is a
mathematical constantA 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
ratioIn 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
circleA 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
diameterIn 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,
scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, and
engineeringEngineering 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 circleThe 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 numberIn 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 nonzero, and is therefore not a rational number....
, which means that its value cannot be expressed exactly as a
fractionA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
having
integerThe integers are formed by the natural numbers together with the negatives of the nonzero 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
repeatsIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 nonmathematical 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 nonmathematicians. Reports on the latest, mostprecise 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
perimeterA 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 SyracuseArchimedes 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 BCThe 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 CeulenLudolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
, who computed a 35digit approximation around the year 1600.
The Greek letter
The Latin name of the
Greek 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 uppercase letter Π is used as a symbol for:...
is pi. When referring to the constant, the symbol is pronounced like the
EnglishEnglish is a West Germanic language that arose in the AngloSaxon kingdoms of England and spread into what was to become southeast 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
GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean 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 EulerLeonhard 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 constantA 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 geometryEuclidean 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
ratioIn 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
circleA 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
circumferenceThe 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
diameterIn 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
similarTwo 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
analyticMathematical 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 functionIn 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 numberIn 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 nonzero, and is therefore not a rational number....
, meaning that it cannot be written as the
ratio of two integersIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
, meaning that there is no
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
with
rationalIn 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
constructibleA 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 circleSquaring 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
truncatedIn 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 placesThe 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 (10
^{12}) digits, elementary
applicationsApplied 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
errorA roundoff 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 universeIn 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 atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positivelycharged proton and a single negativelycharged electron bound to the nucleus by the Coulomb force...
.
Because is an
irrational numberIn 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 nonzero, and is therefore not a rational number....
, its decimal representation does not
repeatIn 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 nonrepeating 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
supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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
base10The 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 computerA personal computer is any generalpurpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an enduser with no intervening computer operator...
.
Estimating the value

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


HexadecimalIn 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 PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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 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 periodicIn 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 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
perimeterA 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
regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
hexagon to the
diameterIn 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
circleA 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 geometrybased approach, attributed to
ArchimedesArchimedes 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
perimeterA 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 polygonA 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 sequenceThe 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
infinityInfinity 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
seriesA 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
trigonometryTrigonometry 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
calculusCalculus 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
sequenceIn 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 bruteforce manner, and
, correct to 9 decimal places. This computation is an example of the
van Wijngaarden transformation.
For many purposes, 3.14 or
^{22}⁄
_{7} is close enough, although engineers often use 3.1416 (5
significant figuresThe 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
^{22}⁄
_{7} and
^{355}⁄
_{113}, with 3 and 7 significant figures respectively, are obtained from the simple
continued fractionIn 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
^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit
numerator and denominatorA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
; the next good approximation
^{52163}⁄
_{16604} (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
^{103993}⁄
_{33102} (3.14159265301...) are used, which are both accurate to 10 significant figures.
History
The
Great PyramidThe 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
MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
c.26132589 BC and later in the pyramids of Abusir c.24532422. 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 rightangled 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 BrahmanaThe 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 IndiaIron 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.
ArchimedesArchimedes 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 polygonA 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 96sided polygons, he proved that
The average of these values is about 3.14185.
PtolemyClaudius 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
AlmagestThe Almagest is a 2ndcentury 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 PergaApollonius 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 HuiLiu 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 3072gon (i.e. a 3072sided 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 96gon, 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 ChongzhiZu 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 12288gon. This value would remain the most accurate approximation of available for the next 900 years.
MaimonidesMoses benMaimon, 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 LambertJohann 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
integralIntegration 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 NivenIvan Morton Niven was a CanadianAmerican 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 CartwrightDame Mary Lucy Cartwright DBE FRS was a leading 20thcentury 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 ADFile:2nd millennium montage.pngFrom 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 seriesA 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
calculusCalculus 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 SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 GregoryJames 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 LeibnizGottfried 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

MadhavaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī 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 CeulenLudolph 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 VièteFrançois 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 NewtonSir 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 MachinJohn 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 Machinlike 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 DaseJohann 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 Machinlike formula to calculate 200 decimals of in his head at the behest of GaussJohann 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 ShanksWilliam 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 LambertJohann 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 AdrienMarie LegendreAdrienMarie Legendre was a French mathematician.The Moon crater Legendre is named after him. Life :...
also proved in 1794 ^{2} to be irrational. When Leonhard EulerLeonhard 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 problemThe 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 numberA 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 transcendentalIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 LindemannCarl 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 NeumannJohn von Neumann was a HungarianAmerican 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 ENIACENIAC was the first generalpurpose electronic computer. It was a Turingcomplete 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 milliondigit 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 transformA 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 RamanujanSrīnivāsa 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 factorialIn mathematics, the factorial of a nonnegative 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 symbolIn mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a nonnegative 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 brothersThe Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their homebuilt 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 BrentRichard 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 JonathanJonathan 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
. The methods have been used by Yasumasa Kanadais 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 T2KTsukuba 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 FEEIn 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 Efunction 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 FEEIn 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 Efunction 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. PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
, F. BellardFabrice 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 FEEIn 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 Efunction 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 formulaThe 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 PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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. BaileyDavid 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
, and Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 hexadecimalIn 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 binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 system is a positional notation with a radix of 2...
digit of without calculating all the preceding ones. Between 1998 and 2000, the distributed computingDistributed 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 PiHexPiHex 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 BellardFabrice 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*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*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! 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 twoquadrillion^{th} (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 23day period. The formula is used to compute a single bit of in a small set of mathematical steps.
In 2006, Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 algorithmAn 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 =
{{ppsemismall=yes}}
{{Two other usesthe numberthe Greek letterPi (letter)}}
{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constantA 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 ratioIn 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 circleA 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 diameterIn 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, scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, and engineeringEngineering 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 circleThe 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 numberIn 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 nonzero, and is therefore not a rational number....
, which means that its value cannot be expressed exactly as a fractionA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
having integerThe integers are formed by the natural numbers together with the negatives of the nonzero 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 repeatsIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 nonmathematical 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 nonmathematicians. Reports on the latest, mostprecise 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 perimeterA 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 SyracuseArchimedes 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 BCThe 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 CeulenLudolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
, who computed a 35digit approximation around the year 1600.
The Greek letter
{{MainPi (letter)}}
The Latin name of the Greek letter {{pi}}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 uppercase letter Π is used as a symbol for:...
is pi. When referring to the constant, the symbol {{pi}} is pronounced like the EnglishEnglish is a West Germanic language that arose in the AngloSaxon kingdoms of England and spread into what was to become southeast 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 GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean 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 neededdate=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 EulerLeonhard 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 constantA 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 geometryEuclidean 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 ratioIn 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 circleA 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 circumferenceThe 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....
{{mathC}} to its diameterIn 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...
{{mathd}}:
The ratio {{mathC/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{mathd}} of another circle it will also have twice the circumference {{mathC}}, preserving the ratio {{mathC/d}}.
This definition depends on results of Euclidean geometry, such as the fact that all circles are similarTwo 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 analyticMathematical 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 {{mathx}} for which the trigonometric functionIn 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({{mathx}}) equals zero.
Irrationality and transcendence
{{MainProof that π is irrational}}
{{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, meaning that it cannot be written as the ratio of two integersIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
, meaning that there is no polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
with rationalIn 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 constructibleA 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 circleSquaring 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 alsoApproximations of π}}
The decimal representation of {{pi}} truncatedIn 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 placesThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
is:
 {{gapslhs={{pi}}3.14159265358979323846264338327950288419716939937510...}}.
Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (10^{12}) digits, elementary applicationsApplied 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 errorA roundoff 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 universeIn 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 atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positivelycharged proton and a single negativelycharged electron bound to the nucleus by the Coulomb force...
.
Because {{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, its decimal representation does not repeatIn 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 nonrepeating 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 base10The 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 computerA personal computer is any generalpurpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an enduser with no intervening computer operator...
.
Estimating the value
{{MainApproximations of π}}

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

{{gaps 
265358979323846264338327950288...}} 
HexadecimalIn 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 
A8885A308D31319...}} 
Sexagesimal (used by ancients, including PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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, {{frac227}}, {{frac333106}}, {{frac355113}}, {{frac5216316604}}, {{frac10399333102}}, ...
(listed in order of increasing accuracy) 
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 periodicIn 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 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 {{num3}}.{{verification faileddate=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 perimeterA 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 regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
hexagon to the diameterIn 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 circleA 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 geometrybased approach, attributed to ArchimedesArchimedes 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 perimeterA 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...
, {{mathP}}_{{{mathn}}}, of a regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
with {{mathn}} sides circumscribed around a circle with diameter {{mathd}}. Then compute the limit of a sequenceThe 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 {{mathn}} increases to infinityInfinity 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 seriesA 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 trigonometryTrigonometry 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 calculusCalculus 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 sequenceIn 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 bruteforce manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.
For many purposes, 3.14 or ^{22}⁄_{7} is close enough, although engineers often use 3.1416 (5 significant figuresThe 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 neededdate=March 2011}} The approximations ^{22}⁄_{7} and ^{355}⁄_{113}, with 3 and 7 significant figures respectively, are obtained from the simple continued fractionIn 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 ^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit numerator and denominatorA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
; the next good approximation ^{52163}⁄_{16604} (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 ^{103993}⁄_{33102} (3.14159265301...) are used, which are both accurate to 10 significant figures.
History
{{See alsoChronology of computation of πApproximations of π}}
The Great PyramidThe 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 MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
c.26132589 BC and later in the pyramids of Abusir c.24532422. 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 rightangled 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 BrahmanaThe 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 IndiaIron 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 {{bibleverse1Kings7:23NKJV}} and {{bibleverse2Chronicles4:2NKJV}} 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
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}}ArchimedesArchimedes 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 polygonA 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 96sided polygons, he proved that The average of these values is about 3.14185.
PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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 PergaApollonius 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 HuiLiu 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 3072gon (i.e. a 3072sided 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 96gon, 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 ChongzhiZu 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 12288gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.
MaimonidesMoses benMaimon, 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 LambertJohann 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 integralIntegration 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 NivenIvan Morton Niven was a CanadianAmerican 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 CartwrightDame Mary Lucy Cartwright DBE FRS was a leading 20thcentury 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 ADFile:2nd millennium montage.pngFrom 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 seriesA 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 calculusCalculus 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 SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 GregoryJames 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 LeibnizGottfried 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

MadhavaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī 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 CeulenLudolph 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 VièteFrançois 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 NewtonSir 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 MachinJohn 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 Machinlike 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 DaseJohann 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 Machinlike formula to calculate 200 decimals of {{pi}} in his head at the behest of GaussJohann 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 ShanksWilliam 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 LambertJohann 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 AdrienMarie LegendreAdrienMarie 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 EulerLeonhard 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 problemThe 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 numberA 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 transcendentalIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 LindemannCarl 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 NeumannJohn von Neumann was a HungarianAmerican 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 ENIACENIAC was the first generalpurpose electronic computer. It was a Turingcomplete 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 milliondigit 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 transformA 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 RamanujanSrīnivāsa 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 {{mathk}}! is the factorialIn mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
of {{mathk}}.
A collection of some others are in the table below:
where

is the Pochhammer symbolIn mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a nonnegative 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 brothersThe Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their homebuilt 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 BrentRichard 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 {{matha}}_{{{mathn}}} and {{mathb}}_{{{mathn}}} 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 JonathanJonathan 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
. The methods have been used by Yasumasa Kanadais 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 T2KTsukuba 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 FEEIn 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 Efunction 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 FEEIn 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 Efunction 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. PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
, F. BellardFabrice 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 FEEIn 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 Efunction 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 formulaThe 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 PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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. BaileyDavid 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
, and Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 hexadecimalIn 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 binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 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 computingDistributed 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 PiHexPiHex 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 BellardFabrice 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 {{mathb}} and {{mathc}} are positive integers and {{mathp}} and {{mathp}} 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 {{mathb}}^{{{mathc}}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{mathk}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*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 {{mathk}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{mathb}} = 2 and {{mathc}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*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 neededdate=March 2011}}
In September 2010, 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 twoquadrillion^{th} (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 23day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.
In 2006, Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 algorithmAn 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 {{mathq}} =
{{ppsemismall=yes}}
{{Two other usesthe numberthe Greek letterPi (letter)}}
{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constantA 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 ratioIn 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 circleA 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 diameterIn 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, scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, and engineeringEngineering 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 circleThe 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 numberIn 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 nonzero, and is therefore not a rational number....
, which means that its value cannot be expressed exactly as a fractionA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
having integerThe integers are formed by the natural numbers together with the negatives of the nonzero 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 repeatsIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 nonmathematical 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 nonmathematicians. Reports on the latest, mostprecise 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 perimeterA 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 SyracuseArchimedes 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 BCThe 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 CeulenLudolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
, who computed a 35digit approximation around the year 1600.
The Greek letter
{{MainPi (letter)}}
The Latin name of the Greek letter {{pi}}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 uppercase letter Π is used as a symbol for:...
is pi. When referring to the constant, the symbol {{pi}} is pronounced like the EnglishEnglish is a West Germanic language that arose in the AngloSaxon kingdoms of England and spread into what was to become southeast 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 GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean 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 neededdate=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 EulerLeonhard 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 constantA 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 geometryEuclidean 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 ratioIn 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 circleA 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 circumferenceThe 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....
{{mathC}} to its diameterIn 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...
{{mathd}}:
The ratio {{mathC/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{mathd}} of another circle it will also have twice the circumference {{mathC}}, preserving the ratio {{mathC/d}}.
This definition depends on results of Euclidean geometry, such as the fact that all circles are similarTwo 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 analyticMathematical 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 {{mathx}} for which the trigonometric functionIn 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({{mathx}}) equals zero.
Irrationality and transcendence
{{MainProof that π is irrational}}
{{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, meaning that it cannot be written as the ratio of two integersIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
, meaning that there is no polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
with rationalIn 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 constructibleA 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 circleSquaring 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 alsoApproximations of π}}
The decimal representation of {{pi}} truncatedIn 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 placesThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
is:
 {{gapslhs={{pi}}3.14159265358979323846264338327950288419716939937510...}}.
Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (10^{12}) digits, elementary applicationsApplied 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 errorA roundoff 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 universeIn 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 atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positivelycharged proton and a single negativelycharged electron bound to the nucleus by the Coulomb force...
.
Because {{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, its decimal representation does not repeatIn 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 nonrepeating 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 base10The 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 computerA personal computer is any generalpurpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an enduser with no intervening computer operator...
.
Estimating the value
{{MainApproximations of π}}

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

{{gaps 
265358979323846264338327950288...}} 
HexadecimalIn 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 
A8885A308D31319...}} 
Sexagesimal (used by ancients, including PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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, {{frac227}}, {{frac333106}}, {{frac355113}}, {{frac5216316604}}, {{frac10399333102}}, ...
(listed in order of increasing accuracy) 
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 periodicIn 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 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 {{num3}}.{{verification faileddate=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 perimeterA 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 regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
hexagon to the diameterIn 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 circleA 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 geometrybased approach, attributed to ArchimedesArchimedes 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 perimeterA 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...
, {{mathP}}_{{{mathn}}}, of a regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
with {{mathn}} sides circumscribed around a circle with diameter {{mathd}}. Then compute the limit of a sequenceThe 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 {{mathn}} increases to infinityInfinity 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 seriesA 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 trigonometryTrigonometry 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 calculusCalculus 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 sequenceIn 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 bruteforce manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.
For many purposes, 3.14 or ^{22}⁄_{7} is close enough, although engineers often use 3.1416 (5 significant figuresThe 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 neededdate=March 2011}} The approximations ^{22}⁄_{7} and ^{355}⁄_{113}, with 3 and 7 significant figures respectively, are obtained from the simple continued fractionIn 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 ^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit numerator and denominatorA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
; the next good approximation ^{52163}⁄_{16604} (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 ^{103993}⁄_{33102} (3.14159265301...) are used, which are both accurate to 10 significant figures.
History
{{See alsoChronology of computation of πApproximations of π}}
The Great PyramidThe 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 MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
c.26132589 BC and later in the pyramids of Abusir c.24532422. 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 rightangled 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 BrahmanaThe 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 IndiaIron 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 {{bibleverse1Kings7:23NKJV}} and {{bibleverse2Chronicles4:2NKJV}} 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
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}}ArchimedesArchimedes 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 polygonA 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 96sided polygons, he proved that The average of these values is about 3.14185.
PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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 PergaApollonius 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 HuiLiu 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 3072gon (i.e. a 3072sided 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 96gon, 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 ChongzhiZu 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 12288gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.
MaimonidesMoses benMaimon, 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 LambertJohann 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 integralIntegration 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 NivenIvan Morton Niven was a CanadianAmerican 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 CartwrightDame Mary Lucy Cartwright DBE FRS was a leading 20thcentury 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 ADFile:2nd millennium montage.pngFrom 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 seriesA 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 calculusCalculus 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 SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 GregoryJames 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 LeibnizGottfried 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

MadhavaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī 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 CeulenLudolph 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 VièteFrançois 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 NewtonSir 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 MachinJohn 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 Machinlike 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 DaseJohann 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 Machinlike formula to calculate 200 decimals of {{pi}} in his head at the behest of GaussJohann 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 ShanksWilliam 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 LambertJohann 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 AdrienMarie LegendreAdrienMarie 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 EulerLeonhard 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 problemThe 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 numberA 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 transcendentalIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 LindemannCarl 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 NeumannJohn von Neumann was a HungarianAmerican 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 ENIACENIAC was the first generalpurpose electronic computer. It was a Turingcomplete 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 milliondigit 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 transformA 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 RamanujanSrīnivāsa 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 {{mathk}}! is the factorialIn mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
of {{mathk}}.
A collection of some others are in the table below:
where

is the Pochhammer symbolIn mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a nonnegative 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 brothersThe Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their homebuilt 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 BrentRichard 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 {{matha}}_{{{mathn}}} and {{mathb}}_{{{mathn}}} 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 JonathanJonathan 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
. The methods have been used by Yasumasa Kanadais 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 T2KTsukuba 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 FEEIn 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 Efunction 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 FEEIn 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 Efunction 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. PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
, F. BellardFabrice 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 FEEIn 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 Efunction 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 formulaThe 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 PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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. BaileyDavid 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
, and Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 hexadecimalIn 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 binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 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 computingDistributed 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 PiHexPiHex 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 BellardFabrice 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 {{mathb}} and {{mathc}} are positive integers and {{mathp}} and {{mathp}} 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 {{mathb}}^{{{mathc}}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{mathk}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*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 {{mathk}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{mathb}} = 2 and {{mathc}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*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 neededdate=March 2011}}
In September 2010, 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 twoquadrillion^{th} (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 23day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.
In 2006, Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 algorithmAn 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 {{mathq}} = {{mathIn 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 {{mathk}} is an odd number, and {{matha}}, {{mathb}}, {{mathc}} are rational numberIn 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 {{mathk}} is of the form 4{{mathm}} + 3, then the formula has the particularly simple form,
for some rational number {{mathp}} 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 fractionIn 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 fractionIn 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 rapidlyconverging expression:
Memorizing digits
{{MainPiphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira HaraguchiAkira 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 RecordsGuinness 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 Guinnessrecognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu ChaoLu 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 24yearold graduate student from ChinaChinese 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 JeansSir 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 CadenzaCadaeic 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 techniquesA 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 piphilologyPiphilology 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 lociThe 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 numberIn 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.
BaileyDavid 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 CrandallRichard 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 formulaThe 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 conjectureA 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 theoryChaos 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 {{mathThe 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 independentIn 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 nontrivial polynomial equation with coefficients in K...
, although Yuri NesterenkoYuri 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
{{ppsemismall=yes}}
{{Two other usesthe numberthe Greek letterPi (letter)}}
{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constantA 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 ratioIn 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 circleA 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 diameterIn 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, scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, and engineeringEngineering 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 circleThe 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 numberIn 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 nonzero, and is therefore not a rational number....
, which means that its value cannot be expressed exactly as a fractionA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
having integerThe integers are formed by the natural numbers together with the negatives of the nonzero 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 repeatsIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 nonmathematical 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 nonmathematicians. Reports on the latest, mostprecise 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 perimeterA 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 SyracuseArchimedes 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 BCThe 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 CeulenLudolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
, who computed a 35digit approximation around the year 1600.
The Greek letter
{{MainPi (letter)}}
The Latin name of the Greek letter {{pi}}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 uppercase letter Π is used as a symbol for:...
is pi. When referring to the constant, the symbol {{pi}} is pronounced like the EnglishEnglish is a West Germanic language that arose in the AngloSaxon kingdoms of England and spread into what was to become southeast 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 GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean 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 neededdate=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 EulerLeonhard 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 constantA 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 geometryEuclidean 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 ratioIn 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 circleA 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 circumferenceThe 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....
{{mathC}} to its diameterIn 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...
{{mathd}}:
The ratio {{mathC/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{mathd}} of another circle it will also have twice the circumference {{mathC}}, preserving the ratio {{mathC/d}}.
This definition depends on results of Euclidean geometry, such as the fact that all circles are similarTwo 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 analyticMathematical 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 {{mathx}} for which the trigonometric functionIn 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({{mathx}}) equals zero.
Irrationality and transcendence
{{MainProof that π is irrational}}
{{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, meaning that it cannot be written as the ratio of two integersIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
, meaning that there is no polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
with rationalIn 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 constructibleA 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 circleSquaring 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 alsoApproximations of π}}
The decimal representation of {{pi}} truncatedIn 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 placesThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
is:
 {{gapslhs={{pi}}3.14159265358979323846264338327950288419716939937510...}}.
Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (10^{12}) digits, elementary applicationsApplied 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 errorA roundoff 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 universeIn 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 atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positivelycharged proton and a single negativelycharged electron bound to the nucleus by the Coulomb force...
.
Because {{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, its decimal representation does not repeatIn 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 nonrepeating 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 base10The 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 computerA personal computer is any generalpurpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an enduser with no intervening computer operator...
.
Estimating the value
{{MainApproximations of π}}

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

{{gaps 
265358979323846264338327950288...}} 
HexadecimalIn 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 
A8885A308D31319...}} 
Sexagesimal (used by ancients, including PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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, {{frac227}}, {{frac333106}}, {{frac355113}}, {{frac5216316604}}, {{frac10399333102}}, ...
(listed in order of increasing accuracy) 
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 periodicIn 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 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 {{num3}}.{{verification faileddate=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 perimeterA 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 regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
hexagon to the diameterIn 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 circleA 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 geometrybased approach, attributed to ArchimedesArchimedes 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 perimeterA 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...
, {{mathP}}_{{{mathn}}}, of a regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
with {{mathn}} sides circumscribed around a circle with diameter {{mathd}}. Then compute the limit of a sequenceThe 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 {{mathn}} increases to infinityInfinity 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 seriesA 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 trigonometryTrigonometry 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 calculusCalculus 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 sequenceIn 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 bruteforce manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.
For many purposes, 3.14 or ^{22}⁄_{7} is close enough, although engineers often use 3.1416 (5 significant figuresThe 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 neededdate=March 2011}} The approximations ^{22}⁄_{7} and ^{355}⁄_{113}, with 3 and 7 significant figures respectively, are obtained from the simple continued fractionIn 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 ^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit numerator and denominatorA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
; the next good approximation ^{52163}⁄_{16604} (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 ^{103993}⁄_{33102} (3.14159265301...) are used, which are both accurate to 10 significant figures.
History
{{See alsoChronology of computation of πApproximations of π}}
The Great PyramidThe 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 MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
c.26132589 BC and later in the pyramids of Abusir c.24532422. 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 rightangled 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 BrahmanaThe 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 IndiaIron 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 {{bibleverse1Kings7:23NKJV}} and {{bibleverse2Chronicles4:2NKJV}} 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
}}ArchimedesArchimedes 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 polygonA 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 96sided polygons, he proved that The average of these values is about 3.14185.
PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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 PergaApollonius 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 HuiLiu 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 3072gon (i.e. a 3072sided 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 96gon, 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 ChongzhiZu 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 12288gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.
MaimonidesMoses benMaimon, 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 LambertJohann 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 integralIntegration 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 NivenIvan Morton Niven was a CanadianAmerican 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 CartwrightDame Mary Lucy Cartwright DBE FRS was a leading 20thcentury 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 ADFile:2nd millennium montage.pngFrom 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 seriesA 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 calculusCalculus 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 SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 GregoryJames 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 LeibnizGottfried 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

MadhavaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī 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 CeulenLudolph 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 VièteFrançois 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 NewtonSir 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 MachinJohn 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 Machinlike 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 DaseJohann 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 Machinlike formula to calculate 200 decimals of {{pi}} in his head at the behest of GaussJohann 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 ShanksWilliam 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 LambertJohann 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 AdrienMarie LegendreAdrienMarie 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 EulerLeonhard 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 problemThe 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 numberA 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 transcendentalIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 LindemannCarl 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 NeumannJohn von Neumann was a HungarianAmerican 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 ENIACENIAC was the first generalpurpose electronic computer. It was a Turingcomplete 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 milliondigit 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 transformA 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 RamanujanSrīnivāsa 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 {{mathk}}! is the factorialIn mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
of {{mathk}}.
A collection of some others are in the table below:
where

is the Pochhammer symbolIn mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a nonnegative 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 brothersThe Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their homebuilt 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 BrentRichard 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 {{matha}}_{{{mathn}}} and {{mathb}}_{{{mathn}}} 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 JonathanJonathan 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
. The methods have been used by Yasumasa Kanadais 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 T2KTsukuba 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 FEEIn 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 Efunction 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 FEEIn 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 Efunction 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. PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
, F. BellardFabrice 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 FEEIn 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 Efunction 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 formulaThe 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 PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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. BaileyDavid 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
, and Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 hexadecimalIn 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 binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 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 computingDistributed 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 PiHexPiHex 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 BellardFabrice 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 {{mathb}} and {{mathc}} are positive integers and {{mathp}} and {{mathp}} 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 {{mathb}}^{{{mathc}}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{mathk}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*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 {{mathk}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{mathb}} = 2 and {{mathc}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*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 neededdate=March 2011}}
In September 2010, 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 twoquadrillion^{th} (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 23day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.
In 2006, Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 algorithmAn 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 {{mathq}} = {{mathIn 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 {{mathk}} is an odd number, and {{matha}}, {{mathb}}, {{mathc}} are rational numberIn 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 {{mathk}} is of the form 4{{mathm}} + 3, then the formula has the particularly simple form,
for some rational number {{mathp}} 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 fractionIn 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 fractionIn 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 rapidlyconverging expression:
Memorizing digits
{{MainPiphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira HaraguchiAkira 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 RecordsGuinness 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 Guinnessrecognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu ChaoLu 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 24yearold graduate student from ChinaChinese 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 JeansSir 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 CadenzaCadaeic 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 techniquesA 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 piphilologyPiphilology 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 lociThe 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 numberIn 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.
BaileyDavid 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 CrandallRichard 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 formulaThe 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 conjectureA 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 theoryChaos 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 {{mathThe 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 independentIn 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 nontrivial polynomial equation with coefficients in K...
, although Yuri NesterenkoYuri 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
{{ppsemismall=yes}}
{{Two other usesthe numberthe Greek letterPi (letter)}}
{{Pi box}}
{{pi}} (sometimes written pi) is a mathematical constantA 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 ratioIn 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 circleA 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 diameterIn 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, scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
, and engineeringEngineering 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 circleThe 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 numberIn 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 nonzero, and is therefore not a rational number....
, which means that its value cannot be expressed exactly as a fractionA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
having integerThe integers are formed by the natural numbers together with the negatives of the nonzero 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 repeatsIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 nonmathematical 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 nonmathematicians. Reports on the latest, mostprecise 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 perimeterA 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 SyracuseArchimedes 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 BCThe 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 CeulenLudolph van Ceulen was a German / Dutch mathematician from Hildesheim. He emigrated to the Netherlands....
, who computed a 35digit approximation around the year 1600.
The Greek letter
{{MainPi (letter)}}
The Latin name of the Greek letter {{pi}}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 uppercase letter Π is used as a symbol for:...
is pi. When referring to the constant, the symbol {{pi}} is pronounced like the EnglishEnglish is a West Germanic language that arose in the AngloSaxon kingdoms of England and spread into what was to become southeast 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 GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean 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 neededdate=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 EulerLeonhard 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 constantA 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 geometryEuclidean 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 ratioIn 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 circleA 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 circumferenceThe 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....
{{mathC}} to its diameterIn 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...
{{mathd}}:
The ratio {{mathC/d}} is constant, regardless of a circle's size. For example, if a circle has twice the diameter {{mathd}} of another circle it will also have twice the circumference {{mathC}}, preserving the ratio {{mathC/d}}.
This definition depends on results of Euclidean geometry, such as the fact that all circles are similarTwo 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 analyticMathematical 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 {{mathx}} for which the trigonometric functionIn 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({{mathx}}) equals zero.
Irrationality and transcendence
{{MainProof that π is irrational}}
{{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, meaning that it cannot be written as the ratio of two integersIn 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 numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
, meaning that there is no polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
with rationalIn 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 constructibleA 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 circleSquaring 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 alsoApproximations of π}}
The decimal representation of {{pi}} truncatedIn 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 placesThe decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
is:
 {{gapslhs={{pi}}3.14159265358979323846264338327950288419716939937510...}}.
Various online web sites provide {{pi}} to many more digits. While the decimal representation of {{pi}} has been computed to more than a trillion (10^{12}) digits, elementary applicationsApplied 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 errorA roundoff 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 universeIn 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 atomA hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positivelycharged proton and a single negativelycharged electron bound to the nucleus by the Coulomb force...
.
Because {{pi}} is an irrational numberIn 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 nonzero, and is therefore not a rational number....
, its decimal representation does not repeatIn 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 nonrepeating 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 base10The 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 computerA personal computer is any generalpurpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an enduser with no intervening computer operator...
.
Estimating the value
{{MainApproximations of π}}

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

{{gaps 
265358979323846264338327950288...}} 
HexadecimalIn 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 
A8885A308D31319...}} 
Sexagesimal (used by ancients, including PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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, {{frac227}}, {{frac333106}}, {{frac355113}}, {{frac5216316604}}, {{frac10399333102}}, ...
(listed in order of increasing accuracy) 
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 periodicIn 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 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 {{num3}}.{{verification faileddate=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 perimeterA 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 regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
hexagon to the diameterIn 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 circleA 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 geometrybased approach, attributed to ArchimedesArchimedes 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 perimeterA 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...
, {{mathP}}_{{{mathn}}}, of a regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
with {{mathn}} sides circumscribed around a circle with diameter {{mathd}}. Then compute the limit of a sequenceThe 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 {{mathn}} increases to infinityInfinity 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 seriesA 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 trigonometryTrigonometry 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 calculusCalculus 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 sequenceIn 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 bruteforce manner, and , correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.
For many purposes, 3.14 or ^{22}⁄_{7} is close enough, although engineers often use 3.1416 (5 significant figuresThe 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 neededdate=March 2011}} The approximations ^{22}⁄_{7} and ^{355}⁄_{113}, with 3 and 7 significant figures respectively, are obtained from the simple continued fractionIn 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 ^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit numerator and denominatorA 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, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
; the next good approximation ^{52163}⁄_{16604} (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 ^{103993}⁄_{33102} (3.14159265301...) are used, which are both accurate to 10 significant figures.
History
{{See alsoChronology of computation of πApproximations of π}}
The Great PyramidThe 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 MeidumLocated about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mudbrick mastabas.Pyramid:...
c.26132589 BC and later in the pyramids of Abusir c.24532422. 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 rightangled 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 BrahmanaThe 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 IndiaIron 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 {{bibleverse1Kings7:23NKJV}} and {{bibleverse2Chronicles4:2NKJV}} 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
}}ArchimedesArchimedes 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 polygonA 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 96sided polygons, he proved that The average of these values is about 3.14185.
PtolemyClaudius 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 AlmagestThe Almagest is a 2ndcentury 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 PergaApollonius 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 HuiLiu 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 3072gon (i.e. a 3072sided 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 96gon, 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 ChongzhiZu 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 12288gon. This value would remain the most accurate approximation of {{pi}} available for the next 900 years.
MaimonidesMoses benMaimon, 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 LambertJohann 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 integralIntegration 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 NivenIvan Morton Niven was a CanadianAmerican 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 CartwrightDame Mary Lucy Cartwright DBE FRS was a leading 20thcentury 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 ADFile:2nd millennium montage.pngFrom 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 seriesA 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 calculusCalculus 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 SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 GregoryJames 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 LeibnizGottfried 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

MadhavaMādhava of Sañgamāgrama was a prominent Kerala mathematicianastronomer 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 alKāshīGhiyāth alDīn Jamshīd Masʾūd alKāshī 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 CeulenLudolph 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 VièteFrançois 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 NewtonSir 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 MachinJohn 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 Machinlike 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 DaseJohann 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 Machinlike formula to calculate 200 decimals of {{pi}} in his head at the behest of GaussJohann 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 ShanksWilliam 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 LambertJohann 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 AdrienMarie LegendreAdrienMarie 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 EulerLeonhard 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 problemThe 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 numberA 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 transcendentalIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a nonconstant 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 LindemannCarl 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 NeumannJohn von Neumann was a HungarianAmerican 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 ENIACENIAC was the first generalpurpose electronic computer. It was a Turingcomplete 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 milliondigit 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 transformA 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 RamanujanSrīnivāsa 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 {{mathk}}! is the factorialIn mathematics, the factorial of a nonnegative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
of {{mathk}}.
A collection of some others are in the table below:
where

is the Pochhammer symbolIn mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a nonnegative 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 brothersThe Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their homebuilt 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 supercomputerA supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculationintensive 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 BrentRichard 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 {{matha}}_{{{mathn}}} and {{mathb}}_{{{mathn}}} 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 JonathanJonathan 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
. The methods have been used by Yasumasa Kanadais 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 T2KTsukuba 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 FEEIn 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 Efunction 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 FEEIn 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 Efunction 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. PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
, F. BellardFabrice 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 FEEIn 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 Efunction 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 formulaThe 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 PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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. BaileyDavid 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 BorweinPeter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a codiscoverer of the BaileyBorweinPlouffe algorithm for computing π.First interest in mathematics:...
, and Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 hexadecimalIn 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 binaryThe binary numeral system, or base2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 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 computingDistributed 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 PiHexPiHex 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 BellardFabrice 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 {{mathb}} and {{mathc}} are positive integers and {{mathp}} and {{mathp}} 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 {{mathb}}^{{{mathc}}} without computing all the preceding digits in that base, in a time just depending on the size of the integer {{mathk}} and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*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 {{mathk}}, and requiring modest computing resources. The previous formula (found by Plouffe for {{pi}} with {{mathb}} = 2 and {{mathc}} = 4, but also found for log(9/10) and for a few other irrational constants), implies that {{pi}} is a SC*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 neededdate=March 2011}}
In September 2010, 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 twoquadrillion^{th} (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 23day period. The formula is used to compute a single bit of {{pi}} in a small set of mathematical steps.
In 2006, Simon PlouffeSimon Plouffe is a Quebec mathematician born on June 11, 1956 in SaintJovite, 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 algorithmAn 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 {{mathq}} = {{mathIn 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 {{mathk}} is an odd number, and {{matha}}, {{mathb}}, {{mathc}} are rational numberIn 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 {{mathk}} is of the form 4{{mathm}} + 3, then the formula has the particularly simple form,
for some rational number {{mathp}} 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 fractionIn 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 fractionIn 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 rapidlyconverging expression:
Memorizing digits
{{MainPiphilology}}
Well before computers were used in calculating {{pi}}, memorizing a record number of digits had become an obsession for some people.
In 2006, Akira HaraguchiAkira 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 RecordsGuinness 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 Guinnessrecognized record for remembered digits of {{pi}} is 67,890 digits, held by Lu ChaoLu 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 24yearold graduate student from ChinaChinese 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 JeansSir 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 CadenzaCadaeic 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 techniquesA 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 piphilologyPiphilology 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 lociThe 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 numberIn 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.
BaileyDavid 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 CrandallRichard 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 formulaThe 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 conjectureA 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 theoryChaos 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 {{mathThe 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 independentIn 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 nontrivial polynomial equation with coefficients in K...
, although Yuri NesterenkoYuri 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}}, {{mathIn 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...
, {{mathIn 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
{{MainList of formulae involving π}}
{{pi}} is ubiquitous in mathematics, science, and engineering.
Geometry and trigonometry
{{See alsoArea of a disk}}
For any circle with radius {{mathr}} and diameter {{mathd}} = 2{{mathr}}, the circumference is {{pi}}{{mathd}} and the area is {{pi}}{{mathr}}^{2}. Further, {{pi}} appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipseIn 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, sphereA sphere is a perfectly round geometrical object in threedimensional 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, conesA cone is an ndimensional 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 toriIn 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 integralsIntegration 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 integralIntegration 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 circleIn 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 revolutionIn 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 unitcircle definition of the trigonometric functionIn 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 {{mathx}} and integers {{mathn}}, sin({{mathx}}) = sin({{mathx}} + 2{{pi}}{{mathn}}) and cos({{mathx}}) = cos({{mathx}} + 2{{pi}}{{mathn}}). Because sin(0) = 0, sin(2{{pi}}{{mathn}}) = 0 for all integers {{mathn}}. 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 {{mathx}} for which sin {{mathx}} = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, {{pi}} can be defined using the inverse trigonometric functionIn 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 numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional 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 analysisComplex 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 functionIn 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 formulaEuler'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 {{mathi}} is the imaginary unitIn 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 {{mathi}}^{2} = −1 and {{mathe}} ≈ 2.71828 is Euler's numberThe 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 {{mathe}} describe rotations on the unit circleIn 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 {{mathφ}} = {{pi}} results in the remarkable Euler's identity
There are {{mathn}} different {{mathn}}th roots of unityIn 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 integralThe Gaussian integral, also known as the EulerPoisson 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 functionIn 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 halfinteger is a rational multiple of √{{pi}}.
Physics
Although not a physical constantA 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 systemIn mathematics, a spherical coordinate system is a coordinate system for threedimensional 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 unitsIn 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
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{{mathx}}) and momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
(Δ{{mathp}}) can not both be arbitrarily small at the same time:


 Einstein's field equation
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 relativityGeneral 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
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 densityEnergy 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 vacuumIn 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...
{{mathρ}}_{vac} via the gravitational constantThe 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...
{{mathG}} as follows:

 Coulomb's law
Coulomb's law or Coulomb's inversesquare 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 forceIn physics, an electric field surrounds electrically charged particles and timevarying 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 chargeElectric 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 ({{mathq}}_{1} and {{mathq}}_{2}) separated by distance {{mathr}} (with {{mathε}}_{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
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 AndréMarie Ampère , French mathematician and physicist, considered the father of electrodynamics...
s (A):

 Kepler's third law constant, relating the 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...
({{mathP}}) and the semimajor axisThe 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...
({{matha}}) to the massMass 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 ({{mathM}} and {{mathm}}) of two coorbiting bodies:
Probability and statistics
In probabilityProbability 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 statisticsStatistics 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 distributionsIn 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
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 meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
{{mathμ}} and standard deviationStandard 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...
{{mathσ}}, due to the Gaussian integralThe Gaussian integral, also known as the EulerPoisson 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
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 {{mathf}}({{mathx}}), the above formulas can be used to produce other integral formulas for {{pi}}.
Buffon's needleIn mathematics, Buffon's needle problem is a question first posed in the 18th century by GeorgesLouis 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 {{mathL}} repeatedly on a surface containing parallel lines drawn {{mathS}} units apart (with {{mathS}} > {{mathL}}). If the needle is dropped {{mathn}} times and {{mathx}} of those times it comes to rest crossing a line ({{mathx}} > 0), then one may approximate {{pi}} using the Monte Carlo methodMonte 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 exponentiallyExponential 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 {{mathL}} and {{mathS}} will transfer directly to an error in the approximated {{pi}}. For example, a difference of a single atomThe 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 10centimeter 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 neededdate=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 straightline from source to mouth length tends to approach {{pi}}. Albert EinsteinAlbert Einstein was a Germanborn 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 shortcircuiting, creating an oxbow 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
{{MainTau (2π)}}
A proposed alternative to {{pi}}, usually represented by the Greek letter tau ({{mathτ}}), 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 angleA 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 {{mathτ}} 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}}", {{mathτ}} 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 nonmathematicians. Reports on the latest, mostprecise calculation of {{pi}} are common news items.
Nobel prizeSince 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 mouselike is the comet's tail!
Pi and its digital representation are often used by selfdescribed "math geekThe 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 technologicallyminded groups. Many schools around the world observe Pi DayPi 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 cheersCheering 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 TechnologyThe Georgia Institute of Technology is a public research university in Atlanta, Georgia, in the United States...
and the Massachusetts Institute of TechnologyThe 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 NortelNortel 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, GoogleGoogle Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internetbased 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, alternativeAlternative 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 BushKate Bush is an English singersongwriter, 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 AerialAerial is the eighth studio album by British singersongwriter 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 SaganCarl 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, coauthor or editor of more than 20 books...
's novel ContactContact 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 filmContact 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 SeriesStar 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" is the bestknown 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 EnterpriseThe USS Enterprise, NCC1701, 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, KirkJames 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 SpockSpock 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 SG1Stargate SG1 is a CanadianAmerican adventure and military science fiction television series and part of MetroGoldwynMayer'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 CarterSamantha "Sam" Carter is a fictional character in the CanadianAmerican military science fiction Stargate franchise, appearing in television series Stargate SG1, Stargate Atlantis, and Stargate Universe. SG1 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 SimpsonsThe 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" 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 FrinkProfessor 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 AronofskyDarren 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 liveaction and animation filmmaking...
's film PiPi, 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 theoristNumber 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 SmithsonianNight 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 AkhmanRa to open the gates to the underworld.
A style of writing called PilishPilish 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
 The 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
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 OnLine Encyclopedia of Integer SequencesThe OnLine 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 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 is an answerengine 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 is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, realtime 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 Cuttheknot
Cuttheknot is a free, advertisementfunded 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}}