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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the irrational numbers (from in-
prefix A prefix is an affix In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) ...
assimilated to ir- (negative prefix,
privative A privative, named from Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be ...
) + rational) are all the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s which are not
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s. That is, irrational numbers cannot be expressed as the ratio of two
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. When the
ratio In mathematics, a ratio indicates 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 ...

ratio
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''
''φ''
, and the
square root of two The square root of 2, or the one-half power of 2, written in mathematics as \sqrt or 2^, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the principal square root of 2, to di ...
. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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 Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

uncountable
and the rationals countable, it follows that
almost all In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
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 Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philos ...
(possibly
Hippasus of Metapontum Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greeks, Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is some ...
), who probably discovered them while identifying sides of the
pentagram A pentagram (sometimes known as a pentalpha, pentangle, pentacle A pentacle (also spelled and pronounced as ''pantacle'' in Thelema, following Aleister Crowley, though that spelling ultimately derived from Éliphas Lévi) "The Pantacle of Fra ...

pentagram
. The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the
hypotenuse In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypotenuse
of an
isosceles right triangle A special right triangle is a right triangle A right triangle (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language nat ...

isosceles right triangle
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

Pythagorean theorem
: ''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
, 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 (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

infinite
. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur. The next step was taken by
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from ...
, 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 Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
, a kind of
reductio ad absurdum In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents sta ...
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 Theodorus of Cyrene Cyrene may refer to: Antiquity * Cyrene (mythology), an ancient Greek mythological figure * Cyrene, Libya, an ancient Greek colony in North Africa (modern Libya) ** Crete and Cyrenaica, a province of the Roman Empire ** Cyrena ...
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 in India. There are references to such calculations in the ''
Samhita Samhita literally means "put together, joined, union", a "collection", and "a methodically, rule-based combination of text or verses".Brahmana The Brahmanas (; Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language of South Asia that belongs to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South As ...
s'', and the ''
Shulba Sutras The ''Shulba Sutras'' or ''Śulbasūtras'' (Sanskrit Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch o ...
'' (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 The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
since the 7th century BC, when
Manava Manava (c. 750 BC – 690 BC) is an author of the text of ''.'' The Manava is not the oldest (the one by is older), nor is it one of the most important, there being at least three Sulbasutras which are considered more important. Histori ...
(c. 750 – 690 BC) believed that the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s of numbers such as 2 and 61 could not be exactly determined. However, historian
Carl Benjamin Boyer Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the " Gibbon of math Mathematics (from Greek: ) includes the study of suc ...
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 the first of the major mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
(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 Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

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 Bhāskara (c. 1114–1185) also known as Bhāskarācārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian people, Indian Indian mathematicians, mathematician and astronomer. He was born in Bij ...
evaluated some of these formulas and critiqued them, identifying their limitations. During the 14th to 16th centuries,
Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma () was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...
and the
Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
discovered the
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
for several irrational numbers such as '''' and certain irrational values of
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric functions
.
Jyeṣṭhadeva Jyeṣṭhadeva (Malayalam Malayalam (; , ) is a Dravidian languages, Dravidian language spoken in the Indian state of Kerala and the union territories of Lakshadweep and Puducherry (union territory), Puducherry (Mahé district) by the Mala ...
provided proofs for these infinite series in the ''
Yuktibhāṣā ''Yuktibhāṣā'' ( ml, യുക്തിഭാഷ, lit=Rationale), also known as (''Compendium of Astronomical Rationale''), is a major treatise A treatise is a formal and systematic written discourse on some subject, generally longer and ...
''.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 history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of ...
, the development of
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
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 A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
" and "
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
" into a more general idea of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, criticized Euclid's idea of
ratio In mathematics, a ratio indicates 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 ...

ratio
s, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of the ''Elements'', the
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, Persian people, the majority ethnic group in Iran, not to be conflated with the Iranian peoples ** Persian language, an Iranian ...

Persian
mathematician
Al-Mahani Abu-Abdullah Muhammad ibn Īsa Māhānī (, flourished c. 860 and died c. 880) was a Persian mathematician and astronomer born in Mahan, (in today Kermān, Iran Iran ( fa, ایران ), also called Persia and officially the Islamic Rep ...
(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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
s as irrational magnitudes. He also introduced an
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
al approach to the concept of irrationality, as he attributes the following to irrational magnitudes: The
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country This is a list of countries located on more than one continent A continent is one of several large landmasses. Generally identi ...

Egypt
ian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to
quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

quadratic equation
s or as
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s in an
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
, often in the form of square roots, cube roots and fourth roots. In the 10th century, the
Iraq Iraq ( ar, الْعِرَاق, translit=al-ʿIrāq; ku, عێراق, translit=Êraq), officially the Republic of Iraq ( ar, جُمْهُورِيَّة ٱلْعِرَاق '; ku, کۆماری عێراق, translit=Komarî Êraq), is a country i ...

Iraq
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 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 in terms of a unit of measu ...
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) The fez (, ), also called tarboosh ( ar, طربوش, translit=ṭarbūš, derived from fa, سرپوش, translit=sarpuš, lit=cap), is a felt headdress in the shape of a short cylindrical peakless hat, usuall ...

Fez
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 technicall ...
during the 12th century, first mentions the use of a fractional bar, where
numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
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 Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

Leonardo Fibonacci
in the 13th century.


Modern period

The 17th century saw
imaginary number An imaginary number is a complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
s become a powerful tool in the hands of
Abraham de Moivre Abraham de Moivre (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex number In mathematics, a complex number is a number that can be expressed in the form , where and are r ...

Abraham de Moivre
, and especially of
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
. The completion of the theory of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s in the 19th century entailed the differentiation of irrationals into algebraic and
transcendental numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
. 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 mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

Karl Weierstrass
(by his pupil Ernst Kossak),
Eduard Heine Heinrich Eduard Heine (16 March 1821 – October 1881) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of German ...
(''
Crelle's Journal ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augu ...
'', 74),
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
(Annalen, 5), and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
. 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 in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by
Paul TanneryImage:Paul Tannery.jpg, Paul Tannery Paul Tannery (20 December 1843 – 27 November 1904) was a France, French Mathematics, mathematician and History of Mathematics, historian of mathematics. He was the older brother of mathematician Jules Tannery, t ...

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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Leopold Kronecker
(Crelle, 101), and Charles Méray.
Continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 Swiss Swiss may refer to: * the adjectival form of Switzerland , french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Sv ...
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 named a ...
(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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s was first established by Liouville (1844, 1851). Later,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
(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 Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently ...
(1873) first proved ''e'' transcendental, and
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
(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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
(1893), and was finally made elementary by
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Adolf Hurwitz
and
Paul Gordan __NOTOC__ Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Gustav Jacob Jacobi, Carl Jacobi at the University of Königsberg before obtaining his Ph.D. at the University of Breslau ...
.


Examples


Square roots

The
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
was the first number proved irrational, and that article contains a number of proofs. The
golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

golden ratio
is another famous quadratic irrational number. The square roots of all natural numbers which 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 number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
. This asserts that every integer has a
unique factorization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English ...
there must be a
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
s. Here is a
proof by contradiction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
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. However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
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 irrational numbers are Transcendental number, 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 set, countable set of real algebraic numbers (essentially defined as the real zero of a function, roots of polynomials 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. rational root theorem, Any rational root of this polynomial equation must be of the form ''r'' /''s'', where ''r'' is a divisor 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 field (mathematics), 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 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'', only ''m'' remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most ''m'' − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats. Conversely, suppose we are faced with a repeating decimal, 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 number, transcendental, hence irrational. This theorem states that if ''a'' and ''b'' are both algebraic numbers, and ''a'' is not equal to 0 or 1, and ''b'' is not a rational number, then any value of ''a''''b'' is a transcendental number (there can be more than one value if Exponentiation#Powers of complex numbers, 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 proof by contradiction, contradiction, 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 number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
(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'' 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 algebraic independence, 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 \gamma are irrational. (Senior Mathematics Seminar, Spring 2008 course) It is not known if either of the tetrations ^n\pi or ^n e is rational for some integer n > 1.


Set of all irrationals

Since the reals form an
uncountable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 d(''x'', ''y'') = , ''x'' − ''y'', , the real numbers are a metric space and hence also a topological space. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed, the induced metric is not complete (topology), complete. However, 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 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 space, zero-dimensional.


See also

* Brjuno number * Computable number * Diophantine approximation * Proof that e is irrational, Proof that is irrational * Proof that π is irrational, Proof that is irrational * Square root of 3 * Square root of 5 * 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 named a ...
, ''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 Irrational numbers, Articles containing proofs