View source for Irrational number

{{Short description|Number that is not a ratio of integers}}
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[[File:Square root of 2 triangle.svg|right|thumb|240px|The number [[Square root of 2|<math>\sqrt{2}</math>]] is irrational.]]

In [[mathematics]], the '''irrational numbers''' (''[[wikt:in-#Etymology_3|in-]]'' + ''rational'') are all the [[real number]]s that are not [[rational number]]s. That is, irrational numbers cannot be expressed as the ratio of two [[integer]]s. When the [[ratio]] of lengths of two line segments is an irrational number, the line segments are also described as being ''[[commensurability (mathematics)|incommensurable]]'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio [[Pi|{{pi}}]] of a circle's circumference to its diameter, Euler's number [[E (mathematical constant)|''e'']], the golden ratio [[Golden ratio|''φ'']], and the [[square root of two]].<ref>[http://sprott.physics.wisc.edu/Pickover/trans.html The 15 Most Famous Transcendental Numbers]. by [[Clifford A. Pickover]]. URL retrieved 24 October 2007.</ref>  In fact, all square roots of [[natural number]]s, other than of [[square number|perfect square]]s, are irrational.<ref>{{Cite journal |last=Jackson |first=Terence |date=2011-07-01 |title=95.42 Irrational square roots of natural numbers — a geometrical approach |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/9542-irrational-square-roots-of-natural-numbers-a-geometrical-approach/6B9D8EBFDCC016013D303AA78973429F |journal=The Mathematical Gazette |language=en |volume=95 |issue=533 |pages=327–330 |doi=10.1017/S0025557200003193 |s2cid=123995083 |issn=0025-5572}}</ref>

Like all real numbers, irrational numbers can be expressed in [[positional notation]], notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor [[repeating decimal|end with a repeating sequence]]. For example, the decimal representation of {{pi}} starts with 3.14159, but no finite number of digits can represent {{pi}} exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics.

Irrational numbers can also be expressed as [[Simple continued fraction#Infinite continued fractions and convergents|non-terminating continued fractions]] (which in some cases are [[Periodic_continued_fraction|periodic]]), and in many other ways.

As a consequence of [[Cantor's diagonal argument|Cantor's proof]] that the real numbers are [[uncountable]] and the [[rational number|rationals]] countable, it follows that [[almost all]] real numbers are irrational.<ref>{{Cite book|last=Cantor|first=Georg|author-link=Georg Cantor|year=1955|orig-year=1915|title=Contributions to the Founding of the Theory of Transfinite Numbers|url=https://archive.org/details/contributionstot003626mbp|editor=Philip Jourdain|editor-link=Philip Jourdain|place=New York|publisher=Dover|isbn= 978-0-486-60045-1 }}</ref>

== History ==
<!-- [[History of irrational numbers]] links here -->
[[File:Set of real numbers (diagram).svg|thumb|An [[Euler diagram]] showing the [[Set (mathematics)|set]] of real numbers (<math>\mathbb {R}</math>), which include the rationals (<math>\mathbb {Q}</math>), which include the integers (<math>\mathbb {Z}</math>), which include the natural numbers (<math>\mathbb {N}</math>). The real numbers also include the irrationals (<math>\mathbb {R}</math>\<math>\mathbb {Q}</math>).]]

=== Ancient Greece ===
The first proof of the existence of irrational numbers is usually attributed to a [[Pythagoreanism|Pythagorean]] (possibly [[Hippasus|Hippasus of Metapontum]]),<ref>{{cite journal|title=The Discovery of Incommensurability by Hippasus of Metapontum|author=Kurt Von Fritz|author-link=Kurt von Fritz|journal=[[Annals of Mathematics]]|year=1945|volume=46|issue= 2|pages= 242–264|doi=10.2307/1969021|jstor=1969021|s2cid=126296119}}</ref> who probably discovered them while identifying sides of the [[pentagram]].<ref>{{cite journal|title=The Pentagram and the Discovery of an Irrational Number|journal=The Two-Year College Mathematics Journal|author=James R. Choike|year=1980|volume=11|issue= 5|pages=312–316|doi=10.2307/3026893|jstor=3026893|s2cid=115390951}}</ref>
The 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. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the [[hypotenuse]] of an [[isosceles right triangle]] was indeed [[Commensurability (mathematics)|commensurable]] with a leg, then one of those lengths measured in 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'' (''a'' = ''b'' since it is isosceles). 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''<sup>2</sup> = ''a''<sup>2</sup>+''b''<sup>2</sup> = ''b''<sup>2</sup>+''b''<sup>2</sup> = 2''b''<sup>2</sup>. (Since the triangle is isosceles, ''a'' = ''b'').
* Since ''c''<sup>2</sup> = 2''b''<sup>2</sup>, ''c''<sup>2</sup> is divisible by 2, and therefore even.
* Since ''c''<sup>2</sup> is even, ''c'' must be even.
* Since ''c'' is even, dividing ''c'' by 2 yields an integer. Let ''y'' be this integer (''c'' = 2''y'').
* Squaring both sides of ''c'' = 2''y'' yields ''c''<sup>2</sup> = (2''y'')<sup>2</sup>, or ''c''<sup>2</sup> = 4''y''<sup>2</sup>.
* Substituting 4''y''<sup>2</sup> for ''c''<sup>2</sup> in the first equation (''c''<sup>2</sup> = 2''b''<sup>2</sup>) gives us 4''y''<sup>2</sup>= 2''b''<sup>2</sup>.
* Dividing by 2 yields 2''y''<sup>2</sup> = ''b''<sup>2</sup>.
* Since ''y'' is an integer, and 2''y''<sup>2</sup> = ''b''<sup>2</sup>, ''b''<sup>2</sup> is divisible by 2, and therefore even.
* Since ''b''<sup>2</sup> is even, ''b'' must be even.
* We have just shown that both ''b'' and ''c'' must be even.   Hence they have a common factor of 2.  However, this contradicts the assumption that they have no common factors.  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.<ref>[[Morris Kline|Kline, M.]] (1990). ''Mathematical Thought from Ancient to Modern Times'', Vol. 1. New York: Oxford University Press (original work published 1972), p. 33.</ref>

[[Greek mathematics|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.'<ref>Kline 1990, p. 32.</ref> 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 numbers 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. This was brought to 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",<ref name="Kline 1990, p. 34">Kline 1990, p. 34.</ref> but 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".<ref name="Kline 1990, p. 34"/> 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. In fact, these divisions of quantity must necessarily be [[Infinity|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 [[Zeno's paradoxes|four paradoxes]], which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary 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".<ref>Kline 1990, p. 48.</ref> 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".<ref>Kline 1990, p. 49.</ref> This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until [[Eudoxus of Cnidus|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.<ref>{{cite book|title=The historical development of the calculus|author=Charles H. Edwards|year=1982|publisher=Springer}}</ref>

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 numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of ''x''<sup>2</sup> and ''x''<sup>3</sup> as ''x'' squared and ''x'' cubed instead of ''x'' to the second power and ''x'' to the 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 underlie that theory. Out of this necessity, Eudoxus developed his [[method of exhaustion]], a kind of [[reductio ad absurdum]] that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof".<ref>Kline 1990, p. 50.</ref> This method of exhaustion is the first step in the creation of calculus.

[[Theodorus of Cyrene]] proved the irrationality of the [[Nth root|surds]] of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of 17.<ref>{{cite journal|title=Theodorus' Irrationality Proofs|author=Robert L. McCabe|journal=Mathematics Magazine|year=1976| volume=49|issue= 4|pages=201–203|doi=10.1080/0025570X.1976.11976579|jstor=2690123|s2cid=124565880}}.</ref>

===India===
Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the [[Vedic period]] in India. There are references to such calculations in the ''[[Samhita]]s'', ''[[Brahmana]]s'', and the ''[[Shulba Sutras]]'' (800 BC or earlier).<ref>{{cite journal|last=Bag|first=Amulya Kumar|title=Ritual Geometry in India and its Parallelism in other Culture Areas|journal=Indian Journal of History of Science|volume=25|date=1990|url=http://repository.ias.ac.in/74671/}}</ref>

It is suggested that the concept of irrationality was implicitly accepted by [[Indian mathematics|Indian mathematicians]] since the 7th century BC, when [[Manava]] (c. 750 &ndash; 690 BC) believed that the [[square root]]s of numbers such as 2 and 61 could not be exactly determined.<ref>T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411&ndash;2, in {{Cite book|title=Mathematics Across Cultures: The History of Non-western Mathematics|editor1-first=Helaine|editor1-last=Selin|editor1-link=Helaine Selin|editor2-first=Ubiratan|editor2-last=D'Ambrosio|year=2000|publisher=[[Springer Science+Business Media|Springer]]|isbn=1-4020-0260-2}}.</ref> Historian [[Carl Benjamin Boyer]], however, writes that "such claims are not well substantiated and unlikely to be true".<ref name="Boyer 208">{{cite book|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of Mathematics|edition=2nd|year=1991|chapter=China and India|quote=It has been claimed also that the first recognition of incommensurables appears in India during the ''Sulbasutra'' period, but such claims are not well substantiated. The case for early Hindu awareness of incommensurable magnitudes is rendered most unlikely by the lack of evidence that Indian mathematicians of that period had come to grips with fundamental concepts.|page=208|publisher=Wiley | isbn=0471093742| oclc=414892}}</ref>

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.<ref>{{cite journal| last1=Datta| first1=Bibhutibhusan| last2=Singh| first2=Awadhesh Narayan| title=Surds in Hindu mathematics| url=https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol28_3_2_BDatta.pdf| journal=Indian Journal of History of Science| volume=28| issue=3| pages=253–264| year=1993| access-date=18 September 2018| archive-url=https://web.archive.org/web/20181003062219/https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol28_3_2_BDatta.pdf| archive-date=2018-10-03| url-status=dead}}</ref>

Mathematicians like [[Brahmagupta]] (in 628 AD) and [[Bhāskara I]] (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th century [[Bhāskara II]] evaluated some of these formulas and critiqued them, identifying their limitations.

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 ''[[pi|π]]'' and certain irrational values of [[trigonometric functions]]. [[Jyeṣṭhadeva]] provided proofs for these infinite series in the ''[[Yuktibhāṣā]]''.<ref>{{Cite journal|last=Katz|first=V. J.|author-link=Victor J. Katz|date=1995|title=Ideas of Calculus in Islam and India|journal=[[Mathematics Magazine]]|volume=63|issue=3|pages=163–174|doi=10.2307/2691411|jstor=2691411}}</ref>

===Islamic World===
In the [[Middle Ages]], the development of [[algebra]] by [[Mathematics in medieval Islam|Muslim mathematicians]] allowed irrational numbers to be treated as ''algebraic objects''.<ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999|mode=cs1}}.</ref> Middle Eastern mathematicians also merged the concepts of "[[number]]" and "[[Magnitude (mathematics)|magnitude]]" into a more general idea of [[real number]]s, criticized Euclid's idea of [[ratio]]s, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.<ref name=Matvievskaya>{{Cite journal|last=Matvievskaya|first=Galina|author-link=Galina Matvievskaya|year=1987|title=The theory of quadratic irrationals in medieval Oriental mathematics|journal=[[New York Academy of Sciences|Annals of the New York Academy of Sciences]]|volume=500|issue=1|pages=253&ndash;277|doi=10.1111/j.1749-6632.1987.tb37206.x|bibcode=1987NYASA.500..253M|s2cid=121416910}} See in particular pp. 254 & 259–260.</ref> In his commentary on Book 10 of the ''Elements'', the [[Persian people|Persian]] mathematician [[Al-Mahani]] (d. 874/884) examined and classified [[quadratic irrational]]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:<ref name=Matvievskaya/>

{{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 root]]s as irrational magnitudes. He also introduced an [[arithmetic]]al approach to the concept of irrationality, as he attributes the following to irrational magnitudes:<ref name=Matvievskaya/>

{{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]]ian mathematician [[Abū Kāmil Shujā ibn Aslam]] (c. 850 &ndash; 930) was the first to accept irrational numbers as solutions to [[quadratic equation]]s or as [[coefficient]]s in an [[equation]] in the form of square roots and [[Nth root|fourth roots]].<ref>Jacques Sesiano, "Islamic mathematics", p. 148, in {{Cite book|title=Mathematics Across Cultures: The History of Non-western Mathematics|first1=Helaine|last1=Selin|first2=Ubiratan|last2=D'Ambrosio|year=2000|publisher=[[Springer Science+Business Media|Springer]]|isbn=1-4020-0260-2}}.</ref> In the 10th century, the [[Iraq]]i mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.<ref name=Matvievskaya/>

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 [[Fes|Fez]] specializing in [[Islamic inheritance jurisprudence]] during the 12th century, first mentions the use of a fractional bar, where [[numerator]]s 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, <math>\frac{3 \quad 1}{5 \quad 3}</math>."<ref>{{cite book|last=Cajori|first=Florian|title=A History of Mathematical Notations (Vol.1)|publisher=The Open Court Publishing Company|year=1928|place=La Salle, Illinois}} pg. 269.</ref> This same fractional notation appears soon after in the work of [[Leonardo Fibonacci]] in the 13th century.<ref>{{harv|Cajori|1928|loc=pg.89}}</ref>

===Modern period===
The 17th century saw [[imaginary number]]s become a powerful tool in the hands of [[Abraham de Moivre]], and especially of [[Leonhard Euler]]. The completion of the theory of [[complex number]]s in the 19th century entailed the differentiation of irrationals into algebraic and [[transcendental numbers]], the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored since [[Euclid]]. The year 1872 saw the publication of the theories of [[Karl Weierstrass]] (by his pupil Ernst Kossak), [[Eduard Heine]] (''[[Crelle's Journal]]'', 74), [[Georg Cantor]] (Annalen, 5), and [[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,<ref>{{cite journal |author = Salvatore Pincherle |title = Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principii del prof. C. Weierstrass |journal = Giornale di Matematiche |year = 1880 |pages=178–254, 317–320 |url=https://archive.org/stream/giornaledimatem07unkngoog#page/n198}}</ref> 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 [[Dedekind cut|cut (Schnitt)]] in the system of all [[rational number]]s, separating them 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]].

[[Continued fraction]]s, 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.

[[Johann Heinrich Lambert]] proved (1761) that π cannot be rational, and that ''e''<sup>''n''</sup> is irrational if ''n'' is rational (unless ''n''&nbsp;=&nbsp;0).<ref>{{cite journal |title = Mémoire sur quelques propriétés remarquables des quantités transcendentes, circulaires et logarithmiques |journal = Mémoires de l'Académie royale des sciences de Berlin |first = J. H.|last= Lambert|author-link=Johann Heinrich Lambert |year = 1761 |pages = 265–322 |url=http://www.kuttaka.org/~JHL/L1768b.pdf |language=fr| archive-url=https://web.archive.org/web/20160428083325/http://www.kuttaka.org/~JHL/L1768b.pdf| archive-date=2016-04-28| url-status=live}}</ref> While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. [[Adrien-Marie Legendre]] (1794), after introducing the [[Bessel&ndash;Clifford function]], provided a proof to show that π<sup>2</sup> is irrational,  whence it follows immediately that π is irrational also. The existence of [[transcendental number]]s was first established by Liouville (1844, 1851).  Later, [[Georg Cantor]] (1873) proved their existence by a [[Georg Cantor's first set theory article|different method]], which 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]]{{citation needed|date=January 2018}} and [[Paul Gordan]].<ref>{{cite journal |title = Transcendenz von ''e'' und π |first = Paul |last = Gordan |journal = [[Mathematische Annalen]] |volume = 43 |number = 2–3 |pages = 222–224 |year = 1893 |publisher = Teubner |doi=10.1007/bf01443647|s2cid = 123203471 |url = https://zenodo.org/record/1428218 }}</ref>

== Examples ==
{{More citations needed|date=May 2023}}

=== Square roots ===
The [[square root of 2]] was likely the first number proved irrational.<ref>{{citation
 | last = Fowler | first = David H.
 | issue = 10
 | journal = Neusis
 | mr = 1891736
 | pages = 45–61
 | title = The story of the discovery of incommensurability, revisited
 | year = 2001}}</ref> The [[golden ratio]] is another famous quadratic irrational number. The square roots of all natural numbers that are not [[perfect squares]] are irrational and a proof may be found in [[quadratic irrational]]s.

===General roots===
The proof for the irrationality of 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 number|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 {{mvar|k}}th power of another integer, then that first integer's [[nth root|{{mvar|k}}th root]] is irrational.

=== Logarithms ===
Perhaps the numbers most easy to prove irrational are certain [[logarithm]]s. Here is a [[proof by contradiction]] that log<sub>2</sub>&nbsp;3 is irrational (log<sub>2</sub>&nbsp;3 ≈&nbsp;1.58&nbsp;>&nbsp;0).

Assume log<sub>2</sub>&nbsp;3 is rational. For some positive integers ''m'' and ''n'', we have

: <math>\log_2 3 = \frac{m}{n}.</math>

It follows that

: <math>2^{m/n}=3</math>

: <math>(2^{m/n})^n = 3^n</math>

: <math>2^m=3^n.</math>

The number 2 raised to any positive integer power must be even (because it is divisible by&nbsp;2) and the number&nbsp;3 raised to any positive integer power must be odd (since none of its [[prime factor]]s will be&nbsp;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<sub>2</sub>&nbsp;3 is rational (and so expressible as a quotient of integers ''m''/''n'' with ''n''&nbsp;≠&nbsp;0). The contradiction means that this assumption must be false, i.e. log<sub>2</sub>&nbsp;3 is irrational, and can never be expressed as a quotient of integers ''m''/''n'' with ''n''&nbsp;≠&nbsp;0.

Cases such as log<sub>10</sub>&nbsp;2 can be treated similarly.

==Types==
An irrational number may be [[Algebraic number|algebraic]], that is a real [[zero of a function|root]] of a [[polynomial]] with integer coefficients. Those that are not algebraic are [[Transcendental number|transcendental]].

=== Algebraic ===
The real [[algebraic number]]s are the real solutions of polynomial equations

:<math>p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\;, </math>

where the coefficients <math>a_i</math> are integers and <math>a_n \ne 0</math>. An example of an irrational algebraic number is ''x''<sub>0</sub>&nbsp;=&nbsp;(2<sup>1/2</sup>&nbsp;+&nbsp;1)<sup>1/3</sup>. It is clearly algebraic since it is the root of an integer polynomial, <math> (x^3 - 1)^2= 2 </math>, which is equivalent to <math> (x^6 - 2x^3-1)= 0 </math>. This polynomial has no rational roots, since the [[rational root theorem]] shows that the only possibilities are&nbsp;±1, but ''x''<sub>0</sub> is greater than 1. So ''x''<sub>0</sub> is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials.

=== Transcendental ===
[[Almost all]] irrational numbers are [[Transcendental number|transcendental]]. Examples are ''e''<sup>&nbsp;''r''</sup> and π<sup>&nbsp;''r''</sup>, which are transcendental for all nonzero rational&nbsp;''r.''

Because the algebraic numbers form a [[field (mathematics)|subfield]] of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3{{pi}}&nbsp;+&nbsp;2, {{pi}}&nbsp;+&nbsp;{{radic|2}} and ''e''{{radic|3}} are irrational (and even transcendental).

==Decimal expansions==
The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there is not a finite number of nonzero digits), unlike any rational number. The same is true for [[Binary numeral system|binary]], [[octal]] or [[hexadecimal]] expansions, and in general for expansions in every [[Positional notation|positional]] [[numeral system|notation]] with [[natural number|natural]] bases.

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'', there can never be a [[remainder]] greater than or equal to ''m''. 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 [[repeating decimal]], we can prove that it is a fraction of two integers. For example, consider:

:<math>A=0.7\,162\,162\,162\,\ldots</math>

Here the repetend is 162 and the length of the repetend 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 repetend. In this example we would multiply by 10 to obtain:

:<math>10A = 7.162\,162\,162\,\ldots</math>

Now we multiply this equation by 10<sup>''r''</sup> where ''r'' is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10<sup>3</sup>:

:<math>10,000A=7\,162.162\,162\,\ldots</math>

The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000''A'' matches the tail end of 10''A'' exactly. Here, both 10,000''A'' and 10''A'' have {{gaps|.162|162|162|...}} after the decimal point.

Therefore, when we subtract the 10''A'' equation from the 10,000''A'' equation, the tail end of 10''A'' cancels out the tail end of 10,000''A'' leaving us with:

:<math>9990A=7155.</math>

Then

:<math>A= \frac{7155}{9990} = \frac{53}{74}</math>

is a ratio of integers and therefore a rational number.

== Irrational powers ==
Dov Jarden gave a simple non-[[constructive proof]] that there exist two irrational numbers ''a'' and ''b'', such that ''a''<sup>''b''</sup> is rational:<ref>{{Cite journal |last=Jarden |first=Dov |date=1953 |title=Curiosa No. 339: A simple proof that a power of an irrational number to an irrational exponent may be rational |url= |journal=Scripta Mathematica |volume=19 |pages=229 |doi=}} [https://queuea9.wordpress.com/2015/01/27/the-square-root-of-two-proof/ copy]</ref><ref>{{cite book |url=http://condor.depaul.edu/mash/atotheamg.pdf|title=Philosophies of mathematics| first1=Alexander |last1=George |first2=Daniel J. |last2=Velleman |isbn=0-631-19544-0 |publisher=Blackwell |year=2002 |pages=3–4}}</ref>

:Consider <math>\sqrt{2}</math><sup><math>\sqrt{2}</math></sup>; if this is rational, then take ''a'' = ''b'' = <math>\sqrt{2}</math>.  Otherwise, take ''a'' to be the irrational number <math>\sqrt{2}</math><sup><math>\sqrt{2}</math></sup> and ''b'' = <math>\sqrt{2}</math>.  Then ''a''<sup>''b''</sup> = (<math>\sqrt{2}</math><sup><math>\sqrt{2}</math></sup>)<sup><math>\sqrt{2}</math></sup> <math>=</math> <math>\sqrt{2}</math><sup><math>\sqrt{2}</math>·<math>\sqrt{2}</math></sup> <math>=</math> <math>\sqrt{2}</math><sup><math>2</math></sup> <math>=</math> <math>2</math>, which is rational.

Although the above argument does not decide between the two cases, the [[Gelfond–Schneider theorem]] shows that <math>\sqrt{2}</math><sup><math>\sqrt{2}</math></sup> is [[Transcendental number|transcendental]], hence irrational. This theorem states that if ''a'' and ''b'' are both [[algebraic number]]s, and ''a'' is not equal to 0 or 1, and ''b'' is not a rational number, then any value of ''a''<sup>''b''</sup> is a transcendental number (there can be more than one value if [[Exponentiation#nth roots of a complex number|complex number exponentiation]] is used).

An example that provides a simple constructive proof is<ref>Lord, Nick, "Maths bite: irrational powers of irrational numbers can be rational", ''Mathematical Gazette'' 92, November 2008, p. 534.</ref>

:<math>\left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3.</math>

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, <math>\log_{\sqrt{2}}3</math>, is irrational. This is so because, by the formula relating logarithms with different bases,

:<math>\log_{\sqrt{2}}3=\frac{\log_2 3}{\log_2 \sqrt{2}}=\frac{\log_2 3}{1/2} = 2\log_2 3</math>

which we can assume, for the sake of establishing a [[proof by contradiction|contradiction]], equals a ratio ''m/n'' of positive integers. Then <math>\log_2 3 = m/2n</math> hence <math>2^{\log_2 3}=2^{m/2n}</math> hence <math>3=2^{m/2n}</math> hence <math>3^{2n}=2^m</math>, which is a contradictory pair of prime factorizations and hence violates the [[fundamental theorem of arithmetic]] (unique prime factorization).

A stronger result is the following:<ref name=Marshall>Marshall, Ash J., and Tan, Yiren, "A rational number of the form ''a''<sup>''a''</sup> with ''a'' irrational", ''[[Mathematical Gazette]]'' 96, March 2012, pp. 106-109.</ref> Every rational number in the interval <math>((1/e)^{1/e}, \infty)</math> can be written either as ''a''<sup>''a''</sup> for some irrational number ''a'' or as ''n''<sup>''n''</sup> for some natural number ''n''. Similarly,<ref name=Marshall/> every positive rational number can be written either as <math>a^{a^a}</math> for some irrational number ''a'' or as <math>n^{n^n}</math> for some natural number ''n''.

== Open questions ==

* Various combinations of {{math|''e''}}, {{math|''π''}} and [[Elementary function|elementary functions]] (such as {{math|''e'' + ''π''}}'', {{mvar|eπ}}'', {{math|''e''<sup>''e''</sup>}}, {{math|''π''<sup>''e''</sup>}}, ''{{mvar|π}}<sup>{{mvar|π}}</sup>'', {{math|ln ''π''}}) are not known to be irrational, in part because {{math|''e''}} and {{math|''π''}} are not known to be [[Algebraic independence|algebraically independent]]. [[Schanuel's conjecture]] would imply that all of the above numbers are irrational and even transcendental.<ref name=":12">{{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}</ref>
* The question about the irrationality of [[Euler's constant]] ''{{mvar|γ}}'' is a long standing [[List of unsolved problems in mathematics|open problem]] in [[number theory]].<ref>{{Cite web |last=Waldschmidt |first=Michel |date=2023 |title=Some of the most famous open problems in number theory |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/OpenPbsNT.pdf}}</ref>
* Other important numbers which are not known to be irrational include [[Particular values of the Riemann zeta function#Odd positive integers|odd zeta constants]] {{math|''ζ''(5), ''ζ''(7), ''ζ''(9), ...}} for {{math|n > 3}} and [[Catalan's constant]] {{math|''β''(2)}}.<ref name=":4">{{Cite web |last=Waldschmidt |first=Michel |date=2022 |title=Transcendental Number Theory: recent results and open problems. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs |website=Michel Waldschmidt}}</ref>

== In constructive mathematics ==

In [[constructive mathematics]], [[excluded middle]] is not valid, so it is not true that every real number is rational or irrational. Thus, the notion of an irrational number bifurcates into multiple distinct notions. One could take the traditional definition of an irrational number as a real number that is not rational.<ref>{{cite book|title=Real Analysis: A Constructive Approach through Interval Arithmetic|author=Mark Bridger|publisher=[[John Wiley & Sons]]|year=2007|isbn=978-1-470-45144-8}}</ref> However, there is a second definition of an irrational number used in constructive mathematics, that a real number <math>r</math> is an irrational number if it is [[Apartness relation|apart]] from every rational number, or equivalently, if the distance <math>\vert r - q \vert</math> between <math>r</math> and every rational number <math>q</math> is positive. This definition is stronger than the traditional definition of an irrational number. This second definition is used in [[Errett Bishop]]'s [[Square root of 2#Constructive proof|proof that the square root of 2 is irrational]].<ref>{{cite book|title=Constructive Analysis|author1=Errett Bishop|author2=Douglas Bridges|year=1985|publisher=[[Springer Science+Business Media|Springer]]|isbn=0-387-15066-8}}</ref>

== Set of all irrationals ==
Since the reals form an [[uncountable]]
set, of which the rationals are a [[Countable set|countable]] subset, the complementary set of
irrationals is uncountable.

Under the usual ([[Euclidean distance|Euclidean]]) distance function <math>d(x, y) = \vert x - y \vert</math>, 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 (topology)|complete]].  Being a [[G-delta set]]&mdash;i.e., a countable intersection of open subsets&mdash;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 G-delta 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.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen groups so the space is [[Zero-dimensional space|zero-dimensional]].

==See also==
* [[Brjuno number]]
* [[Computable number]]
* [[Diophantine approximation]]
* [[Irrationality measure]]
* [[Proof that e is irrational|Proof that {{mvar|e}} is irrational]]
* [[Proof that &pi; is irrational|Proof that {{mvar|&pi;}} is irrational]]
* [[Square root of 3]]
* [[Square root of 5]]
* [[Trigonometric number]]
{{Classification of numbers}}

== References ==
{{Reflist|2}}

== Further reading ==
* [[Adrien-Marie 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 Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyter

== External links ==
{{Commons category|Irrational numbers}}
* [http://www.dm.uniba.it/~psiche/bas2/node5.html Zeno's Paradoxes and Incommensurability] {{Webarchive|url=https://web.archive.org/web/20160513063302/http://www.dm.uniba.it/~psiche/bas2/node5.html |date=2016-05-13 }} (n.d.). Retrieved April 1, 2008

{{Irrational number}}
{{Number systems}}
{{Authority control}}

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