The famous mathematical constant
pi (π) is among the best known irrational numbers and is much represented in popular culture
In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. 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 decimals. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.^{[1]}
When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.
Number which are irrational include the ratio of a circle's circumference to its diameter π, Euler's number e, the golden ratio φ, and the square root of two.^{[2]}^{[3]}^{[4]}
Contents

History 1

Ancient Greece 1.1

India 1.2

Middle Ages 1.3

Modern period 1.4

Example proofs 2

Square roots 2.1

General roots 2.2

Logarithms 2.3

Transcendental and algebraic irrationals 3

Decimal expansions 4

Irrational powers 5

Open questions 6

Set of all irrationals 7

See also 8

References 9

Further reading 10

External links 11
History
The number \scriptstyle\sqrt{2} is irrational.
It has been suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined.^{[5]} However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".^{[6]}
Ancient Greece
The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),^{[7]} who probably discovered them while identifying sides of the pentagram.^{[8]} The thencurrent 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 of an isosceles right triangle was indeed commensurable with a leg, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:


Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by c:b.

Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).

By the Pythagorean theorem: c^{2} = a^{2}+b^{2} = b^{2}+b^{2} = 2b^{2}. (Since the triangle is isosceles, a = b).

Since c^{2} = 2b^{2}, c^{2} is divisible by 2, and therefore even.

Since c^{2} is even, c must be even.

Since c and b have no common factors, and c is even, b must be odd (if b were even, b and c would have a common factor of 2).

Since c is even, dividing c by 2 yields an integer. Let y be this integer (c = 2y).

Squaring both sides of c = 2y yields c^{2} = (2y)^{2}, or c^{2} = 4y^{2}.

Substituting 4y^{2} for c^{2} in the first equation (c^{2} = 2b^{2}) gives us 4y^{2}= 2b^{2}.

Dividing by 2 yields 2y^{2} = b^{2}.

Since y is an integer, and 2y^{2} = b^{2}, b^{2} is divisible by 2, and therefore even.

Since b^{2} is even, b must be even.

However, we have already asserted that b must be odd, and b cannot be both odd and even. This contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers.^{[9]}
Greek mathematicians 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.”^{[10]} 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, who 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.”^{[11]} 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.”^{[11]} 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 comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes, 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, 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.”^{[12]} 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.”^{[13]} 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 x^{2} or x^{3} 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 that 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

Zeno's Paradoxes and Incommensurability (n.d.). Retrieved April 1, 2008

Weisstein, Eric W., "Irrational Number", MathWorld.

Square root of 2 is irrational
External links

AdrienMarie Legendre, Éléments de Géometrie, Note IV, (1802), Paris

Rolf Wallisser, "On Lambert's proof of the irrationality of π", in Algebraic Number Theory and Diophantine Analysis, Franz HalterKoch and Robert F. Tichy, (2000), Walter de Gruyer
Further reading

^ Cantor, Georg (1955) [1915].

^ The 15 Most Famous Transcendental Numbers. by Clifford A. Pickover. URL retrieved 24 October 2007.

^ http://www.mathsisfun.com/irrationalnumbers.html URL retrieved 24 October 2007.

^ Weisstein, Eric W., "Irrational Number", MathWorld. URL retrieved 26 October 2007.

^ T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, in .

^

^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.

^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The TwoYear College Mathematics Journal. .

^ Kline, M. (1990). Mathematical Thought from Ancient to Modern Times, Vol. 1. New York: Oxford University Press. (Original work published 1972). p.33.

^ Kline 1990, p. 32.

^ ^{a} ^{b} Kline 1990, p.34.

^ Kline 1990, p.48.

^ Kline 1990, p.49.

^ Kline 1990, p.50.

^ Robert L. McCabe (1976). "Theodorus' Irrationality Proofs". Mathematics Magazine. .

^ Charles H. Edwards (1982). The historical development of the calculus. Springer.

^ Katz, V. J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Mathematical Association of America) 68 (3): 163–74.

^ ..

^ Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". .

^ ^{a} ^{b} Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences 500: 253–277 [259].

^ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000). Mathematics Across Cultures: The History of Nonwestern Mathematics. .

^ Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences 500: 253–277 [260]. .

^ Matvievskaya, Galina (1987). "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics". Annals of the New York Academy of Sciences 500: 253–277 [261]. .

^ Cajori, Florian (1928), A History of Mathematical Notations (Vol.1), La Salle, Illinois: The Open Court Publishing Company pg. 269.

^ (Cajori 1928, pg.89)

^ Salvatore Pincherle (1880). "Saggio di una introduzione alla teorica delle funzioni analitiche secondo i principi del prof. Weierstrass". Giornale di Matematiche.

^ J. H. Lambert (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques". Histoire de l'Académie Royale des Sciences et des BellesLettres der Berlin: 265–276.

^ George, Alexander; Velleman, Daniel J. (2002). Philosophies of mathematics. Blackwell. pp. 3–4.

^ Lord, Nick, "Maths bite: irrational powers of irrational numbers can be rational", Mathematical Gazette 92, November 2008, p. 534.

^ ^{a} ^{b} Marshall, Ash J., and Tan, Yiren, "A rational number of the form a^{a} with a irrational", Mathematical Gazette 96, March 2012, pp. 106109.

^ Weisstein, Eric W., "Pi", MathWorld.

^ Weisstein, Eric W., "Irrational Number", MathWorld.

^ Some unsolved problems in number theory
References
See also
Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals have a basis of clopen sets so the space is zerodimensional.
Under the usual (Euclidean) distance function d(x, y) = x − y, the real numbers are a metric space and hence also a topological space. 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 Gdelta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is completely metrizable: 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 Gdelta sets: the continued fraction 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.
Since the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable.
Set of all irrationals
It is not known whether πe, π/e, 2^{e}, e^{e}, e^{ee}, π^{e}, π^{√2}, ln π, Catalan's constant, or the Euler–Mascheroni gamma constant γ are irrational.^{[31]}^{[32]}^{[33]} It is not known if ^{n}π or ^{n}e is rational for any positive integer n.
It is not known whether π + e or π − e is irrational or not. In fact, there is no pair of nonzero 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 over Q.
Open questions
A stronger result is the following:^{[30]} Every rational number in the interval ((1/e)^{1/e}, \infty) can be written either as a^{a} for some irrational number a or as n^{n} for some natural number n. Similarly,^{[30]} every positive rational number can be written either as a^{a^a} for some irrational number a or as n^{n^n} for some natural number n.
which we can assume, for the sake of establishing a contradiction, equals a ratio m/n of positive integers. Then \log_2 3 = m/2n hence 2^{\log_2 3}=2^{m/2n} hence 3=2^{m/2n} hence 3^{2n}=2^m, which is a contradictory pair of prime factorizations and hence violates the fundamental theorem of arithmetic (unique prime factorization).

\log_{\sqrt{2}}3=\frac{\log_2 3}{\log_2 \sqrt{2}}=\frac{\log_2 3}{1/2} = 2\log_2 3
The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, \log_{\sqrt{2}}3, is irrational. This is so because, by the formula relating logarithms with different bases,

\left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3.
An example that provides a simple constructive proof is^{[29]}
Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √2^{√2} is transcendental, hence irrational. This theorem states that if a and b are both algebraic numbers, and a is not equal to 0 or 1, and b is not a rational number, then any value of a^{b} is a transcendental number (there can be more than one value if complex number exponentiation is used).
Indeed, if √2^{√2} is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2^{√2} and b = √2. Then a^{b} = (√2^{√2})^{√2} = √2^{√2·√2} = √2^{2} = 2, which is rational.
Dov Jarden gave a simple nonconstructive proof that there exist two irrational numbers a and b, such that a^{b} is rational.^{[28]}
Irrational powers
(135 is the greatest common divisor of 7155 and 9990). 53/74 is a quotient of integers and therefore a rational number.

A= \frac{7155}{9990} = \frac{135 \times 53}{135 \times 74} = \frac{53}{74},
Then

9990A=7155.
Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:
The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A and 10A have .162162162 ... at the end.

10,000A=7\,162.162\,162\,\cdots .
Now we multiply this equation by 10^{r} where r is the length of the repitend. This has the effect of moving the decimal point to be in front of the "next" repitend. In our example, multiply by 10^{3}:

10A = 7.162\,162\,162\,\cdots .
Here the repitend is 162 and the length of the repitend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repitend. In this example we would multiply by 10 to obtain:

A=0.7\,162\,162\,162\,\cdots .
Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:
To show this, suppose we divide integers n by m (where m is nonzero). When long division 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.
The decimal expansion of an irrational number never repeats or terminates, unlike a rational number. Similarly for binary, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.
Decimal expansions
Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. For example 3π + 2, π + √2 and e√3 are irrational (and even transcendental).
where the coefficients a_{i} are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and a_{n} is nonzero, then because of the intermediate value theorem). The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor of a_{0} and s is a divisor of a_{n}; there are only finitely so many such candidates you can 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 = (2^{1/2} + 1)^{1/3} is irrational: we have (x^{3} − 1)^{2} = 2 and hence x^{6} − 2x^{3} − 1 = 0, and this latter polynomial does not have any rational roots (the only candidates to check are ±1).

p(x) = a_nx^n + a_{n1}x^{n1} + \cdots + a_1x + a_0 = 0 \,
Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials with integer coefficients: start with a polynomial equation
Almost all irrational numbers are transcendental and all real transcendental numbers are irrational (there are also complex transcendental numbers): the article on transcendental numbers lists several examples. e^{ r} and π^{ r} are irrational if r ≠ 0 is rational; e^{π} is irrational.
Transcendental and algebraic irrationals
Cases such as log_{10} 2 can be treated similarly.
However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log_{2} 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. log_{2} 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.

2^m=3^n.\,

(2^{m/n})^n = 3^n\,

2^{m/n}=3\,
It follows that

\log_2 3 = \frac{m}{n}.
Assume log_{2} 3 is rational. For some positive integers m and n, we have
Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction (reductio ad absurdum) that log_{2} 3 is irrational. Notice that log_{2} 3 ≈ 1.58 > 0.
Logarithms
The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic. 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 in the denominator that does not divide into the numerator whatever power each is raised to. Therefore if an integer is not an exact k^{th} power of another integer then its k^{th} root is irrational.
General roots
The square root of 2 was the first number proved irrational, and that article contains a number of proofs. The golden ratio is another famous quadratic irrational and there is a simple proof of its irrationality in its article. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.
Square roots
Example proofs
different method, that showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (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 (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan.
Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of Joseph Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
The 17th century saw Richard Dedekind. 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 in 1880,^{[26]} and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray.
Modern period
Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century. AlHassār, a Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence 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 threefifths and a third of a fifth, write thus, \frac{3 \quad 1}{5 \quad 3}." ^{[24]} This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.^{[25]}
"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."
The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.^{[21]} In the 10th century, the Iraqi mathematician AlHashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.^{[22]} Iranian mathematician, Abū Ja'far alKhāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite quantity is:^{[23]}
"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."
In contrast to Euclid's concept of magnitudes as lines, AlMahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:^{[20]}
"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 the Middle ages, the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects.^{[18]} Middle Eastern mathematicians also merged the concepts of "number" and "magnitude" into a more general idea of real numbers, criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.^{[19]} In his commentary on Book 10 of the Elements, the Persian mathematician AlMahani (d. 874/884) examined and classified quadratic irrationals 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:^{[20]}
Middle Ages
During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions. Jyesthadeva provided proofs for these infinite series in the Yuktibhāṣā.^{[17]}
Mathematicians like Brahmagupta (in 628 AD) and Bhaskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century Bhaskara II evaluated some of these formulas and critiqued them, identifying their limitations.
Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots. (See Datta, Singh, Indian Journal of History of Science, 28(3), 1993).
It is suggested that Aryabhata (5th century AD) in calculating a value of pi to 5 significant figures, he used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational).
Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India and there are references to such calculations in the Samhitas, Brahmanas and more notably in the Sulbha sutras (800 BC or earlier). (See Bag, Indian Journal of History of Science, 25(14), 1990).
India
Theodorus of Cyrene proved the irrationality of the surds 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.^{[15]} It wasn't until Eudoxus 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.^{[16]}
This method of exhaustion is the first step in the creation of calculus.
[14]
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.