The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1.

The square root of 2, also known as Pythagoras' constant, often denoted by Image File history File links Illustration for square root of 2. ... Image File history File links Illustration for square root of 2. ... A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

is the positive real number that, when multiplied by itself, gives the number 2. Its numerical value approximated to 65 decimal places (sequence A002193 in OEIS) is: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article does not cite any references or sources. ... For other uses, see Decimal (disambiguation). ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799.

The square root of 2 was probably the first known irrational number. Geometrically, it is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. On basic calculators with no square root function, the quick approximation for the square root of two is better than the quick approximation for pi, probably the most widely known irrational number. In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... 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. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... When a circles diameter is 1, its circumference is π. Pi or π is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... 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. ...

List of numbers - Irrational numbers ζ(3) - - φ - √3 - √5 - α - e - π - δ
Binary	1.0110101000001001111...
Decimal	1.4142135623730950488...
Hexadecimal	1.6A09E667F3BCC908B2F...
Continued fraction

The silver ratio is This is a list of articles about numbers (not about numerals). ... 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. ... In mathematics, Apérys constant is a curious number that occurs in a variety of situations. ... 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. ... The square root of 3 is equal to the length across the flat sides of a regular hexagon with sides of length 1. ... The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. ... The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... 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. ... The Feigenbaum constants are two mathematical constants named after the mathematician Mitchell Feigenbaum. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... For other uses, see Decimal (disambiguation). ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... The silver ratio is a mathematical constant. ...

History

Babylonian clay tablet YBC 7289 with annotations.
(Image by Bill Casselman)

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an approximation of in four sexagesimal figures, which is about six decimal figures:^[1] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Babylonia was an ancient state in Iraq), combining the territories of Sumer and Akkad. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... For other uses, see Decimal (disambiguation). ...

Another early close approximation of this number is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BCE) as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth.^[2] That is, The History of India begins with the Indus Valley Civilization, which flourished in the north-western part of the Indian subcontinent from 3300 to 1700 BCE. This Bronze Age civilization was followed by the Iron Age Vedic period, which witnessed the rise of major kingdoms known as the Mahajanapadas. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ...

This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of In mathematics, the Pell numbers and companion Pell numbers (Pell-Lucas numbers) are both sequences of integers. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. According to one legend, 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 he sentenced Hippasus to death by drowning. [1] Other legends report that Hippasus was drowned by fanatical Pythagoreans [2], or merely expelled from their circle. [3] 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. ... The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ...

Computation algorithm

There are a number of algorithms for approximating the square root of 2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method^[3] of computing square roots, which is one of many methods of computing square roots. It goes as follows: This article presents and explains several methods which can be used to calculate square roots. ...

First, pick an arbitrary guess, F₀; the guess doesn't matter, as it only affects how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation: See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...

The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better approximation of the square root of 2 is achieved.

The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997. Yasumasa Kanada (金田 康正) is a Japanese mathematician most known for his numerous world records over the past two decades for calculating digits of Ï€. Kanada is a professor in the Department of Information Science at the University of Tokyo in Tokyo, Japan. ...

In February of 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 200,000,000,000 decimal places in slightly over 13 days and 14 hours using a 3.6GHz PC with 16GB of memory.^{[citation needed]}

Among mathematical constants with nonrepeating decimal expansions, only π has been calculated more accurately. [4] 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. ...

Proofs of irrationality

Proof by infinite descent

One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, which means the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, which means that the proposition must be true. In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ... 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, meaning that there exists an integer a and an integer b such that a / b = √2.
2. Then √2 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)² = 2.
3. It follows that a² / b² = 2 and a² = 2 b².
4. Therefore a² is even because it is equal to 2 b² which is obviously even.
5. It follows that a must be even (as squares of odd integers are also odd).
6. Because a is even, there exists an integer k that fulfills: a = 2k.
7. Inserting (6) into the last equation of (3): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k².
8. Because 2k² is even it follows that b² is also even which means that b is even because odd integers have odd squares.
9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

quod erat demonstrandum

Since there is a contradiction, the assumption (1) that √2 is a rational number must be false. The opposite is proven: √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 −1, or equivalently, if their greatest common divisor is 1. ... For other meanings of the abbreviation QED, see QED. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

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. ...

Proof by unique factorization

An alternative proof uses the same approach with the unique factorization theorem: In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. ...

1. Assume that √2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = √2.
2. Then √2 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)² = 2.
3. It follows that a² / b² = 2 and a² = 2 b².
4. By the unique factorization theorem, both a and b have a unique prime factorization, such that a = 2^xk and b = 2^ym for the nonnegative integers x, y, and the nonnegative odd integers m and k.
5. Therefore, a² = 2^2xk² and b² = 2^2ym².
6. Inserting back into (3) we get that 2^2xk² = 2·2^2ym² = 2^2y+1m².
7. This states that a prime factorization with an even power of 2 (2x) is equal to one with an odd power of 2 (2y+1). But this contradicts the unique factorization theorem. Therefore the original statement must be false.

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 −1, or equivalently, if their greatest common divisor is 1. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. ...

Geometric proof

Another reductio ad absurdum showing that √2 is irrational is less well-known.^[4] It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. Image File history File links Irrationality_of_sqrt2. ... 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... In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ... 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. ...

Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = √2. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... The integers are commonly denoted by the above symbol. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... 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...

Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and the ∠BAC and ∠DAE coincide. Therefore the triangles ABC and ADE are congruent by SAS. See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

Since ∠EBF is a right angle and ∠BEF is half a right angle, BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n. By symmetry, DF = m − n, and FDC is also a right isosceles triangle. It also follows that FC = n − (m − n) = 2n − m.

Hence we have an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore m and n cannot be both integers, hence √2 is irrational.

Properties of the square root of two

One-half of √2, approximately 0.70710 67811 86548, is a common quantity in geometry and trigonometry, due to the fact that the unit vector that makes a 45° angle with the axes in a plane has the coordinates Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

This number satisfies

One interesting property of the square root of two is as follows:

This is a result of a property of silver means. The silver ratio is a mathematical constant. ...

The square root of two can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations: In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

and

Series and product representations

The identity cos(π/4) = sin(π/4) = √2/2, along with the infinite product representations for the sine and cosine, leads to products such as

and

or equivalently,

The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos(π/4) gives As the degree of the Taylor series rises, it approaches the correct function. ...

The Taylor series of √(1+x) with x = 1 gives

The convergence of this series can be accelerated with an Euler transform, producing In mathematics, in the area of combinatorics, the binomial transform is a transformation of sequence by computing its forward differences. ...

It is not known whether √2 can be represented with a BBP-type formula. BBP-type formulas are known for π√2 and √2 ln(1+√2), however. [5] In mathematics, the Bailey-Borwein-Plouffe formula (BBP formula) is a Ï€ summation formula discovered in 1995 by Simon Plouffe. ...

Continued Fraction Representation

The square root of two has the following continued fraction representation: In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

See also

• Square root of 3
• Square root of 5
• The square root of two is the aspect ratio of paper sizes under ISO 216.
• The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music.
• The square root of two also forms the relationship of f stops in photographic lenses.

The square root of 3 is equal to the length across the flat sides of a regular hexagon with sides of length 1. ... The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. ... The aspect ratio of a two-dimensional shape is the ratio of its longer dimension to its shorter dimension. ... Comparison of the most common paper sizes. ... ISO 216 specifies international standard (ISO) paper sizes, used in most countries in the world today. ... In music theory, the term interval describes the difference in pitch between two notes. ... For other uses, see Tritone (disambiguation). ... An equal temperament is a musical temperament — that is, a system of tuning intended to approximate some form of just intonation — in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios). ... F number redirects here. ...

Notes

1. ^ Fowler and Robson, p. 368.
   Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
   High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
2. ^ Henderson.
3. ^ Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of √2 seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures.
   Fowler and Robson, p. 376. Flannery, p. 32, 158.
4. ^ Apostol (2000), p. 841

References

• Apostol, Tom M. (November 2000). "Irrationality of The Square Root of Two—A Geometric Proof". The American Mathematical Monthly 107 (9): 841-842. 
• Flannery, David (2005). The Square Root of Two. Springer. ISBN 0-387-20220-X. 
• Fowler, David; Eleanor Robson (November 1998). "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context". Historia Mathematica 25 (4): 366-378. 
• Gourdon, X. & Sebah, P. Pythagoras' Constant: √2. Includes information on how to compute digits of .
• Henderson, David W., Square Roots in the Sulbasutra
• Eric W. Weisstein, Pythagoras's Constant at MathWorld.

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

• The Square Root of Two to 5 million digits by Jerry Bonnell and Robert Nemiroff. May, 1994.
• Square root of 2 is irrational, a collection of proofs
• √2.net, enthusiast site with realtime computation