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The 'square root of 2', also known as 'Pythagoras' constant', often denoted by
:$sqrt\{2\},$
is the positive real number that, when multiplied by itself, gives the number 2. Its numerical value approximated to 65 decimal places is:
: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 $frac\{99\}\{70\}$ for the square root of two is better than the quick approximation $frac\{22\}\{7\}$ for pi, probably the most widely known irrational number.
{| border="1" style="float: right; border-collapse: collapse;"
| colspan="2" align="center" | List of numbers - Irrational numbers
γ - ζ(3) - $sqrt\{2\}$ - √3 - √5 - φ - α - e - π - δ
|-
|Binary
| 1.0110101000001001111...
|-
| Decimal
| 1.4142135623730950488...
|-
| Hexadecimal
| 1.6A09E667F3BCC908B2F...
|-
| Continued fraction
|
|}
The silver ratio is
:$1+sqrt\{2\}.,$

Contents

History

Computation algorithm

Proofs of irrationality

Proof by infinite descent

Proof by unique factorization

Geometric proof

Properties of the square root of two

Series and product representations

Continued Fraction Representation

See also

Notes

References

External links

History

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an approximation of $sqrt\{2\}$ 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. 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 $sqrt\{2\}.$
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]

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:
First, pick an arbitrary guess, $F\_0$; 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.
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. [4]

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.
# Assume that √2 is a rational number, meaning that there exists an integer ''a'' and an integer ''b'' such that ''a'' / ''b'' = √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.
# It follows that ''a''² / ''b''² = 2 and ''a''² = 2 ''b''².
# Therefore ''a''² is even because it is equal to 2 ''b''² which is obviously even.
# It follows that ''a'' must be even (as squares of odd integers are also odd).
# Because ''a'' is even, there exists an integer ''k'' that fulfills: ''a'' = 2''k''.
# Inserting (6) into the last equation of (3): 2''b''² = (2''k'')² is equivalent to 2''b''² = 4''k''² is equivalent to ''b''² = 2''k''².
# Because 2''k''² is even it follows that ''b''² is also even which means that ''b'' is even because odd integers have odd squares.
# 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.
This proof can be generalized to show that any root of any natural number is either a natural number or irrational.

Proof by unique factorization

An alternative proof uses the same approach with the unique factorization theorem:
# Assume that √2 is a rational number, meaning that there exists an integer ''a'' and an integer ''b'' such that ''a'' / ''b'' = √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.
# It follows that ''a''² / ''b''² = 2 and ''a''² = 2 ''b''².
# By the unique factorization theorem, both ''a'' and ''b'' have a unique prime factorization, such that ''a'' = 2^x''k'' and ''b'' = 2^y''m'' for the nonnegative integers ''x'', ''y'', and the nonnegative odd integers ''m'' and ''k''.
# Therefore, ''a''² = 2^2x''k''² and ''b''² = 2^2y''m''².
# Inserting back into (3) we get that 2^2x''k''² = 2·2^2y''m''² = 2^2y+1''m''².
# This states that a prime factorization with an even power of 2 (2''x'') is equal to one with an odd power of 2 (2''y''+1). But this contradicts the unique factorization theorem. Therefore the original statement must be false.

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.
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.
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'') = 2''n'' − ''m''.
Hence we have an even smaller right isosceles triangle, with hypotenuse length 2''n'' − ''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
:
ight).
This number satisfies
:
One interesting property of the square root of two is as follows:
:$!\; \{1\; over\; \{sqrt\{2\}\; -\; 1\}\}\; =\; sqrt\{2\}\; +\; 1.$
This is a result of a property of silver means.
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

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
:
left(1-rac{1}{(4k+2)^2}
ight) =
left(1-rac{1}{4}
ight)
left(1-rac{1}{36}
ight)
left(1-rac{1}{100}
ight) cdots
and
:$sqrt\{2\}\; =$
prod_{k=0}^infty
rac{(4k+2)^2}{(4k+1)(4k+3)} =
left(rac{2 cdot 2}{1 cdot 3}
ight)
left(rac{6 cdot 6}{5 cdot 7}
ight)
left(rac{10 cdot 10}{9 cdot 11}
ight)
left(rac{14 cdot 14}{13 cdot 15}
ight) cdots
or equivalently,
:$sqrt\{2\}\; =$
prod_{k=0}^infty
left(1+rac{1}{4k+1}
ight)
left(1-rac{1}{4k+3}
ight)
=
left(1+rac{1}{1}
ight)
left(1-rac{1}{3}
ight)
left(1+rac{1}{5}
ight)
left(1-rac{1}{7}
ight) cdots.
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos(π/4) gives
:
ight)^{2k}}{(2k)!}.
The Taylor series of √(1+''x'') with ''x'' = 1 gives
:
1 + rac{1}{2} - rac{1}{2cdot4} + rac{1cdot3}{2cdot4cdot6} -
rac{1cdot3cdot5}{2cdot4cdot6cdot8} + cdots.
The convergence of this series can be accelerated with an Euler transform, producing
:
rac{15}{64} + rac{35}{256} + rac{315}{4096} + rac{693}{16384} + cdots.
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]

Continued Fraction Representation

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

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.

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 $sqrt\{2\}$ 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

★ Irrationality of The Square Root of Two—A Geometric Proof, , Tom M., Apostol, The American Mathematical Monthly, 2000

★ The Square Root of Two, , David, Flannery, Springer, 2005, ISBN 0-387-20220-X

★ Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context, , David, Fowler, Historia Mathematica, 1998

★ Gourdon, X. & Sebah, P. Pythagoras' Constant: √2. Includes information on how to compute digits of $sqrt\{2\}$.

★ Henderson, David W., ''Square Roots in the Sulbasutra''

★

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