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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the irrational numbers (from in-
prefix A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. Particul ...
assimilated to ir- (negative prefix,
privative A privative, named from Latin '' privare'', "to deprive", is a particle that negates or inverts the value of the stem of the word. In Indo-European languages many privatives are prefixes; but they can also be suffixes, or more independent elemen ...
) + rational) are all the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s that are not
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. That is, irrational numbers cannot be expressed as the ratio of two
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. When the
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of lengths of two line segments is an irrational number, the line segments are also described as being '' 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 of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which th ...
, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of starts with 3.14159, but no finite number of digits can represent 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 non-terminating continued fractions and many other ways. As a consequence of Cantor's proof that the real numbers are
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
and the rationals countable, it follows that
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
real numbers are irrational.


History


Ancient Greece

The first proof of the existence of irrational numbers is usually attributed to a
Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...
(possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aro ...
. 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. Hippasus, in the 5th century BC, however, 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 In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
of an
isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
was indeed 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''. 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
: ''c''2 = ''a''2+''b''2 = ''b''2+''b''2 = 2''b''2. (Since the triangle is isosceles, ''a'' = ''b''). * Since ''c''2 = 2''b''2, ''c''2 is divisible by 2, and therefore even. * Since ''c''2 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''2 = (2''y'')2, or ''c''2 = 4''y''2. * Substituting 4''y''2 for ''c''2 in the first equation (''c''2 = 2''b''2) gives us 4''y''2= 2''b''2. * Dividing by 2 yields 2''y''2 = ''b''2. * Since ''y'' is an integer, and 2''y''2 = ''b''2, ''b''2 is divisible by 2, and therefore even. * Since ''b''2 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. 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." 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. This was brought into light by
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
, 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",Kline 1990, p. 34. but Zeno found that in fact " uantitiesin general are not discrete collections of units; this is why ratios of incommensurable uantitiesappear... . antities 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 Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
. 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 Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...
, 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". This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not 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. 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 geometric terms. This may account for why we still conceive of ''x''2 and ''x''3 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 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 ...
, a kind of
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
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". This method of exhaustion is the first step in the creation of calculus. Theodorus of Cyrene proved the irrationality of the 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.


India

Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the
Vedic period The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betw ...
in India. There are references to such calculations in the ''
Samhita Saṃhitā literally means "put together, joined, union", a "collection", and "a methodically, rule-based combination of text or verses".Brahmana The Brahmanas (; Sanskrit: , ''Brāhmaṇam'') are Vedic śruti works attached to the Samhitas (hymns and mantras) of the Rig, Sama, Yajur, and Atharva Vedas. They are a secondary layer or classification of Sanskrit texts embedded within ...
s'', and the ''
Shulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. Purpose and origins The ...
'' (800 BC or earlier). (See Bag, Indian Journal of History of Science, 25(1-4), 1990). It is suggested that the concept of irrationality was implicitly accepted by
Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...
since the 7th century BC, when
Manava Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of ''Sulba Sutras.'' The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasut ...
(c. 750 – 690 BC) believed that the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s of numbers such as 2 and 61 could not be exactly determined. Historian Carl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true". It is also suggested that
Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (whic ...
(5th century AD), in calculating a value of pi to 5 significant figures, used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). 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. Mathematicians like
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tre ...
(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 Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shirom ...
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 The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta ...
discovered the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
for several irrational numbers such as '' π'' and certain irrational values of
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
. Jyeṣṭhadeva provided proofs for these infinite series in the '' Yuktibhāṣā''.Katz, V. J. (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' (Mathematical Association of America) 68 (3): 163–74.


Middle Ages

In the
Middle ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
, the development of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
by Muslim mathematicians allowed irrational numbers to be treated as ''algebraic objects''. Middle Eastern mathematicians also merged the concepts of "
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
" and " magnitude" into a more general idea of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, criticized Euclid's idea of
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. See in particular pp. 254 & 259–260. In his commentary on Book 10 of the ''Elements'', the
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...
mathematician Al-Mahani (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: 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 In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
s as irrational magnitudes. He also introduced an
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
al approach to the concept of irrationality, as he attributes the following to irrational magnitudes: The
Egypt Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Medit ...
ian mathematician
Abū Kāmil Shujā ibn Aslam Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, ar, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as ''Al-ḥāsib al-miṣrī''—lit. "the Egyptian reckoner") (c. 850 – ...
(c. 850 – 930) was the first to accept irrational numbers as solutions to
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...
s or as
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s in an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, often in the form of square roots, cube roots and fourth roots. In the 10th century, the
Iraq Iraq,; ku, عێراق, translit=Êraq officially the Republic of Iraq, '; ku, کۆماری عێراق, translit=Komarî Êraq is a country in Western Asia. It is bordered by Turkey to the north, Iran to the east, the Persian Gulf a ...
i mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions. Iranian mathematician, Abū Ja'far al-Khāzin (900–971) provides a definition of rational and irrational magnitudes, stating that if a definite
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
is: Many of these concepts were eventually accepted by European mathematicians sometime after the
Latin translations of the 12th century Latin translations of the 12th century were spurred by a major search by European scholars for new learning unavailable in western Europe at the time; their search led them to areas of southern Europe, particularly in central Spain and Sicily, ...
. Al-Hassār, a Moroccan mathematician from
Fez Fez most often refers to: * Fez (hat), a type of felt hat commonly worn in the Ottoman Empire * Fez, Morocco (or Fes), the second largest city of Morocco Fez or FEZ may also refer to: Media * ''Fez'' (Frank Stella), a 1964 painting by the moder ...
specializing in
Islamic inheritance jurisprudence Islamic Inheritance jurisprudence is a field of Islamic jurisprudence ( ar, فقه) 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 ...
during the 12th century, first mentions the use of a fractional bar, where
numerator A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
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, \frac." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.


Modern period

The 17th century saw
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...
s become a powerful tool in the hands of Abraham de Moivre, and especially of
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
. The completion of the theory of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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 Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
. The year 1872 saw the publication of the theories of
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
(by his pupil Ernst Kossak),
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) 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 Le ...
('' Crelle's Journal'', 74),
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
(Annalen, 5), and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
. 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, 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 all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, separating them into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass,
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
(Crelle, 101), and Charles Méray.
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 intege ...
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 Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJohann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subje ...
proved (1761) that π cannot be rational, and that ''e''''n'' is irrational if ''n'' is rational (unless ''n'' = 0). 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 Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are nam ...
(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 number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classe ...
s was first established by Liouville (1844, 1851). Later,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
(1873) proved their existence by a different method, which showed that every interval in the reals contains transcendental numbers.
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
(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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
(1893), and was finally made elementary by
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
and Paul Gordan.


Examples


Square roots

The
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the prin ...
was likely the first number proved irrational. The
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ...
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 irrationals.


General roots

The proof above for the square root of two can be generalized using the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...
. 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact th power of another integer, then that first integer's th root is irrational.


Logarithms

Perhaps the numbers most easy to prove irrational are certain
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s. Here is a
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known ...
that log2 3 is irrational (log2 3 ≈ 1.58 > 0). Assume log2 3 is rational. For some positive integers ''m'' and ''n'', we have : \log_2 3 = \frac. It follows that : 2^=3 : (2^)^n = 3^n : 2^m=3^n. 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 factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s 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 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.


Types

* number theoretic distinction : transcendental/algebraic * normal/ abnormal (non-normal)


Transcendental/algebraic

Almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
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. So ''e'' ''r'' and π ''r'' are irrational for all nonzero rational ''r'', and, e.g., ''e''π is irrational, too. Irrational numbers can also be found within the
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
set of real
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the ...
s (essentially defined as the real
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s with integer coefficients), i.e., as real solutions of polynomial equations :p(x) = a_nx^n + a_x^ + \cdots + a_1x + a_0 = 0\;, where the coefficients a_i are integers and a_n \ne 0. Any rational root of this polynomial equation must be of the form ''r'' /''s'', where ''r'' is a
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of ''a''0 and ''s'' is a divisor of ''a''''n''. If a real root x_0 of a polynomial p is not among these finitely many possibilities, it must be an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that ''x''0  = (21/2 + 1)1/3 is an irrational root of a polynomial with integer coefficients: it satisfies (''x''3 − 1)2 = 2 and hence ''x''6 − 2''x''3 − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and ''x''0, being greater than 1, is neither of these), so ''x''0 is an irrational algebraic number. Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 + 2,  +  and ''e'' are irrational (and even transcendental).


Decimal expansions

The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), unlike any rational number. The same is true for
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...
,
octal The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 numbe ...
or
hexadecimal In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
expansions, and in general for expansions in every positional
notation In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefor ...
with
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
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 In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In alge ...
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 A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if ...
, we can prove that it is a fraction of two integers. For example, consider: :A=0.7\,162\,162\,162\,\ldots 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: :10A = 7.162\,162\,162\,\ldots Now we multiply this equation by 10''r'' 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 103: :10,000A=7\,162.162\,162\,\ldots 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 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: :9990A=7155. Then :A= \frac = \frac 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''''b'' is rational: Consider ; if this is rational, then take ''a'' = ''b'' = . Otherwise, take ''a'' to be the irrational number and ''b'' = . Then ''a''''b'' = () = · = 2 = 2, which is rational. Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that is transcendental, hence irrational. This theorem states that if ''a'' and ''b'' are both
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the ...
s, 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). An example that provides a simple constructive proof is :\left(\sqrt\right)^=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_3, is irrational. This is so because, by the formula relating logarithms with different bases, :\log_3=\frac=\frac = 2\log_2 3 which we can assume, for the sake of establishing a
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
, equals a ratio ''m/n'' of positive integers. Then \log_2 3 = m/2n hence 2^=2^ hence 3=2^ hence 3^=2^m, which is a contradictory pair of prime factorizations and hence violates the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...
(unique prime factorization). A stronger result is the following:Marshall, Ash J., and Tan, Yiren, "A rational number of the form ''a''''a'' with ''a'' irrational", ''
Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
'' 96, March 2012, pp. 106-109.
Every rational number in the interval ((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, every positive rational number can be written either as a^ for some irrational number ''a'' or as n^ for some natural number ''n''.


Open questions

It is not known if \pi+e (or \pi-e) is irrational. In fact, there is no pair of non-zero integers m, n for which it is known whether m\pi+ n e is irrational. Moreover, it is not known if the set \ is algebraically independent over \Q. It is not known if \pi e,\ \pi/e,\ 2^e,\ \pi^e,\ \pi^\sqrt,\ \ln\pi, Catalan's constant, or the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natura ...
\gamma are irrational. It is not known if either of the
tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as re ...
s ^n\pi or ^n e is rational for some integer n > 1.


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. However, there is a second definition of an irrational number used in constructive mathematics, that a real number r is an irrational number if it is
apart Apart may refer to * "Apart" (The Cure song), by The Cure on their 1992 album ''Wish'' * ''Apart'' (album) (1995), by Paul Schütze * "Apart" (Brandy song), by Brandy on her 2002 album ''Full Moon'' * ''Apart'' (film), 2011 American drama * ' ...
from every rational number, or equivalently, if the distance \vert r - q \vert between r and every rational number q is positive. This definition is stronger than the traditional definition of an irrational number. This second definition is used in Errett Bishop's proof that the square root of 2 is irrational.


Set of all irrationals

Since the reals form an
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
set, of which the rationals are a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
subset, the complementary set of irrationals is uncountable. Under the usual ( Euclidean) distance function d(x, y) = \vert x - y \vert, the real numbers are a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
and hence also a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
. 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. Being a G-delta 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 G-delta sets: the
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 intege ...
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 sets so the space is zero-dimensional.


See also

* Brjuno number * Computable number *
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
* Proof that is irrational * Proof that is irrational *
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 denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...
*
Square root of 5 The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This numbe ...
* Trigonometric number


References


Further reading

*
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are nam ...
, ''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 Gruyer


External links


Zeno's Paradoxes and Incommensurability
(n.d.). Retrieved April 1, 2008 * {{DEFAULTSORT:Irrational Number Articles containing proofs Sets of real numbers