The square root of 2, also known as Pythagoras' constant, often denoted by

or √2

but can also be written as

2^{1 / 2},

is the positive real number that, when multiplied by itself, gives the number 2. In Mathematics, the real numbers may be described informally in several different ways In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 Its numerical value approximated to 65 decimal places (sequence A002193 in OEIS) is:

1. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences 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. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose 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 Geometrically, it is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry 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. 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 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

List of numbers - Irrational numbers ζ(3) - - φ - √3 - √5 - α - e - π - δ
Binary	1. 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 In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations 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 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 Continued fraction It can be expressed as the Continued fraction 4 4 4 4 4 The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 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 binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. 0110101000001001111. . .
Decimal	1. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 4142135623730950488. . .
Hexadecimal	1. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a 6A09E667F3BCC908B2F. . .
Continued fraction

The silver ratio is

Contents 1 History 2 Computation algorithm 3 Proofs of irrationality 3.1 Proof by infinite descent 3.2 Proof by unique factorization 3.2.1 Another proof 3.3 Geometric proof 4 Properties of the square root of two 5 Series and product representations 6 Continued fraction representation 7 Paper size 8 See also 9 Notes 10 References 11 External links

History

The Babylonian clay tablet YBC 7289 (c. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital 1800–1600 BCE) gives an approximation of in four sexagesimal figures, which is about six decimal figures:^[1]

Another early close approximation of this number is given in ancient Indian mathematical texts, the Sulbasutras (c. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. This article is about the history of South Asia prior to the Partition of British India in 1947 The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the 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,

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

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. In Mathematics, the Pell numbers and companion Pell numbers (Pell-Lucas numbers are both sequences of Integers that have been known since ancient 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\}}}} 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 Hippasus of Metapontum (Ίππασος b c 500 BC in Magna Graecia, was a Greek Philosopher. 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. ^[3] Other legends report that Hippasus was drowned by some Pythagoreans,^[4] or merely expelled from their circle. ^[5]

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^[6] of computing square roots, which is one of many methods of computing square roots. This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good It goes as follows:

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:

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. is a Japanese Mathematician most known for his numerous world records over the past two decades for calculating digits of π.

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.

Among mathematical constants with nonrepeating decimal expansions, only π has been calculated more accurately. 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 ^[7]

Proofs of irrationality

Proof by infinite descent

One proof of the number's irrationality is the following proof by infinite descent. In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. 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. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile

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 a / b such that a and b are coprime integers and (a / b)² = 2. An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than
3. It follows that a² / b² = 2 and a² = 2 b². ((a / b)ⁿ = aⁿ / bⁿ)
4. Therefore a² is even because it is equal to 2 b². (2 b² is necessarily even because it's divisible by 2—that is, (2 b²)/2 = b² — and numbers divisible by two are even by definition. )
5. It follows that a must be even as (squares of odd integers are also odd, referring to b) or (only even numbers have even squares, referring to a).
6. Because a is even, there exists an integer k that fulfills: a = 2k.
7. Substituting 2k from (6) for a in the second equation of (3): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k².
8. Because 2k² is divisible by two and therefore even, and because 2k² = b², it follows that b² is also even which means that b is even.
9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

Q.E.D.

Since there is a contradiction, the assumption (1) that √2 is a rational number must be false. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" The opposite is proven: √2 is irrational.

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 (also called counting number) can mean either an element of the set (the positive Integers or an

Proof by unique factorization

An alternative proof uses the same approach with the unique factorization theorem:

1. Assume that √2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = √2. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written
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. An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than
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. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written
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 (the exponent is 2x) is equal to one with an odd power of 2 (the exponent is 2y + 1). But this contradicts the unique factorization theorem. Therefore the original statement must be false.

Another proof

The following reductio ad absurdum argument showing the irrationality of √2 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
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.

Geometric proof

Another reductio ad absurdum showing that √2 is irrational is less well-known. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile ^[8] It is also an example of proof by infinite descent. In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles

Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = √2. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Suppose m and n are integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Let m:n be a ratio given in its lowest terms. A ratio is an expression which compares quantities relative to each other An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator

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.

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 because the unit vector that makes a 45° angle with the axes in a plane has the coordinates

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. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length The silver ratio is a mathematical Constant. Its name is an allusion to the Golden ratio; analogously to the way the golden ratio is the limiting ratio

Another interesting property of the square root of two:

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:

and

The square root of two is also the only real number whose infinite tetrate is equal to its square. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Mathematics, tetration (also known as hyper -4

Series and product representations

The identity cos(π/4) = sin(π/4) = 1/√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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives For example, the series for cos(π/4) gives

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

The convergence of this series can be accelerated with an Euler transform, producing

It is not known whether √2 can be represented with a BBP-type formula. In Combinatorial Mathematics the binomial transform is a Sequence transformation (ie a transform of a Sequence) that computes its Forward BBP-type formulas are known for π√2 and √2 ln(1+√2), however. [1]

Continued fraction representation

The square root of two has the following continued fraction representation:

The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks due to their use in approximating the ratio between the sides and diagonal of a square). 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\}}}} A convergent is one of a sequence of values obtained by evaluating successive truncations of a Continued fraction. In Mathematics, the Pell numbers and companion Pell numbers (Pell-Lucas numbers are both sequences of Integers that have been known since ancient

Paper size

The square root of two is the aspect ratio of paper sizes under ISO 216. The aspect ratio of a Shape is the ratio of its longer Dimension to its shorter dimension There have been many standard sizes of Paper at different times and in different countries but today there are two widespread systems in use the international standard (A4 A series Paper in the A series format has a 1\sqrt{2} aspect ratio although this is rounded to the nearest millimetre This ratio guarantees that cutting in half a sheet by a line parallel to its short side results in two sheets having the same ratio.

Indeed, if a rectangle has sides x and , its half has sides x and , the latter being the same as . Therefore, the proportion between the long side (x) and the short side () is again .

See also

• Square root of 3
• Square root of 5
• The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music. 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 Continued fraction It can be expressed as the Continued fraction 4 4 4 4 4 In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical The tritone ( Tri - or three and tone) is a Musical interval that spans three whole tones. Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio.
• The square root of two also forms the relationship of f stops in photographic lenses.

Notes

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. 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

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

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