Inmathematics, theirrational numbers are all thereal numbers that are notrational numbers. That is, irrational numbers cannot be expressed as the ratio of twointegers. When theratio of lengths of two line segments is an irrational number, the line segments are also described as beingincommensurable, 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 ratioπ of a circle's circumference to its diameter, Euler's numbere, the golden ratioφ, and thesquare root of two.[1] In fact, all square roots ofnatural numbers, other than ofperfect squares, are irrational.[2]
Like all real numbers, irrational numbers can be expressed inpositional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, norend with a repeating sequence. For example, the decimal representation ofπ starts with 3.14159, but no finite number of digits can representπ 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.
AnEuler diagram showing theset of real numbers (), which include the rationals(), which include the integers (), which include the natural numbers (). The real numbers also include the irrationals(\).
The first proof of the existence of irrational numbers is usually attributed to aPythagorean (possiblyHippasus of Metapontum),[4] who probably discovered them while identifying sides of thepentagram.[5]The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if thehypotenuse of anisosceles right triangle was indeedcommensurable 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 integersa,b, andc (a =b since it is isosceles). The ratio of the hypotenuse to a leg is represented byc:b.
Assumea,b, andc are in the smallest possible terms (i.e. they have no common factors).
By thePythagorean theorem:c2 =a2+b2 =b2+b2 = 2b2. (Since the triangle is isosceles,a =b).
Sincec2 = 2b2,c2 is divisible by 2, and therefore even.
Sincec2 is even,c must be even.
Sincec is even, dividingc by 2 yields an integer. Lety be this integer (c = 2y).
Substituting 4y2 forc2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.
Dividing by 2 yields 2y2 =b2.
Sincey is an integer, and 2y2 =b2,b2 is divisible by 2, and therefore even.
Sinceb2 is even,b must be even.
We have just shown that bothb andc 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 thatc andb cannot both be integers and thus the existence of a number that cannot be expressed as a ratio of two integers.[6]
Greek mathematicians termed this ratio of incommensurable magnitudesalogos, 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.'[7] Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory.
The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This was brought to light byZeno 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",[8] but Zeno found that in fact "[quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear... .[Q]uantities are, in other words, continuous".[8] What this means is that contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily beinfinite. 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 formulatingfour paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore, further investigation had to occur.
The next step was taken byEudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5".[9] 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".[10] This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not untilEudoxus 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.[11]
As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive ofx2 andx3 asx squared andx cubed instead ofx to the second power andx 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 hismethod of exhaustion, a kind ofreductio ad absurdum that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof".[12] This method of exhaustion is the first step in the creation of calculus.
Theodorus of Cyrene proved the irrationality of thesurds 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.[13]
Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during theVedic period in India. There are references to such calculations in theSamhitas,Brahmanas, and theShulba Sutras (800 BC or earlier).[14]
It is suggested that the concept of irrationality was implicitly accepted byIndian mathematicians since the 7th century BC, whenManava (c. 750 – 690 BC) believed that thesquare roots of numbers such as 2 and 61 could not be exactly determined.[15] HistorianCarl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true".[16]
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.[17]
Mathematicians likeBrahmagupta (in 628 AD) andBhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In the 12th centuryBhāskara II evaluated some of these formulas and critiqued them, identifying their limitations.
In theMiddle Ages, the development ofalgebra byMuslim mathematicians allowed irrational numbers to be treated asalgebraic objects.[19] Middle Eastern mathematicians also merged the concepts of "number" and "magnitude" into a more general idea ofreal numbers, criticized Euclid's idea ofratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude.[20] In his commentary on Book 10 of theElements, thePersian mathematicianAl-Mahani (d. 874/884) examined and classifiedquadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows:[20]
"It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubesetc."
In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots andcube roots as irrational magnitudes. He also introduced anarithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes:[20]
"their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it."
TheEgyptian mathematicianAbū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions toquadratic equations or ascoefficients in anequation in the form of square roots andfourth roots.[21] In the 10th century, theIraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions.[20]
Many of these concepts were eventually accepted by European mathematicians sometime after theLatin translations of the 12th century.Al-Hassār, a Moroccan mathematician fromFez specializing inIslamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, wherenumerators 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,."[22] This same fractional notation appears soon after in the work ofLeonardo Fibonacci in the 13th century.[23]
The 17th century sawimaginary numbers become a powerful tool in the hands ofAbraham de Moivre, and especially ofLeonhard Euler. The completion of the theory ofcomplex numbers in the 19th century entailed the differentiation of irrationals into algebraic andtranscendental numbers, the proof of the existence of transcendental numbers, and the resurgence of the scientific study of the theory of irrationals, largely ignored sinceEuclid. The year 1872 saw the publication of the theories ofKarl Weierstrass (by his pupil Ernst Kossak),Eduard Heine (Crelle's Journal, 74),Georg Cantor (Annalen, 5), andRichard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth bySalvatore Pincherle in 1880,[24] and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement byPaul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of acut (Schnitt) in the system of allrational numbers, separating them into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass,Leopold Kronecker (Crelle, 101), andCharles Méray.
Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings ofJoseph-Louis Lagrange. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
Johann Heinrich Lambert proved (1761) that π cannot be rational, and thaten is irrational ifn is rational (unlessn = 0).[25] While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous.Adrien-Marie Legendre (1794), after introducing theBessel–Clifford function, provided a proof to show that π2 is irrational, whence it follows immediately that π is irrational also. The existence oftranscendental numbers was first established by Liouville (1844, 1851). Later,Georg Cantor (1873) proved their existence by adifferent method, which showed that every interval in the reals contains transcendental numbers.Charles Hermite (1873) first provede transcendental, andFerdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further byDavid Hilbert (1893), and was finally made elementary byAdolf Hurwitz[citation needed] andPaul Gordan.[26]
The proof for the irrationality of the square root of two can be generalized using thefundamental theorem of arithmetic. This asserts that every integer has aunique 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 inlowest terms there must be aprime in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exactkth power of another integer, then that first integer'skth root is irrational.
Perhaps the numbers most easy to prove irrational are certainlogarithms. Here is aproof by contradiction that log2 3 is irrational (log2 3 ≈ 1.58 > 0).
Assume log2 3 is rational. For some positive integersm andn, we have
It follows that
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 itsprime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integersm/n withn ≠ 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 integersm/n withn ≠ 0.
The realalgebraic numbers are the real solutions of polynomial equations
where the coefficients are integers and. An example of an irrational algebraic number isx0 = (21/2 + 1)1/3. It is clearly algebraic since it is the root of an integer polynomial,, which is equivalent to. This polynomial has no rational roots, since therational root theorem shows that the only possibilities are ±1, butx0 is greater than 1. Sox0 is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials.
Almost all irrational numbers aretranscendental. Examples areer and πr, which are transcendental for all nonzero rational r.
Because the algebraic numbers form asubfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3π + 2,π + √2 ande√3 are irrational (and even transcendental).
The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there is not a finite number of nonzero digits), unlike any rational number. The same is true forbinary,octal orhexadecimal expansions, and in general for expansions in everypositionalnotation withnatural bases.
To show this, suppose we divide integersn bym (wherem is nonzero). Whenlong division is applied to the division ofn bym, there can never be aremainder greater than or equal tom. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at mostm − 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 arepeating decimal, we can prove that it is a fraction of two integers. For example, consider:
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:
Now we multiply this equation by 10r wherer 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:
The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A and 10A have.162162162... after the decimal point.
Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:
Then
is a ratio of integers and therefore a rational number.
Dov Jarden gave a simple non-constructive proof that there exist two irrational numbersa andb, such thatab is rational:[28][29]
Consider; if this is rational, then takea =b =. Otherwise, takea to be the irrational number andb =. Thenab = ()·, which is rational.
Although the above argument does not decide between the two cases, theGelfond–Schneider theorem shows that istranscendental, hence irrational. This theorem states that ifa andb are bothalgebraic numbers, anda is not equal to 0 or 1, andb is not a rational number, then any value ofab is a transcendental number (there can be more than one value ifcomplex number exponentiation is used).
An example that provides a simple constructive proof is[30]
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,, is irrational. This is so because, by the formula relating logarithms with different bases,
which we can assume, for the sake of establishing acontradiction, equals a ratiom/n of positive integers. Then hence hence hence, which is a contradictory pair of prime factorizations and hence violates thefundamental theorem of arithmetic (unique prime factorization).
A stronger result is the following:[31] Every rational number in the interval can be written either asaa for some irrational numbera or asnn for some natural numbern. Similarly,[31] every positive rational number can be written either as for some irrational numbera or as for some natural numbern.
Various combinations ofe,π andelementary functions (such ase +π,eπ,ee,πe,ππ,lnπ) are not known to be irrational, in part becausee andπ are not known to bealgebraically independent.Schanuel's conjecture would imply that all of the above numbers are irrational and even transcendental.[32]
Inconstructive 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.[35] However, there is a second definition of an irrational number used in constructive mathematics, that a real number is an irrational number if it isapart from every rational number, or equivalently, if the distance between and every rational number is positive. This definition is stronger than the traditional definition of an irrational number. This second definition is used inErrett Bishop'sproof that the square root of 2 is irrational.[36]
Since the reals form anuncountableset, of which the rationals are acountable subset, the complementary set ofirrationals is uncountable.
Under the usual (Euclidean) distance function, the real numbers are ametric space and hence also atopological 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 notcomplete. Being aG-delta set—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals iscompletely 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: thecontinued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.
Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen groups so the space iszero-dimensional.
^James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number".The Two-Year College Mathematics Journal.11 (5):312–316.doi:10.2307/3026893.JSTOR3026893.S2CID115390951.
^Kline, M. (1990).Mathematical Thought from Ancient to Modern Times, Vol. 1. New York: Oxford University Press (original work published 1972), p. 33.
^T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 411–2, inSelin, Helaine; D'Ambrosio, Ubiratan, eds. (2000).Mathematics Across Cultures: The History of Non-western Mathematics.Springer.ISBN1-4020-0260-2..
^Boyer (1991). "China and India".A History of Mathematics (2nd ed.). Wiley. p. 208.ISBN0471093742.OCLC414892.It has been claimed also that the first recognition of incommensurables appears in India during theSulbasutra period, but such claims are not well substantiated. The case for early Hindu awareness of incommensurable magnitudes is rendered most unlikely by the lack of evidence that Indian mathematicians of that period had come to grips with fundamental concepts.
^Datta, Bibhutibhusan; Singh, Awadhesh Narayan (1993)."Surds in Hindu mathematics"(PDF).Indian Journal of History of Science.28 (3):253–264. Archived fromthe original(PDF) on 2018-10-03. Retrieved18 September 2018.
^Jacques Sesiano, "Islamic mathematics", p. 148, inSelin, Helaine; D'Ambrosio, Ubiratan (2000).Mathematics Across Cultures: The History of Non-western Mathematics.Springer.ISBN1-4020-0260-2..
^Cajori, Florian (1928).A History of Mathematical Notations (Vol.1). La Salle, Illinois: The Open Court Publishing Company. pg. 269.
^Fowler, David H. (2001), "The story of the discovery of incommensurability, revisited",Neusis (10):45–61,MR1891736
^Jarden, Dov (1953). "Curiosa No. 339: A simple proof that a power of an irrational number to an irrational exponent may be rational".Scripta Mathematica.19: 229.copy
Rolf Wallisser, "On Lambert's proof of the irrationality of π", inAlgebraic Number Theory and Diophantine Analysis, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyter
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