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Encyclopedia > Irrational number

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. 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. Perhaps[1][2] the most well known irrational numbers are π [3] and $scriptstylesqrt{2}$ [4]. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... The integers consist of the positive natural numbers (1, 2, 3, &#8230;) the negative natural numbers (&#8722;1, &#8722;2, &#8722;3, ...) and the number zero. ... In mathematics, the phrase almost all has a number of specialised uses. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek Ï€ÎµÏÎ¹Ï†Î­ÏÎµÎ¹Î±, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ...

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 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. This article is about the mathematical concept. ... Commensurability in general Generally, two quantities are commensurable if both can be measured in the same units. ...

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC. It was known that the diagonal and side of a square are incommensurable with each other[5]. The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Centuries: 10th century BC - 9th century BC - 8th century BC Decades: 850s BC 840s BC 830s BC 820s BC 810s BC - 800s BC - 790s BC 780s BC 770s BC 760s BC 750s BC Events and Trends 804 BC - Hadad-nirari IV of Assyria conquers Damascus. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC - 490s BC - 480s BC - 470s BC - 460s BC - 450s BC Events and Trends 509 BC - Foundation of the Roman Republic 508 BC - Office of pontifex maximus created...

The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontum[6], a Pythagorean who probably discovered them while identifying sides of the pentagram[7]. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. 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[8]. It wasn't until Eudoxus developed a theory of irrational ratios that a strong mathematical foundation of irrational numbers was created[9]. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ... The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... A pentagram A pentagram (sometimes known as a pentalpha or pentangle or, more formally, as a star pentagon) is the shape of a five-pointed star drawn with five straight strokes. ... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... This article is about Theodorus the mathematician from Cyrene. ... In mathematics, an nth root of a number a is a number b such that bn=a. ... Another article concerns Eudoxus of Cyzicus. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

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. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ...

Lambert proved (1761) that π cannot be rational, and that en is irrational if n is rational (unless n = 0)[11]. 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. 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. 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. Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 &#8211; September 25, 1777), was a mathematician, physicist and astronomer. ... Adrien-Marie Legendre (September 18, 1752&#8211;January 10, 1833) was a French mathematician. ... In mathematical analysis, the Bessel-Clifford function is a an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. ... Charles Hermite (pronounced in IPA, ) (December 24, 1822 â€“ January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that &#960; is a transcendental number, i. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Paul Albert Gordan (April 27, 1837 &#8211; December 21, 1912) was a German mathematician. ...

## Example proofs

### The square root of 2

One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the contrary and showing that doing so leads to a contradiction (hence the proposition must be true). The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption...

1. Assume that $scriptstylesqrt{2}$ is a rational number. This assumption implies 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 (the fraction is shortened as much as possible). This means that m and n are coprime integers, i.e., they have no common factor greater than 1.
3. From m/n = √2 it follows that m = n√2, and so m2 = (n√2)2 = 2n2.
4. So m2 is an even number, because it is equal to 2n2, which is even.
5. It follows that m itself is even (since only even numbers have even squares).
6. Because m is even, there exists an integer k satisfying m = 2k.
7. We may therefore substitute 2k for m in the last equation of (3), thereby obtaining the equation (2k)2 = 2n2, which is equivalent to 4k2 = 2n2 and may be simplified to 2k2 = n2.
8. Because 2k2 is even, it now follows that n2 is also even, which means that n is even (recall that only even numbers have even squares).
9. Then, by (5) and (8), m and n are both even, which contradicts the property stated in (2) that m/n is irreducible.

Since we have found a contradiction, the initial assumption (1) that √2 is a rational number is false; that is to say, √2 is irrational. An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and âˆ’1, or equivalently, if their greatest common divisor is 1. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ...

This proof can be generalized to show that any root of any natural number is either a natural number or irrational. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

#### Another proof

The following reductio ad absurdum argument showing the irrationality of √2 is less well-known. It uses the additional information √2 > 1. Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption...

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 $sqrt{2} = frac{sqrt{2}cdot n(sqrt{2}-1)}{n(sqrt{2}-1)} = frac{2n-sqrt{2}n}{sqrt{2}n-n} = frac{2n-m}{m-n}$, because $sqrt{2}n=m$.
4. Since √2 > 1, it follows that m > n, which in turn implies that m > 2nm.
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. 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. 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. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers. An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...

### The square root of 10

If $sqrt{10}$ is a 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 not itself a square, every square ends in an even numbers of zeros, whence √10r in radix r is irrational, that is, √r is irrational. 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.)

### 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 Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ...

$varphi={1+sqrt{5} over 2}.$

Assume this is a rational number n/m in lowest terms. 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

${n over m}={mathrm{whole} over mathrm{longer} mathrm{part}} ={mathrm{longer} mathrm{part} over mathrm{shorter} mathrm{part}} ={m over n-m}.$

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. Here is a proof by reductio ad absurdum that log23 is irrational: Logarithms to various bases: is to base e, is to base , and is to base . ... Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption...

• Assume log23 is rational. For some positive integers m and n, we have log23 = m/n.
• It follows that 2m/n = 3.
• Raise each side to the n power, find 2m = 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. er and πr are irrational if r ≠ 0 is rational; eπ is also irrational. In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials with integer coefficients: start with a polynomial equation In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

p(x) = an xn + an-1 xn−1 + ... + a1 x + a0 = 0

where the coefficients ai are integers. Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem). 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 an; there are only finitely many such candidates which you can all check by hand. 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 = (21/2 + 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). In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers. For example 3π+2, π + √2 and e3 are irrational (and even transcendental). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

## 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. 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! In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

Conversely, suppose we are faced with a recurring decimal, we can prove that it is a fraction of two integers. For example: A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...

$A=0.7,162,162,162,dots$

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

$1000A=7,16.2,162,162,dots$

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:

$999A=715.5,.$

Then

$A=frac{715.5}{999}=frac{7155}{9990} = frac{135 times 53}{135 times 74} = frac{53}{74},$

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

## Miscellaneous

It has been shown that there exists two irrational numbers a and b, such that ab 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 ab = √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. 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. When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. This means that for every finite sequence Î±1, ..., Î±n of elements of S, no two the...

It is not known whether 2e, πe, π√2, Catalan's constant, or the Euler-Mascheroni gamma constant γ are irrational. Catalans constant K, which occasionally appears in estimates in combinatorics, is defined by or equivalently along with where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Intriguingly, the constant is also given by the integral: Its value is approximately &#947; &#8776; 0. ...

## 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. In mathematics, an uncountable set is a set which is not countable. ... In mathematics, a countable set is a set with the same cardinality (i. ...

Under the usual (Euclidean) distance function d(xy) = |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. However, being a G-delta set -- i.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. 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, the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In the mathematical field of topology a G-delta set or GÎ´ set is a set in a topological space which is in a certain sense simple. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

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

In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards... In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... Although the mathematical constant known as Ï€ (pi) has been studied since ancient times, as has the concept of irrational number, it was not until the 18th century that Ï€ was proved to be irrational. ... In mathematics, an nth root of a number a is a number b such that bn=a. ... The square root of 3 is equal to the length across the flat sides of a regular hexagon with sides of length 1. ...

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