A collection of some others are in the table below:


where


is the Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 for the falling factorial.

The related one found by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....

 in 1987 is


which delivers 14 digits per term. The Chudnovskys used this formula to set several {{pi}} computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for {{pi}} calculating software that runs on personal computers, as opposed to the supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s used to set modern records.
On August 6, 2010, PhysOrg.com reported that Japanese and American computer experts Shigeru Kondo and Alexander Yee said they've calculated the value of {{pi}} to 5 trillion decimal places on a personal computer, double the previous record.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist, born in 1946. He holds the position of Distinguished Professor of Mathematics and Computer Science with a joint appointment in the Mathematical Sciences Institute and the College of Engineering and Computer Science at...

 and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting


and iterating


until {{math|a}}{{math|n}} and {{math|b}}{{math|n}} are close enough. Then the estimate for {{pi}} is given by


Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...

 and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

. The methods have been used by Yasumasa Kanada
Yasumasa Kanada
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of π. He has set the record 11 of the past 21 times....

 and team to set most of the {{pi}} calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record was almost 2.7 trillion digits. This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

Another method for fast calculation of the constant {{pi}} is the method for fast summing series of special form FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

. To calculate the {{pi}} it's possible to use the Euler formula
and apply the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 to sum the Taylor series for
One can apply the same procedure also to the other special series approximating the constant {{pi}}. Besides the formulas representing the {{pi}} via arctangents, the new formulas for {{pi}} derived in the 1990s by S. Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, F. Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 and some other computer scientists, are good for fast summing via the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 and fast computation of the constant {{pi}}.

An important recent development was the Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 (BBP formula), discovered by Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 and named after the authors of the paper in which the formula was first published, David H. Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

, Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

, and Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

. The formula,


is remarkable because it allows extracting any individual hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

 or binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 digit of {{pi}} without calculating all the preceding ones. Between 1998 and 2000, the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

 project PiHex
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of Pi, the greatest calculation of Pi ever successfully attempted. 1,246 contributors used idle time slices on almost two thousand computers to make its calculations...

 used a modification of the BBP formula due to Fabrice Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 to compute the quadrillionth (1,000,000,000,000,000:th) bit of {{pi}}, which turned out to be 0.

If a formula of the form


were found where {{math|b}} and {{math|c}} are positive integers and {{math|p}} and {{math|p}} are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of {{pi}} at any position in base {{math|b}}{{math|c}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{math|k}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer {{math|k}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{math|b}} = 2 and {{math|c}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

 number.{{Citation needed|date=March 2011}}

In September 2010, Yahoo!
Yahoo!
Yahoo! Inc. is an American multinational internet corporation headquartered in Sunnyvale, California, United States. The company is perhaps best known for its web portal, search engine , Yahoo! Directory, Yahoo! Mail, Yahoo! News, Yahoo! Groups, Yahoo! Answers, advertising, online mapping ,...

 employee Nicholas Sze used the company's Hadoop production application to compute 256 bits of {{pi}} starting at a position a little before the two-quadrillionth (2,000,000,000,000,000th) bit, doubling the previous record by PiHex. The record was broken on 1,000 of Yahoo!'s computers over a 23-day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.

In 2006, Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, using the integer relation algorithm
Integer relation algorithm
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such thata_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,...

 PSLQ, found a series of formulas. Let {{math|q}} = {{math
Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...

 (Gelfond's constant), then



and others of form,


where {{math|k}} is an odd number, and {{math|a}}, {{math|b}}, {{math|c}} are rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

In the previous formula, if {{math|k}} is of the form 4{{math|m}} + 3, then the formula has the particularly simple form,


for some rational number {{math|p}} where the denominator is a highly factorable number. General expressions for these kinds of sums are known.

Representation as a continued fraction


The sequence of partial denominators of the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 of {{pi}} does not show any obvious pattern:
or

However, there are generalized continued fraction
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....

s for {{pi}} with a perfectly regular structure, such as:

Combining the last continued fraction with Machin's arctangent formula provides an even more rapidly-converging expression:

Memorizing digits


{{Main|Piphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira Haraguchi
Akira Haraguchi
Akira Haraguchi , a retired Japanese engineer, currently working as a mental health counsellor and business consultant in Mobara City, is known for memorizing and reciting digits of Pi....

, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records
Guinness World Records
Guinness World Records, known until 2000 as The Guinness Book of Records , is a reference book published annually, containing a collection of world records, both human achievements and the extremes of the natural world...

. The Guinness-recognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu Chao
Lu Chao
Lu Chao from China is the recognized Guinness record holder for reciting digits of Pi. He successfully recited 67,890 digits of pi in 24 hours and 4 minutes with an error at the 67,891st digit, saying it was a "5", when it was actually a "0". He stated he had 100,000 memorized, and was going to...

, a 24-year-old graduate student from China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of {{pi}} without an error.

There are many ways to memorize {{pi}}, including the use of "piems", which are poems that represent {{pi}} in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem, originally devised by Sir James Jeans
James Hopwood Jeans
Sir James Hopwood Jeans OM FRS MA DSc ScD LLD was an English physicist, astronomer and mathematician.-Background:...

: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. The Cadaeic Cadenza
Cadaeic Cadenza
Cadaeic Cadenza is a 1996 short story by Mike Keith. It is an example of constrained writing, a book with restrictions on how it can be written. It is also one of the most prodigious examples of piphilology, being written in "pilish"....

 contains the first 3835 digits of {{pi}} in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 to remember the digits of {{pi}}, known as piphilology
Piphilology
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant . The word is a play on Pi itself and the linguistic field of philology....

. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of {{pi}}. Other methods include remembering patterns in the numbers and the method of loci
Method of loci
The method of loci , also called the memory palace, is a mnemonic device introduced in ancient Roman rhetorical treatises . It relies on memorized spatial relationships to establish, order and recollect memorial content...

.

Open questions


One open question about {{pi}} is whether it is a normal number
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...

—whether any digit block occurs in the expansion of {{pi}} just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every integer base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of {{pi}}, although it is clear that at least two such digits must occur infinitely often, since otherwise {{pi}} would be rational, which it is not.

Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

 and Crandall
Richard Crandall
Richard E. Crandall is an American Physicist and computer scientist who has made contributions to computational number theory.He is most notable for the development of the irrational base discrete weighted transform, an important method of finding very large primes. He has, at various times, been...

 showed in 2000 that the existence of the above mentioned Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 and similar formulas imply that the normality in base 2 of {{pi}} and various other constants can be reduced to a plausible conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 of chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

.

It is also unknown whether {{pi}} and {{math
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

 are algebraically independent
Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K...

, although Yuri Nesterenko
Yuri Valentinovich Nesterenko
Yuri Valentinovich Nesterenko is a mathematician who has written papers in algebraic independence theory and transcendental number theory.In 1997 he was awarded the Ostrowski Prize for his proof that the numbers π and eπ are algebraically independent...

 proved the algebraic independence of
{{pp-semi|small=yes}}
{{Two other uses|the number|the Greek letter|Pi (letter)}}


{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 that is the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of any circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

. {{pi}} is approximately equal to 3.14. Many formulae in mathematics, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 involve {{pi}}, which makes it one of the most important mathematical constants. For instance, the area of a circle
Area of a circle
The area of a circle is πr2 when the circle has radius r. Here the symbol π denotes, as usual, the constant ratio of the circumference of a circle to its diameter...

 is equal to {{pi}} times the square of the radius of the circle.

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, which means that its value cannot be expressed exactly as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

 having integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s in both the numerator and denominator (unlike 22/7). Consequently, its decimal representation never ends and never repeats
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can render its value; proving this fact was a significant mathematical achievement of the 19th century.

Throughout the history of mathematics, there has been much effort to determine {{pi}} more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. Probably because of the simplicity of its definition, {{pi}} has become more entrenched in popular culture than almost any other mathematical concept, and is firm common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of {{pi}} are common news items; the record as of September 2011, if verified, stands at 5 trillion decimal digits.

The Greek letter {{pi}} was first adopted for the number as an abbreviation of the Greek word for perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 (περίμετρος), or as an abbreviation for "periphery/diameter", by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 who provided an approximation of the number during the 3rd century BC
3rd century BC
The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period.-Overview:...

, although this name is uncommon today. Even rarer is the name Ludolphine number or Ludolph's Constant, after Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

, who computed a 35-digit approximation around the year 1600.

The Greek letter


{{Main|Pi (letter)}}
The Latin name of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

 is pi. When referring to the constant, the symbol {{pi}} is pronounced like the English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...

 word "pie", which is also the conventional English pronunciation of the Greek letter. The constant is named "{{pi}}" because "{{pi}}" is the first letter of the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 word περιφέρεια "periphery"{{Citation needed|date=May 2011}} (or perhaps περίμετρος "perimeter", referring to the ratio of the perimeter to the diameter, which is constant for all circles). William Jones was the first to use the Greek letter in this way, in 1706, and it was later popularized by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1737. William Jones wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to  ...  3.14159, etc. = {{pi}} ...


When used as a symbol for the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

, the Greek letter ({{pi}}) is not capitalized at the beginning of a sentence. The capital letter {{PI}} (Pi) has a completely different mathematical meaning; it is used for expressing the product of a sequence.

Geometric definition


In Euclidean plane geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, {{pi}} is defined as the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

's circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

  {{math|C}} to its diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

  {{math|d}}:


The ratio {{math|C/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{math|d}} of another circle it will also have twice the circumference {{math|C}}, preserving the ratio {{math|C/d}}.

This definition depends on results of Euclidean geometry, such as the fact that all circles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

, which can be a problem when {{pi}} occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define {{pi}} without reference to geometry, instead selecting one of its analytic
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 properties as a definition. A common choice is to define {{pi}} as twice the smallest positive {{math|x}} for which the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

 cos({{math|x}}) equals zero.


Irrationality and transcendence


{{Main|Proof that π is irrational}}

{{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, meaning that it cannot be written as the ratio of two integers
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. {{pi}} is also a transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, meaning that there is no polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 with rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 coefficients for which {{pi}} is a root. An important consequence of the transcendence of {{pi}} is the fact that it is not constructible
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity. Many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.{{cite book
| author=Schlager, Neil; Lauer, Josh | year=2001 | page=185
| title=Science and Its Times: Understanding the Social Significance of Scientific Discovery
| volume=1 | series=Science and Its Times
| publisher=Gale Group | isbn=0787639338 }}

Decimal representation


{{See also|Approximations of π}}

The decimal representation of {{pi}} truncated
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

 to 50 decimal places
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 is:

{{gaps|lhs={{pi}}|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510|...}}.


Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (1012) digits, elementary applications
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of {{pi}} truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error
Round-off error
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

 of less than one millimetre, and the decimal representation of {{pi}} truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 with precision comparable to the radius of a hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

.

Because {{pi}} is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

, its decimal representation does not repeat
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating {{pi}}'s properties. Despite much analytical work, and supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

 calculations that have determined over 1 trillion digits of the decimal representation of {{pi}}, no simple base-10
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 pattern in the digits has ever been found. Digits of the decimal representation of {{pi}} are available on many web pages, and there is software for calculating the decimal representation of {{pi}} to billions of digits on any personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...

.

Estimating the value


{{Main|Approximations of π}}
Numeral system Approximation of {{pi}}
Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

{{gaps 26535|89793|23846|26433|83279|50288...}}
Hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

{{gaps A8885|A308D|31319...}}
Sexagesimal (used by ancients, including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

)
{{gaps 3 ; 8′ 30″}} = 377/120
Rational approximations 3, {{frac|22|7}}, {{frac|333|106}}, {{frac|355|113}}, {{frac|52163|16604}}, {{frac|103993|33102}}, ...
(listed in order of increasing accuracy)
Continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

{{nowrap|[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...]}}
(This fraction is not periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

. Shown in linear notation)
Generalized continued fraction expression

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999
Feynman point
The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

The earliest numerical approximation of {{pi}} is almost certainly the value {{num|3}}.{{verification failed|date=November 2011}}

In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

 of an inscribed regular
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 hexagon to the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...

 of the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

{{pi}} can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

, is to calculate the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, {{math|P}}{{math|n}}, of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

 with {{math|n}} sides circumscribed around a circle with diameter {{math|d}}. Then compute the limit of a sequence
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 as {{math|n}} increases to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

:


This sequence converges because the more sides the polygon has, the smaller its maximum distance from the circle. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:


{{pi}} can also be calculated using purely mathematical methods. Due to the transcendental nature of {{pi}}, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulas for calculating π using elementary arithmetic typically include series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to {{pi}}. The more terms included in a calculation, the closer to {{pi}} the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 and calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. However, some are quite simple, such as this form of the Gregory–Leibniz series:


While that series is easy to write and calculate, it is not immediately obvious why it yields {{pi}}. In addition, this series converges so slowly that nearly 300 terms are needed to calculate {{pi}} correctly to two decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...




and then define


then computing will take similar computation time to computing 150 terms of the original series in a brute-force manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.

For many purposes, 3.14 or 227 is close enough, although engineers often use 3.1416 (5 significant figures
Significant figures
The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...

) or 3.14159 (6 significant figures) for more precision.{{Citation needed|date=March 2011}} The approximations 227 and 355113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 expansion of {{pi}}. The approximation 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit numerator and denominator
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

; the next good approximation 5216316604 (3.141592387...), which is also accurate to 7 significant figures, requires much bigger numbers, due to the large number 292 in the continued fraction expansion of {{pi}}. For extremely accurate approximations, either Ramanujan's approximation of (3.14159265258...) or 10399333102 (3.14159265301...) are used, which are both accurate to 10 significant figures.

History


{{See also|Chronology of computation of π|Approximations of π}}
The Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...

 at Giza, constructed c.2589–2566 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits giving the ratio 1760/280 ≈ 2{{pi}}. The same apotropaic proportions were used earlier at the Pyramid of Meidum
Meidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...

 c.2613-2589 BC and later in the pyramids of Abusir c.2453-2422. Some Egyptologists consider this to have been the result of deliberate design proportion. Verner wrote, "We can conclude that although the ancient Egyptians could not precisely define the value of {{pi}}, in practice they used it". Petrie, author of Pyramids and Temples of Gizeh concluded: "but these relations of areas and of circular ratio are so systematic that we should grant that they were in the builders design". Others have argued that the Ancient Egyptians had no concept of {{pi}} and would not have thought to encode it in their monuments. They argued that creation of the pyramid may instead be based on simple ratios of the sides of right-angled triangles (the seked).

The early history of {{pi}} from textual sources roughly parallels the development of mathematics as a whole.

Antiquity


The earliest known textually evidenced approximations of pi date from around 1900 BC. They are found in the Egyptian Rhind Papyrus 256/81 ≈ 3.160 and on Babylonian tablets 25/8 = 3.125, both within 1 percent of the true value.

The Indian text Shatapatha Brahmana
Shatapatha Brahmana
The Shatapatha Brahmana is one of the prose texts describing the Vedic ritual, associated with the Shukla Yajurveda. It survives in two recensions, Madhyandina and Kanva , with the former having the eponymous 100 adhyayas,7624 kandikas in 14 books, and the latter 104 adhyayas,6806 kandikas in 17...

 (composed between the 8th to 6th centuries BCE, Iron Age India
Iron Age India
Iron Age India, the Iron Age in the Indian subcontinent, succeeds the Late Harappan culture, also known as the last phase of the Indus Valley Tradition...

) gives {{pi}} as 339/108 ≈ 3.139. It has been suggested that passages in the {{bibleverse|1|Kings|7:23|NKJV}} and {{bibleverse|2|Chronicles|4:2|NKJV}} discussing a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits show that the writers considered {{pi}} to have had an approximate value of three, which various authors have tried to explain away through various suggestions such as a hexagonal pool or an outward curving rim.

{{multiple image
| direction = vertical
| width = 200
| image1 = Cutcircle2.svg
| caption1 = Estimating {{pi}} with inscribed polygons
| image2 = Archimedes pi.svg
| caption2 = Estimating {{pi}} with circumscribed and inscribed polygons
}}Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 (287–212 BC) was the first to estimate {{pi}} rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s and calculating the outer and inner polygons' respective perimeters: By using the equivalent of 96-sided polygons, he proved that The average of these values is about 3.14185.

Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...

, in his Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

, gives a value of 3.1416, which he may have obtained from Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

.

Around AD 265, the Wei Kingdom mathematician Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

 provided a simple and rigorous iterative algorithm to calculate {{pi}} to any degree of accuracy. He himself carried through the calculation to a 3072-gon (i.e. a 3072-sided polygon) and obtained an approximate value for {{pi}} of 3.1416. Later, Liu Hui invented a quick method of calculating {{pi}} and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 demonstrated that {{pi}} ≈ 355/113 (≈ 3.1415929
), and showed that 3.1415926 < {{pi}} < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.

Maimonides
Maimonides
Moses ben-Maimon, called Maimonides and also known as Mūsā ibn Maymūn in Arabic, or Rambam , was a preeminent medieval Jewish philosopher and one of the greatest Torah scholars and physicians of the Middle Ages...

 mentions with certainty the irrationality of {{pi}} in the 12th century. This was proved in 1768 by Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 calculus. One of those, due to Ivan Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...

, is widely known. A somewhat earlier similar proof is by Mary Cartwright
Mary Cartwright
Dame Mary Lucy Cartwright DBE FRS was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England...

.

Second millennium AD


Until the second millennium AD
2nd millennium
File:2nd millennium montage.png|From left, clockwise: In 1492, Christopher Columbus; The American Revolution; The French Revolution; The Atomic Bomb from World War II; An alternate source of light, the Light Bulb; For the first time, a human being sets foot on the moon in 1969 during the Apollo 11...

, estimations of {{pi}} were accurate to fewer than 10 decimal digits. The next major advances in the study of {{pi}} came with the development of infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 and subsequently with the discovery of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, which permit the estimation of {{pi}} to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400, Madhava of Sangamagrama
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 found the first known such series:


This is now known as the Madhava–Leibniz series or Gregory–Leibniz series since it was rediscovered by James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into


Madhava
Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

 was able to estimate {{pi}} as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...

, who gave an estimate {{pi}} that is correct to 16 decimal digits.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen
Ludolph van Ceulen
Ludolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....

 (1540–1610), who used a geometric method to give an estimate of {{pi}} that is correct to 35 decimal digits. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. {{pi}} is sometimes called "Ludolph's Constant", though not as often as it is called "Archimedes' Constant."

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

 in 1593. Another famous result is Wallis' product,


by John Wallis in 1655. Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 derived the arcsin series for {{pi}} in 1665–66 and calculated 15 digits:


although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1706 John Machin
John Machin
John Machin, , a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.Machin's formula is:...

 was the first to compute 100 decimals of {{pi}}, using the arctan series in the formula


with


Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating {{pi}} well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase
Zacharias Dase
Johann Martin Zacharias Dase was a German mental calculator.He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on him. He used to spend a lot of time playing dominoes, and suggested that this played a significant role in developing...

, who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

. The best value at the end of the 19th century was due to William Shanks
William Shanks
William Shanks was a British amateur mathematician.Shanks is famous for his calculation of π to 707 places, accomplished in 1873, which, however, was only correct up to the first 527 places. This error was highlighted in 1944 by D. F...

, who took 15 years to calculate {{pi}} with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about {{pi}}'s nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

 proved the irrationality of {{pi}} in 1761, and Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 also proved in 1794 {{pi}}2 to be irrational. When Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

 in 1735 solved the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

, finding the exact value of the Riemann zeta function of 2,


which is {{pi}}2/6, he established a deep connection between {{pi}} and the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. Both Legendre and Euler speculated that {{pi}} might be transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

, which was finally proved in 1882 by Ferdinand von Lindemann
Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

.

Computation in the computer age


Practically, one needs only 39 digits of {{pi}} to make a circle the size of the observable universe accurate to the size of a hydrogen atom.

The advent of digital computers in the 20th century led to an increased rate of new {{pi}} calculation records. John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

 et al. used ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 to compute 2037 digits of {{pi}} in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...

 (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 found many new formulas for {{pi}}, some remarkable for their elegance, mathematical depth and rapid convergence. One of his formulas is the series,


where {{math|k}}! is the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 of {{math|k}}.

A collection of some others are in the table below:


where


is the Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...

 for the falling factorial.

The related one found by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....

 in 1987 is


which delivers 14 digits per term. The Chudnovskys used this formula to set several {{pi}} computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for {{pi}} calculating software that runs on personal computers, as opposed to the supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s used to set modern records.
On August 6, 2010, PhysOrg.com reported that Japanese and American computer experts Shigeru Kondo and Alexander Yee said they've calculated the value of {{pi}} to 5 trillion decimal places on a personal computer, double the previous record.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent
Richard Brent (scientist)
Richard Peirce Brent is an Australian mathematician and computer scientist, born in 1946. He holds the position of Distinguished Professor of Mathematics and Computer Science with a joint appointment in the Mathematical Sciences Institute and the College of Engineering and Computer Science at...

 and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting


and iterating


until {{math|a}}{{math|n}} and {{math|b}}{{math|n}} are close enough. Then the estimate for {{pi}} is given by


Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan
Jonathan Borwein
Jonathan Michael Borwein is a Scottish mathematician who holds an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Noted for his prolific and creative work throughout the international mathematical community, he is a close associate of David H...

 and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

. The methods have been used by Yasumasa Kanada
Yasumasa Kanada
is a Japanese mathematician most known for his numerous world records over the past three decades for calculating digits of π. He has set the record 11 of the past 21 times....

 and team to set most of the {{pi}} calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record was almost 2.7 trillion digits. This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.

Another method for fast calculation of the constant {{pi}} is the method for fast summing series of special form FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

. To calculate the {{pi}} it's possible to use the Euler formula
and apply the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 to sum the Taylor series for
One can apply the same procedure also to the other special series approximating the constant {{pi}}. Besides the formulas representing the {{pi}} via arctangents, the new formulas for {{pi}} derived in the 1990s by S. Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, F. Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 and some other computer scientists, are good for fast summing via the FEE
FEE method
In mathematics, the FEE method is the method of fast summation of series of a special form. It was constructed in 1990 by E. A. Karatsuba and was called FEE—Fast E-function Evaluation—because it makes it possible fast computations of the Siegel E -functions, and in particular, e^x.A class of...

 and fast computation of the constant {{pi}}.

An important recent development was the Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 (BBP formula), discovered by Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 and named after the authors of the paper in which the formula was first published, David H. Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

, Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...

, and Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

. The formula,


is remarkable because it allows extracting any individual hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...

 or binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

 digit of {{pi}} without calculating all the preceding ones. Between 1998 and 2000, the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...

 project PiHex
PiHex
PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of Pi, the greatest calculation of Pi ever successfully attempted. 1,246 contributors used idle time slices on almost two thousand computers to make its calculations...

 used a modification of the BBP formula due to Fabrice Bellard
Fabrice Bellard
Fabrice Bellard is a computer programmer who is best known as the creator of the FFmpeg and QEMU software projects. He has also developed a number of other programs, including the Tiny C Compiler....

 to compute the quadrillionth (1,000,000,000,000,000:th) bit of {{pi}}, which turned out to be 0.

If a formula of the form


were found where {{math|b}} and {{math|c}} are positive integers and {{math|p}} and {{math|p}} are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of {{pi}} at any position in base {{math|b}}{{math|c}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{math|k}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer {{math|k}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{math|b}} = 2 and {{math|c}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*
SC (complexity)
In computational complexity theory, SC is the complexity class of problems solvable by a deterministic Turing machine in polynomial time and polylogarithmic space...

 number.{{Citation needed|date=March 2011}}

In September 2010, Yahoo!
Yahoo!
Yahoo! Inc. is an American multinational internet corporation headquartered in Sunnyvale, California, United States. The company is perhaps best known for its web portal, search engine , Yahoo! Directory, Yahoo! Mail, Yahoo! News, Yahoo! Groups, Yahoo! Answers, advertising, online mapping ,...

 employee Nicholas Sze used the company's Hadoop production application to compute 256 bits of {{pi}} starting at a position a little before the two-quadrillionth (2,000,000,000,000,000th) bit, doubling the previous record by PiHex. The record was broken on 1,000 of Yahoo!'s computers over a 23-day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.

In 2006, Simon Plouffe
Simon Plouffe
Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

, using the integer relation algorithm
Integer relation algorithm
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such thata_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\,...

 PSLQ, found a series of formulas. Let {{math|q}} = {{math
Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...

 (Gelfond's constant), then



and others of form,


where {{math|k}} is an odd number, and {{math|a}}, {{math|b}}, {{math|c}} are rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s.

In the previous formula, if {{math|k}} is of the form 4{{math|m}} + 3, then the formula has the particularly simple form,


for some rational number {{math|p}} where the denominator is a highly factorable number. General expressions for these kinds of sums are known.

Representation as a continued fraction


The sequence of partial denominators of the simple continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 of {{pi}} does not show any obvious pattern:
or

However, there are generalized continued fraction
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....

s for {{pi}} with a perfectly regular structure, such as:

Combining the last continued fraction with Machin's arctangent formula provides an even more rapidly-converging expression:

Memorizing digits


{{Main|Piphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira Haraguchi
Akira Haraguchi
Akira Haraguchi , a retired Japanese engineer, currently working as a mental health counsellor and business consultant in Mobara City, is known for memorizing and reciting digits of Pi....

, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records
Guinness World Records
Guinness World Records, known until 2000 as The Guinness Book of Records , is a reference book published annually, containing a collection of world records, both human achievements and the extremes of the natural world...

. The Guinness-recognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu Chao
Lu Chao
Lu Chao from China is the recognized Guinness record holder for reciting digits of Pi. He successfully recited 67,890 digits of pi in 24 hours and 4 minutes with an error at the 67,891st digit, saying it was a "5", when it was actually a "0". He stated he had 100,000 memorized, and was going to...

, a 24-year-old graduate student from China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of {{pi}} without an error.

There are many ways to memorize {{pi}}, including the use of "piems", which are poems that represent {{pi}} in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem, originally devised by Sir James Jeans
James Hopwood Jeans
Sir James Hopwood Jeans OM FRS MA DSc ScD LLD was an English physicist, astronomer and mathematician.-Background:...

: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. The Cadaeic Cadenza
Cadaeic Cadenza
Cadaeic Cadenza is a 1996 short story by Mike Keith. It is an example of constrained writing, a book with restrictions on how it can be written. It is also one of the most prodigious examples of piphilology, being written in "pilish"....

 contains the first 3835 digits of {{pi}} in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 to remember the digits of {{pi}}, known as piphilology
Piphilology
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant . The word is a play on Pi itself and the linguistic field of philology....

. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of {{pi}}. Other methods include remembering patterns in the numbers and the method of loci
Method of loci
The method of loci , also called the memory palace, is a mnemonic device introduced in ancient Roman rhetorical treatises . It relies on memorized spatial relationships to establish, order and recollect memorial content...

.

Open questions


One open question about {{pi}} is whether it is a normal number
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...

—whether any digit block occurs in the expansion of {{pi}} just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every integer base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of {{pi}}, although it is clear that at least two such digits must occur infinitely often, since otherwise {{pi}} would be rational, which it is not.

Bailey
David H. Bailey
David Harold Bailey is a mathematician and computer scientist. He received his B.S. in mathematics from Brigham Young University in 1972 and his Ph.D. in mathematics from Stanford University in 1976...

 and Crandall
Richard Crandall
Richard E. Crandall is an American Physicist and computer scientist who has made contributions to computational number theory.He is most notable for the development of the irrational base discrete weighted transform, an important method of finding very large primes. He has, at various times, been...

 showed in 2000 that the existence of the above mentioned Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula provides a spigot algorithm for the computation of the nth binary digit of π. This summation formula was discovered in 1995 by Simon Plouffe. The formula is named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein,...

 and similar formulas imply that the normality in base 2 of {{pi}} and various other constants can be reduced to a plausible conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 of chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

.

It is also unknown whether {{pi}} and {{math
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

 are algebraically independent
Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K...

, although Yuri Nesterenko
Yuri Valentinovich Nesterenko
Yuri Valentinovich Nesterenko is a mathematician who has written papers in algebraic independence theory and transcendental number theory.In 1997 he was awarded the Ostrowski Prize for his proof that the numbers π and eπ are algebraically independent...

 proved the algebraic independence of {{{pi}}, {{math
Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is eπ, that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by the Gelfond–Schneider theorem and noting the fact that...

, {{math
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

(1/4)} in 1996.

Use in mathematics and science


{{Main|List of formulae involving π}}

{{pi}} is ubiquitous in mathematics, science, and engineering.

Geometry and trigonometry


{{See also|Area of a disk}}

For any circle with radius {{math|r}} and diameter {{math|d}} = 2{{math|r}}, the circumference is {{pi}}{{math|d}} and the area is {{pi}}{{math|r}}2. Further, {{pi}} appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

s, sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s, cones
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

, and tori
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

. Accordingly, {{pi}} appears in definite integrals
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the unit disk is given by the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

:
and
gives half the circumference of the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

. More complicated shapes can be integrated as solids of revolution
Solid of revolution
In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane....

.

From the unit-circle definition of the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s also follows that the sine and cosine have period 2{{pi}}. That is, for all {{math|x}} and integers {{math|n}}, sin({{math|x}}) = sin({{math|x}} + 2{{pi}}{{math|n}}) and cos({{math|x}}) = cos({{math|x}} + 2{{pi}}{{math|n}}). Because sin(0) = 0, sin(2{{pi}}{{math|n}}) = 0 for all integers {{math|n}}. Also, the angle measure of 180° is equal to {{pi}} radians. In other words, 1° = ({{pi}}/180) radians.

In modern mathematics, {{pi}} is often defined using trigonometric functions, for example as the smallest positive {{math|x}} for which sin {{math|x}} = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, {{pi}} can be defined using the inverse trigonometric function
Inverse trigonometric function
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...

s, for example as {{pi}} = 2 arccos(0) or {{pi}} = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for {{pi}}.

Complex numbers and calculus


A complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

  can be expressed in polar coordinates as follows:


The frequent appearance of {{pi}} in complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

 can be related to the behavior of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 of a complex variable, described by Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...




where {{math|i}} is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

 satisfying {{math|i}}2 = −1 and {{math|e}} ≈ 2.71828 is Euler's number
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

. This formula implies that imaginary powers of {{math|e}} describe rotations on the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 in the complex plane; these rotations have a period of 360° = 2{{pi}}. In particular, the 180° rotation {{math|φ}} = {{pi}} results in the remarkable Euler's identity

There are {{math|n}} different {{math|n}}-th roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...



The Gaussian integral
Gaussian integral
The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...




A consequence is that the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 of a half-integer is a rational multiple of √{{pi}}.

Physics


Although not a physical constant
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

, {{pi}} appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate system
Spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...

s. Using units such as Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...

 can sometimes eliminate {{pi}} from formulae.
  • Heisenberg's uncertainty principle
    Uncertainty principle
    In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

    , which shows that the uncertainty in the measurement of a particle's position (Δ{{math|x}}) and momentum
    Momentum
    In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

     (Δ{{math|p}}) can not both be arbitrarily small at the same time:
  • Einstein's field equation
    Einstein field equations
    The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

     of general relativity
    General relativity
    General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

    :
  • The cosmological constant
    Cosmological constant
    In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...

     Λ from Einstein's field equation is related to the intrinsic energy density
    Energy density
    Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...

     of the vacuum
    Vacuum
    In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

     {{math|ρ}}vac via the gravitational constant
    Gravitational constant
    The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

      {{math|G}} as follows:
  • Coulomb's law
    Coulomb's law
    Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

     for the electric force
    Electric field
    In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

    , describing the force between two electric charge
    Electric charge
    Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

    s ({{math|q}}1 and {{math|q}}2) separated by distance {{math|r}} (with {{math|ε}}0 representing the vacuum permittivity of free space):
  • Magnetic permeability of free space relates the production of a magnetic field in a vacuum by an electric current in units of Newtons (N) and Ampere
    Ampere
    The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

    s (A):
  • Kepler's third law constant, relating the orbital period
    Orbital period
    The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...

     ({{math|P}}) and the semi-major axis
    Semi-major axis
    The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

     ({{math|a}}) to the mass
    Mass
    Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

    es ({{math|M}} and {{math|m}}) of two co-orbiting bodies:

Probability and statistics


In probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, there are many distributions
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 whose formulas contain {{pi}}, including:
  • the probability density function
    Probability density function
    In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

     for the normal distribution with mean
    Mean
    In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

     {{math|μ}} and standard deviation
    Standard deviation
    Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

     {{math|σ}}, due to the Gaussian integral
    Gaussian integral
    The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...

    :

  • the probability density function for the (standard) Cauchy distribution
    Cauchy distribution
    The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

    :



Note that since for any probability density function {{math|f}}({{math|x}}), the above formulas can be used to produce other integral formulas for {{pi}}.

Buffon's needle
Buffon's needle
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...

 problem is sometimes quoted as an empirical approximation of {{pi}} in "popular mathematics" works. Consider dropping a needle of length {{math|L}} repeatedly on a surface containing parallel lines drawn {{math|S}} units apart (with {{math|S}} > {{math|L}}). If the needle is dropped {{math|n}} times and {{math|x}} of those times it comes to rest crossing a line ({{math|x}} > 0), then one may approximate {{pi}} using the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

:


Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of {{pi}} by experiment. Reliably getting just three digits (including the initial "3") right requires millions of throws, and the number of throws grows exponentially
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...

 with the number of digits desired. Furthermore, any error in the measurement of the lengths {{math|L}} and {{math|S}} will transfer directly to an error in the approximated {{pi}}. For example, a difference of a single atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

 in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.{{Citation needed|date=January 2011}}

Geomorphology and chaos theory


Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the ratio between the actual length of a river and its straight-line from source to mouth length tends to approach {{pi}}. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist and so on. However, increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting, creating an ox-bow lake. The balance between these two opposing factors leads to an average ratio of {{pi}} between the actual length and the direct distance between source and mouth.

Tau as alternative notation for two pi


{{Main|Tau (2π)}}
A proposed alternative to {{pi}}, usually represented by the Greek letter tau ({{math|τ}}), is the ratio of a circle's circumference to its radius (instead of its diameter). This constant is the number of radians in a full circle, so the central angle
Central angle
A central angle is an angle which vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is equal to the central angle itself...

 of a fraction of a full circle is that same fraction of {{math|τ}} radians. Tau's proponents argue this direct relation makes learning about angles expressed in radians easier than with {{pi}}, where the fraction must be doubled. Though it has conventionally been written as the product "2{{pi}}", {{math|τ}} appears in many commonly used formulas.

In popular culture



Probably because of the simplicity of its definition, the concept of {{pi}} has become entrenched in popular culture to a degree far greater than almost any other mathematical construct. It is, perhaps, the most common ground between mathematicians and non-mathematicians. Reports on the latest, most-precise calculation of {{pi}} are common news items.

Nobel prize
Nobel Prize in Literature
Since 1901, the Nobel Prize in Literature has been awarded annually to an author from any country who has, in the words from the will of Alfred Nobel, produced "in the field of literature the most outstanding work in an ideal direction"...

 winning poet Wisława Szymborska wrote a poem about {{pi}}, and here is an excerpt:


The caravan of digits that is pi

does not stop at the edge of the page,

but runs off the table and into the air,

over the wall, a leaf, a bird's nest, the clouds, straight into the sky,

through all the bloatedness and bottomlessness.

Oh how short, all but mouse-like is the comet's tail!


Pi and its digital representation are often used by self-described "math geek
Geek
The word geek is a slang term, with different meanings ranging from "a computer expert or enthusiast" to "a carnival performer who performs sensationally morbid or disgusting acts", with a general pejorative meaning of "a peculiar or otherwise dislikable person, esp[ecially] one who is perceived to...

s" for inside jokes among mathematically and technologically-minded groups. Many schools around the world observe Pi Day
Pi Day
Pi Day is a holiday commemorating the mathematical constant π . Pi Day is celebrated on March 14 , since 3, 1 and 4 are the three most significant digits of π in the decimal form...

 (March 14, from 3.14). Several college cheers
Cheering
Cheering is the uttering or making of sounds encouraging, stimulating or exciting to action, indicating approval or acclaiming or welcoming persons, announcements of events and the like....

 at the Georgia Institute of Technology
Georgia Institute of Technology
The Georgia Institute of Technology is a public research university in Atlanta, Georgia, in the United States...

 and the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

 include "3.14159!" During the 2011 auction for Nortel
Nortel
Nortel Networks Corporation, formerly known as Northern Telecom Limited and sometimes known simply as Nortel, was a multinational telecommunications equipment manufacturer headquartered in Mississauga, Ontario, Canada...

's portfolio of valuable technology patents, Google
Google
Google Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internet-based services and products, and generates profit primarily from advertising through its AdWords program...

 made a series of unusually specific bids based on mathematical and scientific constants, including pi.

On November 7, 2005, alternative
Alternative rock
Alternative rock is a genre of rock music and a term used to describe a diverse musical movement that emerged from the independent music underground of the 1980s and became widely popular by the 1990s...

 musician Kate Bush
Kate Bush
Kate Bush is an English singer-songwriter, musician and record producer. Her eclectic musical style and idiosyncratic vocal style have made her one of the United Kingdom's most successful solo female performers of the past 30 years.In 1978, at the age of 19, Bush topped the UK Singles Chart...

 released the album Aerial
Aerial (album)
Aerial is the eighth studio album by British singer-songwriter and musician Kate Bush.-Overview:Aerial is Bush's first double album, and was released after a twelve year absence from the music industry during which Bush devoted her time to family and the rearing of her son, Bertie...

. The album contains the song "Pi" whose lyrics consist principally of Bush singing the digits of {{pi}} to music, beginning with "3.14".

In Carl Sagan
Carl Sagan
Carl Edward Sagan was an American astronomer, astrophysicist, cosmologist, author, science popularizer and science communicator in astronomy and natural sciences. He published more than 600 scientific papers and articles and was author, co-author or editor of more than 20 books...

's novel Contact
Contact (novel)
Contact is a science fiction novel written by Carl Sagan and published in 1985. It deals with the theme of contact between humanity and a more technologically advanced, extraterrestrial life form. It ranked No. 7 on the 1985 U.S. bestseller list....

, {{pi}} played a key role in the story. The novel suggested that there was a message buried deep within the digits of {{pi}} placed there by the creator of the universe. This part of the story was omitted from the film
Contact (film)
Contact is a 1997 American science fiction drama film adapted from the Carl Sagan novel of the same name and directed by Robert Zemeckis. Both Sagan and wife Ann Druyan wrote the story outline for the film adaptation of Contact....

 adaptation of the novel.

In the Star Trek: The Original Series
Star Trek: The Original Series
Star Trek is an American science fiction television series created by Gene Roddenberry, produced by Desilu Productions . Star Trek was telecast on NBC from September 8, 1966, through June 3, 1969...

 episode "Wolf in the Fold", after a murderous alien entity (which had once been Jack the Ripper
Jack the Ripper
"Jack the Ripper" is the best-known name given to an unidentified serial killer who was active in the largely impoverished areas in and around the Whitechapel district of London in 1888. The name originated in a letter, written by someone claiming to be the murderer, that was disseminated in the...

) takes over the Enterprise
USS Enterprise (NCC-1701)
The USS Enterprise, NCC-1701, is a fictional starship in the Star Trek media franchise. The original Star Trek series depicts her crew's mission "to explore strange new worlds; to seek out new life and new civilizations; to boldly go where no man has gone before" under the command of Captain James...

s main computer with the intention of using it to slowly kill the crew, Kirk
James T. Kirk
James Tiberius "Jim" Kirk is a character in the Star Trek media franchise. Kirk was first played by William Shatner as the principal lead character in the original Star Trek series. Shatner voiced Kirk in the animated Star Trek series and appeared in the first seven Star Trek movies...

 and Spock
Spock
Spock is a fictional character in the Star Trek media franchise. First portrayed by Leonard Nimoy in the original Star Trek series, Spock also appears in the animated Star Trek series, two episodes of Star Trek: The Next Generation, seven of the Star Trek feature films, and numerous Star Trek...

 draw the entity out of the computer by forcing it to compute pi to the nonexistent last digit, causing the creature to abandon the computer, allowing it to be beamed into space.

In the Stargate SG-1
Stargate SG-1
Stargate SG-1 is a Canadian-American adventure and military science fiction television series and part of Metro-Goldwyn-Mayer's Stargate franchise. The show, created by Brad Wright and Jonathan Glassner, is based on the 1994 feature film Stargate by Dean Devlin and Roland Emmerich...

 season 2 episode "Thor's Chariot", Daniel Jackson and Samantha Carter
Samantha Carter
Samantha "Sam" Carter is a fictional character in the Canadian-American military science fiction Stargate franchise, appearing in television series Stargate SG-1, Stargate Atlantis, and Stargate Universe. SG-1 and Atlantis are both about a military team exploring the galaxy via a network of alien...

 and Cimmeria local Gairwyn are transported to the Hall of Thor's Might, in which one of the walls has four runes, while another has four simple geometric figures. After Daniel Jackson mentions the fact that the runes on the wall also represented the numbers 3, 14, 15 and 9, Samantha Carter realizes that this sequence of numbers corresponds to {{pi}}. The team then correctly solves this puzzle by marking the radius on the circle on the second wall.

In The Simpsons
The Simpsons
The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a satirical parody of a middle class American lifestyle epitomized by its family of the same name, which consists of Homer, Marge, Bart, Lisa and Maggie...

 season 12 episode "Bye Bye Nerdie
Bye Bye Nerdie
"Bye Bye Nerdie" is the sixteenth episode of the twelfth season of the American animated sitcom The Simpsons. It originally aired on the Fox network in the United States on March 11, 2001. In the episode, when she becomes the target of a female bully, Lisa discovers a scientific reason as to why...

", Professor Frink
Professor Frink
Professor John Nerdelbaum Frink, Jr., or simply Professor Frink, is a recurring character in the animated television series The Simpsons. He is voiced by Hank Azaria, and first appeared in the 1991 episode "Old Money". Frink is Springfield's nerdy scientist and professor and is extremely...

 exclaims "pi is exactly three!" to get the attention of the attendees to the "12th Annual Big Science Thing" contest.

Darren Aronofsky
Darren Aronofsky
Darren Aronofsky is an American film director, screenwriter and film producer. He attended Harvard University to study film theory and the American Film Institute to study both live-action and animation filmmaking...

's film Pi
Pi (film)
Pi, also titled ,WorldCat gives the title as [Pi] and provides a note which states, "Title is the mathematical symbol for Pi." . Amazon gives the title as Pi with no notation concerning the math symbol . is a 1998 American psychological thriller film written and directed by Darren Aronofsky...

 deals with a number theorist
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.

In the fictional movie, Night at the Museum: Battle of the Smithsonian
Night at the Museum: Battle of the Smithsonian
Night at the Museum: Battle of the Smithsonian is an American adventure comedy film directed by Shawn Levy, and starring Ben Stiller, Hank Azaria, Amy Adams, Owen Wilson, Robin Williams, and Steve Coogan. The film is a sequel to Night at the Museum...

, {{pi}} is the answer to the combination that will allow the Tablet of Akh-man-Ra to open the gates to the underworld.

A style of writing called Pilish
Pilish
Pilish is a style of writing in which the lengths of consecutive words match the digits of the number . The following sentence is an example which matches the first fifteen digits of :...

 has been developed, in which the lengths of consecutive words match the digits of the number {{pi}}.

See also

{{Irrational numbers}}
  • The Feynman point
    Feynman point
    The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...

    , a sequence of six 9s that appears at the 762nd through 767th decimal places of {{pi}}
  • Indiana Pi Bill
    Indiana Pi Bill
    The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most famous attempts to establish scientific truth by legislative fiat...

  • List of topics related to pi
  • Proof that 22/7 exceeds π

External links


{{Commons category}} Decimal expansions of Pi and related links at the On-Line Encyclopedia of Integer Sequences
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...

  • Formulas for {{pi}} at MathWorld
    MathWorld
    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...

  • Representations of Pi at Wolfram Alpha
    Wolfram Alpha
    Wolfram Alpha is an answer-engine developed by Wolfram Research. It is an online service that answers factual queries directly by computing the answer from structured data, rather than providing a list of documents or web pages that might contain the answer as a search engine might...

  • Pi at PlanetMath
    PlanetMath
    PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...

  • Determination of {{pi}} at Cut-the-knot
    Cut-the-knot
    Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...

  • The Tau Manifesto, physicist Michael Hartl outlines a proposal to replace {{pi}} with (tau).

{{Good article}}