Irrational number

# Irrational number

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an irrational number is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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

.

Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

s. As a consequence of Cantor's proof
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...

that the real numbers are uncountable (and the rationals countable) it follows that almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

real numbers are irrational.

## Examples

Perhaps the best-known irrational numbers are:
• π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

• e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

• Φ
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

• {{pp-move-indef}}
In
mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an irrational number is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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

.

Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

s. As a consequence of Cantor's proof
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...

that the real numbers are uncountable (and the rationals countable) it follows that almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

real numbers are irrational.

## Examples

Perhaps the best-known irrational numbers are:
• π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

• e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

• Φ
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

• {{pp-move-indef}}
In
mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an irrational number is any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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

.

Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...

s. As a consequence of Cantor's proof
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...

that the real numbers are uncountable (and the rationals countable) it follows that almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

real numbers are irrational.

## Examples

Perhaps the best-known irrational numbers are:
• π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

• e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

• Φ
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

• √{{overline
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

.

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

of lengths of two line segments is irrational, the line segments are also described as being incommensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...

, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as i.

## History

It has been suggested that the concept of irrationality was implicitly accepted by Indian mathematicians
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

since the 7th century BC, when Manava
Manava
Manava is an author of the Indian geometric text of Sulba Sutras.The Manava Sulbasutra is not the oldest , nor is it one of the most important, there being at least three Sulbasutras which are considered more important...

(c. 750–690 BC) believed that the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

s of numbers such as 2 and 61 could not be exactly determined, but such claims are not well substantiated and unlikely to be true.

### Ancient Greece

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean
Pythagoreanism
Pythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...

(possibly Hippasus of Metapontum
Hippasus
Hippasus of Metapontum in Magna Graecia, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers.-Life:...

), who probably discovered them while identifying sides of the pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...

.
The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse
Hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...

of an isosceles right triangle was indeed commensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...

with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:
• The ratio of the hypotenuse to an arm of an isosceles right triangle is c:b expressed in the smallest units possible.
• By the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

: c2 = a2+b2 = 2b2. (Since the triangle is isosceles, a = b.)
• Since c2 is even, c must be even.
• Since c:b is in its lowest terms, b must be odd (if it were also even, then both c and b would be divisible by 2, therefore not in lowest terms).
• Since c is even, let c = 2y.
• Then c2 = 4y2 = 2b2
• b2 = 2y2 so b2 must be even, therefore b is even.
• However we have already asserted b must be odd. Since b cannot be both odd and even, here is the contradiction.

Greek mathematicians
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. Brought into light by Zeno of Elea
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...

, he questioned the conception that quantities are discrete, and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for “whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects.” However Zeno found that in fact “[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear….[Q]uantities are, in other words, continuous.” What this means is that, contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure will come to zero, but it will never reach exactly zero. This is exactly what Zeno sought to prove. He sought to prove this by formulating four paradoxes
Zeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...

, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno’s paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.

The next step was taken by Eudoxus of Cnidus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...

, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude “was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.” Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. “Eudoxus’ theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.” Book 10 is dedicated to classification of irrational magnitudes.

As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometrical terms. This may account for why we still conceive of x2 or x3 as x squared and x cubed instead of x second power and x third power. Also crucial to Zeno’s work with incommensurable magnitudes was the fundamental focus on deductive reasoning which resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that comprised that theory. Out of this necessity Eudoxus developed his method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

, a kind of reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

which “established the deductive organization on the basis of explicit axioms…” as well as “…reinforced the earlier decision to rely on deductive reasoning for proof.” This method of exhaustion is said to be the first step in the creation of calculus.

Theodorus of Cyrene
Theodorus of Cyrene
Theodorus of Cyrene was a Greek mathematician of the 5th century BC. The only first-hand accounts of him that we have are in two of Plato's dialogues: the Theaetetus and the Sophist...

proved the irrationality of the surds
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

of whole numbers up to 17, but stopped there probably because the algebra he used couldn't be applied to the square root of 17.
It wasn't until Eudoxus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...

developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.

### Middle Ages

In the Middle ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...

, the development of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

by Muslim mathematicians allowed irrational numbers to be treated as "algebraic objects". Muslim mathematicians also merged the concepts of "number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

" and "magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

" into a more general idea of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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

s, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of the Elements, the Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

mathematician Al-Mahani
Al-Mahani
Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian mathematician and astronomer from Mahan, Kermān, Persia.A series of observations of lunar and solar eclipses and planetary conjunctions, made by him from 853 to 866, was in fact used by Ibn Yunus....

(d. 874/884) examined and classified quadratic irrational
In mathematics, a quadratic irrational is an irrational number that is the solution to some quadratic equation with rational coefficients...

s and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:

{{quote|"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc."}}

In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

al approach to the concept of irrationality, as he attributes the following to irrational magnitudes:

{{quote|"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."}}

The Egypt
Egypt
Egypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...

ian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s or as coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

s in an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

, often in the form of square roots, cube roots and fourth roots
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

. In the 10th century, the Iraq
Iraq
Iraq ; officially the Republic of Iraq is a country in Western Asia spanning most of the northwestern end of the Zagros mountain range, the eastern part of the Syrian Desert and the northern part of the Arabian Desert....

i mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions. Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

is:

{{quote|"contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of roots."}}

Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. Al-Hassār, a Moroccan mathematician from Fez
Fes
Fes or Fez is the second largest city of Morocco, after Casablanca, with a population of approximately 1 million . It is the capital of the Fès-Boulemane region....

specializing in Islamic inheritance jurisprudence
Islamic inheritance jurisprudence
Islamic Inheritance jurisprudence is a field of Islamic Jurisprudence that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called Mīrāth, and its branch of Islamic law is technically known as ʿulm al-farāʾiḍ...

during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, ." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.

During the 14th to 16th centuries, Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

and the Kerala school of astronomy and mathematics discovered the infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

for several irrational numbers such as π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

and certain irrational values of trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s. Jyesthadeva provided proofs for these infinite series in the Yuktibhāṣā
Yuktibhasa
Yuktibhāṣā also known as Gaṇitanyāyasaṅgraha , is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530...

.

### Modern period

The 17th century saw imaginary number
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

s become a powerful tool in the hands of Abraham de Moivre
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...

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

. The completion of the theory of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s in the nineteenth century entailed the differentiation of irrationals into algebraic and transcendental numbers, the proof of the existence of transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

s, and the resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

. The year 1872 saw the publication of the theories of Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

(by his pupil Ernst Kossak), Eduard Heine
Eduard Heine
Heinrich Eduard Heine was a German mathematician.Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions . He also investigated basic hypergeometric series...

(Crelle's Journal
Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...

, 74), Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

(Annalen, 5), and Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle
Salvatore Pincherle
Salvatore Pincherle was an Italian mathematician. He contributed significantly to the field of functional analysis, established the Italian Mathematical Union , and was president of the Third International Congress of Mathematicians...

in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery
Paul Tannery
Paul Tannery was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose Notions Mathématiques he contributed an historical chapter...

(1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt)
Dedekind cut
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....

in the system of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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

s into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

(Crelle, 101), and Charles Méray
Charles Méray
Hugues Charles Robert Méray was a French mathematician. He is noted as the first to publish an arithmetical theory of irrational numbers. His work did not have much of a role in the history of mathematics because France, at that time, was less interested in such matters than Germany.-References:...

.

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

s, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

Johann Heinrich Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

proved (1761) that π cannot be rational, and that en is irrational if n is rational (unless n = 0). While Lambert's proof is often said to be incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

(1794), after introducing the Bessel–Clifford function, provided a proof to show that π2 is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method
Cantor's first uncountability proof
Georg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. This proof differs from the more familiar proof that uses his diagonal argument...

, that showed that every interval in the reals contains transcendental numbers. Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

(1873) first proved e transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

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

(1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

and Paul Gordan.

### Square roots

The square root of 2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...

was the first number to be proved irrational{{citation needed|date=January 2011}} and that article contains a number of proofs. The golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

is the next most famous quadratic irrational and there is a simple proof of its irrationality in its article. The square roots of all numbers which are not perfect squares are irrational and a proof may be found in quadratic irrational
In mathematics, a quadratic irrational is an irrational number that is the solution to some quadratic equation with rational coefficients...

s.

The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, called an argument by reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

. The following argument appeals twice to the fact that the square of an odd integer is always odd.

If √2 is rational it has the form m/n for integers m, n not both even. Then m2 = 2n2, hence m is even, say m = 2p. Thus 4p2 = 2n2 so 2p2 = n2, hence n is also even, a contradiction.

### General roots

The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...

which was proved by Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

in 1798. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

in the denominator which does not divide into the numerator whatever power each is raised to. Therefore if an integer is not an exact kth power of another integer then its kth root is irrational.

### Logarithms

Perhaps the numbers most easily proved to be irrational are certain logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

s. Here is a proof by reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

that log2 3 is irrational. Notice that log2 3 ≈ 1.58 > 0.

Assume log2 3 is rational. For some positive integers m and n, we have

It follows that

However, the number 2 raised to any positive integer power must be even (because it will be divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...

s will be 2). Clearly, an integer can not be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log2 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.

Cases such as log10 2 can be treated similarly.

## Transcendental and algebraic irrationals

Almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

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

s are irrational: the article on transcendental numbers lists several examples. e r and π r are irrational if r ≠ 0 is rational; eπ is irrational.

Another way to construct irrational numbers is as irrational algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

s, i.e. as zeros of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

where the coefficients ai are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....

). The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

of a0 and s is a divisor of an; there are only finitely many such candidates which you can all check by hand. If neither of them is a root of p, then x must be irrational. For example, this technique can be used to show that x = (21/2 + 1)1/3 is irrational: we have (x3 − 1)2 = 2 and hence x6 − 2x3 − 1 = 0, and this latter polynomial does not have any rational roots (the only candidates to check are ±1).

Because the algebraic numbers form a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, many irrational numbers can be constructed by combining transcendental and algebraic numbers. For example 3π + 2, π + √2 and e3 are irrational (and even transcendental).

## Decimal expansions

The decimal expansion of an irrational number never repeats or terminates, unlike a rational number.

To show this, suppose we divide integers n by m (where m is nonzero). When long division
Long division
In arithmetic, long division is a standard procedure suitable for dividing simple or complex multidigit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a...

is applied to the division of n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats.

Conversely, suppose we are faced with a recurring decimal, we can prove that it is a fraction of two integers. For example:

Here the length of the repitend is 3. We multiply by 103:

Note that since we multiplied by 10 to the power of the length of the repeating part, we shifted the digits to the left of the decimal point by exactly that many positions. Therefore, the tail end of 1000A matches the tail end of A exactly. Here, both 1000A and A have repeating 162 at the end.

Therefore, when we subtract A from both sides, the tail end of 1000A cancels out of the tail end of A:

Then

(135 is the greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

of 7155 and 9990). Alternatively, since 0.5 = 1/2, one can clear fractions by multiplying the numerator and denominator by 2:

(27 is the greatest common divisor of 1431 and 1998).

53/74 is a quotient of integers and therefore a rational number.

## Irrational powers

Dov Jarden gave a simple non-constructive proof
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...

that there exist two irrational numbers a and b, such that ab is rational.

Indeed, if √22 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √22 and b = √2. Then ab = (√22)2 = √22·√2 = √22 =
2 which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...

implies that √22 is transcendental, hence irrational. This theorem states that all non-rational algebraic powers of algebraic numbers other than 0 or 1 are transcendental.

## Open questions

It is not known whether π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

+ e or π − e is irrational or not. In fact, there is no pair of non-zero integers m and n for which it is known whether mπ + ne is irrational or not. Moreover, it is not known whether the set {π, e} is algebraically independent
Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K...

over Q.

It is not known whether πe, π/e, 2e, πe, π2, ln π, Catalan's constant, or the Euler–Mascheroni gamma constant γ are irrational.

## The set of all irrationals

Since the reals form an uncountable
set, of which the rationals are a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

subset, the complementary set of
irrationals is uncountable.

Under the usual (Euclidean
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

) distance function d(xy) = |x − y|, the real numbers are a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

and hence also a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed,
the induced metric is not complete. However, being a G-delta set
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...

—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is topologically complete: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals have a basis of clopen set
Clopen set
In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...

s so the space is zero-dimensional.

• Dedekind cut
Dedekind cut
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....

• Proof that e is irrational
Proof that e is irrational
In mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...

• Proof that π is irrational
• Trigonometric number
Trigonometric number
In mathematics, a trigonometric number is an irrational number produced by taking the cosine of a rational multiple of a circle, or equivalently, the cosine in radians of a rational multiple of π, or the cosine of a rational number of degrees....

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

• nth root
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

• Square root of 3
Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property...

• Computable number
Computable number
In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...