In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals, but the idea is more profound than that. While it may seem strange at first hearing, almost all real numbers are irrational, in a sense which is defined more precisely below. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses Perhaps^[1][2] the most well known irrational numbers are π and √2. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 ^[3]

When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A ratio is an expression which compares quantities relative to each other In Mathematics, two non- Zero Real numbers a and b are said to be commensurable Iff a / b A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

Contents 1 History 2 Example proofs 2.1 The square root of 2 2.1.1 Another proof 2.2 The square root of 10 and beyond 2.3 The golden ratio 2.4 Logarithms 3 Transcendental and algebraic irrationals 4 Decimal expansions 5 Miscellaneous 6 Open questions 7 The set of all irrationals 8 See also 9 External links 10 References

History

The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontum^[4], a Pythagorean who probably discovered them while identifying sides of the pentagram^[5]. Hippasus of Metapontum (Ίππασος b c 500 BC in Magna Graecia, was a Greek Philosopher. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so, as legend had it, he had Hippasus drowned. Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used couldn't be applied to the square root of 17^[6]. Theodorus of Cyrene was a Greek Mathematician of the 5th century BC who was admired by Plato (who mentions him in several of his works In Mathematics, an n th root of a Number a is a number b such that bn = a. It wasn't until Eudoxus developed a theory of irrational ratios that a strong mathematical foundation of irrational numbers was created^[7]. Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

The sixteenth century saw the acceptance of negative, integral and fractional numbers. A negative number is a Number that is less than zero, such as −2 The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. The next hundred years saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre, and especially of Leonhard Euler. "Moivre" redirects here for the French commune see Moivre Marne. The completion of the theory of complex numbers in the nineteenth 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Kossak is the surname of several persons Juliusz Kossak, Polish painter (1824–99 Wojciech Kossak, Polish painter (1857–1942 Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880,^[8] and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Paul Tannery (1894). Salvatore Pincherle ( March 11, 1853 &mdash July 10, 1936) was an Italian Mathematician. Paul Tannery (1843–1904 was a French mathematician and historian of mathematics. Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued

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 nineteenth century were brought into prominence through the writings of Lagrange. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

Lambert proved (1761) that π cannot be rational, and that eⁿ is irrational if n is rational (unless n = 0)^[9]. Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and While Lambert's proof is often said to be incomplete, modern assessments support it as satisfactory, and in fact for its time unusually rigorous. Legendre (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. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. In mathematical analysis the Bessel-Clifford function is an Entire function of two Complex variables which can be used to provide an alternative development of the The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method, that showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed the same for π. Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Albert Gordan. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi

Example proofs

The square root of 2

The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, called an argument by reductio ad absurdum. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile The following argument appeals twice to the fact that the square of an odd integer is always odd.

If √2 is rational it has the form m/n for integers m, n not both even. Then m² = 2n² whence m is even, say m = 2p. Thus 4p² = 2n² so 2p² = n² whence n is also even, a contradiction.

Another proof

The following reductio ad absurdum argument is less well-known. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile It uses the additional information √2 > 1.

1. Assume that √2 is a rational number. This would mean that there exist integers m and n with n ≠ 0 such that m/n = √2.
2. Then √2 can also be written as an irreducible fraction m/n with positive integers, because √2 > 0.
3. Then , because .
4. Since √2 > 1, it follows that m > n, which in turn implies that m > 2n – m.
5. So the fraction m/n for √2, which according to (2) is already in lowest terms, is represented by (3) in strictly lower terms. An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator This is a contradiction, so the assumption that √2 is rational must be false.

Similarly, assume an isosceles right triangle whose leg and hypotenuse have respective integer lengths n and m. By the Pythagorean theorem, the ratio m/n equals √2. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry It is possible to construct by a classic compass and straightedge construction a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m − n and 2n − m. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.

The square root of 10 and beyond

If √10 is rational, say m/n, then m² = 10n². However, in decimal notation, every square ends in an even number of zeros. So then m² and 10n² in decimal must end in respectively an even and odd number of zeros, a contradiction.

More generally, in any radix r that is square-free, every square ends in an even numbers of zeros, whence √10_r in radix r is irrational, that is, √r  is irrational. In Mathematics, an element r of a Unique factorization domain R is called square-free if it is not divisible by a non-trivial square It follows that the only integers with rational square roots are squares. As a case in point, 2 is not a square, and 2 in binary is 10₂. (Note the convention of subscripting nondecimal numerals with their radix, to avoid ambiguity. As part of that convention the subscripts are understood to be in decimal, not being subscripted themselves. )

To go even further, we can consider m^k = r × n^k for any integers r and k. If r ≠ u^k for any integer u, then r has at least one prime factor p raised to an exponent that is not divisible by k. As all the exponents in the prime factorization of m^k are divisible by k, for the equation to hold, the prime factorization of n^k must contain p raised to a power that is also not divisible by k. But this is clearly impossible. Thus, for any integers r and k, ^k√r  is irrational if r ≠ u^k for any integer u. This result also follows from the fact that raising a non-integral rational number to an integral power can never equal an integer besides 1.

The golden ratio

When a line segment is divided into two disjoint subsegments in such a way that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part, then that ratio is the golden ratio, equal to

Assume this is a rational number n/m in lowest terms. In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the Take n to be the length of the whole and m the length of the longer part. Then n > m, and the length of the shorter part is n − m. Then we have

However, this puts a fraction already in lowest terms into lower terms—a contradiction. Therefore the initial assumption, that the golden ratio is rational, is false.

Logarithms

Perhaps the numbers most easily proved to be irrational are certain logarithms. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce Here is a proof by reductio ad absurdum that log₂3 is irrational:

• Assume log₂3 is rational. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile For some positive integers m and n, we have log₂3 = m/n.
• It follows that 2^m/n = 3.
• Raise each side to the n power, find 2^m = 3ⁿ.
• But 2 to any integer power greater than 0 is even (because at least one of its prime factors is 2) and 3 to any integer power greater than 0 is odd (because none of its prime factors is 2), so the original assumption is false.

Cases such as log₁₀2 can be treated similarly.

Transcendental and algebraic irrationals

Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lists several examples. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation e^r and π^r are irrational if r ≠ 0 is rational; e^π is also irrational.

Another way to construct irrational numbers is as irrational algebraic numbers, i. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or e. as zeros of polynomials with integer coefficients: start with a polynomial equation

p(x) = a_n xⁿ + a_n-1 xⁿ⁻¹ + . In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations . . + a₁ x + a₀ = 0

where the coefficients a_i are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and a_n is non-zero, then because of the intermediate value theorem). In Mathematical analysis, the intermediate value theorem states that for each value between the upper and lower bounds of the image of a Continuous function The only possible rational roots of this polynomial equation are of the form r/s where r is a divisor of a₀ and s is a divisor of a_n; there are only finitely many such candidates which you can all check by hand. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without If neither of them is a root of p, then x must be irrational. For example, this technique can be used to show that x = (2^1/2 + 1)^1/3 is irrational: we have (x³ − 1)² = 2 and hence x⁶ − 2x³ − 1 = 0, and this latter polynomial does not have any rational roots (the only candidates to check are ±1).

Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division For example 3π+2, π + √2 and e√3 are irrational (and even transcendental).

Decimal expansions

The decimal expansion of an irrational number never repeats or terminates, unlike a rational number.

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. Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 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 recurring decimal, we can prove that it is a fraction of two integers. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is For example:

Here the length of the repitend is 3. We multiply by 10³:

Note that since we multiplied by 10 to the power of the length of the repeating part, we shifted the digits to the left of the decimal point by exactly that many positions. Therefore, the tail end of 1000A matches the tail end of A exactly. Here, both 1000A and A have repeating 162 at the end.

Therefore, when we subtract A from both sides, the tail end of 1000A cancels out of the tail end of A:

Then

which is a quotient of integers and therefore a rational number.

Miscellaneous

It has been shown that there exist two irrational numbers a and b, such that a^b is rational. Here is an example:

If √2^√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2^√2 and b = √2. Then a^b = (√2^√2)^√2 = √2^√2·√2 = √2² = 2 which is rational.

Open questions

It is not known whether π + e or π − e is irrational or not. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems In fact, there is no pair of non-zero integers m and n for which it is known whether mπ + ne is irrational or not. Moreover, it is not known whether the set {π, e} is algebraically independent over Q. In Abstract algebra, a Subset S of a field L is algebraically independent over a subfield K if the elements

It is not known whether 2^e, π^e, π^√2, Catalan's constant, or the Euler-Mascheroni gamma constant γ are irrational. In Mathematics, Catalan's constant G, which occasionally appears in estimates in Combinatorics, is defined by G = \beta(2 = \sum_{n=0}^{\infty} The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually

The set of all irrationals

Since the reals form an uncountable set of which the rationals are a countable subset, the complementary set of irrationals is uncountable.

Under the usual (Euclidean) distance function d(x, y) = |x − y|, the real numbers are a metric space and hence also a topological space. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has However, being a G-delta set -- i. In the Mathematical field of Topology, a Gδ set, is a Subset of a Topological space that is a countable intersection of open sets e. , a countable intersection of open subsets -- in a complete metric space, the space of irrationals is topologically complete: 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. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has 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. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}}

Furthermore, the set of all irrationals is a disconnected metric space.

See also

• Dedekind cut
• Proof that e is irrational
• Proof that π is irrational
• Trigonometric number
• nth root
• Square root of 3

External links

• Eric W. Weisstein, Irrational Number at MathWorld. In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove Although the Mathematical constant known as &pi (pi has been studied since ancient times and so has the concept of Irrational number, it was not until the 18th In mathematics a trigonometric number is an Irrational number produced by taking the Cosine of a rational multiple of a circle or equivalently the cosine In Mathematics, an n th root of a Number a is a number b such that bn = a. Geometry If an equilateral triangle ( Equilateral polygon with three sides with sides of length 1 is cut into two equal halves by bisecting an internal angle across Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Square root of 2 is irrational
• All square roots that are not natural, are irrational

References

1. ^ http://sprott.physics.wisc.edu/Pickover/trans.html; URL retrieved 24 October 2007
2. ^ http://www.mathsisfun.com/irrational-numbers.html; URL retrieved 24 October 2007
3. ^ Eric W. Weisstein, Irrational Number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W URL retrieved 26 October 2007.
4. ^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.  
5. ^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal.  
6. ^ Robert L. McCabe (1976). "Theodorus' Irrationality Proofs". Mathematics Magazine.  
7. ^ Charles H. Edwards (1982). The historical development of the calculus. Springer.  
8. ^ Salvatore Pincherle (1880). Salvatore Pincherle ( March 11, 1853 &mdash July 10, 1936) was an Italian Mathematician. "Saggio di una introduzione alla teorica delle funzioni analitiche secondo i principi del prof. Weierstrass". Giornale di Matematiche.  
9. ^ J. H. Lambert (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques". Histoire de l'Académie Royale des Sciences et des Belles-Lettres der Berlin: 265-276.  

• Adrien-Marie Legendre, Éléments de Géometrie, Note IV, (1802), Paris
• Rolf Wallisser, "On Lambert's proof of the irrationality of π", in Algebraic Number Theory and Diophantine Analysis, Franz Halter-Koch and Robert F. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. Tichy, (2000), Walter de Gruyer

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