The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places is: :1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799....

The quick approximation 99/70 for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000.

	1.0110101000001001111...
Decimal	1.4142135623730950488...
Hexadecimal	1.6A09E667F3BCC908B2F...
Continued fraction	$1\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash ddots\}\}\}\}$

History

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an approximation of $\backslash sqrt\{2\}$ in four sexagesimal figures, which is about six decimal figures:

:$1\; +\; \backslash frac\{24\}\{60\}\; +\; \backslash frac\{51\}\{60^2\}\; +\; \backslash frac\{10\}\{60^3\}\; =\; 1.41421\backslash overline\{296\}.$

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.'' That is,

:$1\; +\; \backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{3\; \backslash cdot\; 4\}\; -\; \backslash frac\{1\}\{3\; \backslash cdot4\; \backslash cdot\; 34\}\; =\; \backslash frac\{577\}\{408\}\; \backslash approx\; 1.414215686.$

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 $\backslash sqrt\{2\}.$

Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, ancient Greeks treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.

Computation algorithms

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 of computing square roots, which is one of many methods of computing square roots. It goes as follows:

First, pick a guess, ''a''₀ > 0; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:

:$a\_\{n+1\}\; =\; \backslash frac\{a\_n\; +\; \backslash frac\{2\}\{a\_n\}\}\{2\}=\backslash frac\{a\_n\}\{2\}+\backslash frac\{1\}\{a\_n\}.$

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. Each iteration approximately doubles the number of correct digits. Starting with ''a''₀ = 1 the next approximations are

3/2 = 1.5

17/12 = 1.416...

577/408 = 1.414215...

665857/470832 = 1.4142135623746...

The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997.

In February 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 3.6 GHz PC with 16 GiB of memory.

Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely.

Proofs of irrationality

A short proof of this result is to obtain it from rational root theorem, that if ''p''(''x'') is a monic polynomial with integer coefficients, then any rational root of ''p''(''x'') is necessarily an integer. Applying this to the polynomial ''p''(''x'') = ''x''² − 2, it follows that √2 is either an integer or irrational. Because √2 is not an integer (2 is not a perfect square), √2 must therefore be irrational.

See quadratic irrational or infinite descent#Irrationality of √k if it is not an integer for a proof that the square root of any non-square natural number is irrational.

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, thereby implying that the proposition must be true.

# Assume that √2 is a rational number, meaning that there exists an integer ''a'' and an integer ''b'' in general such that ''a'' / ''b'' = √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. # It follows that ''a''² / ''b''² = 2 and ''a''² = 2 ''b''².   ( (''a'' / ''b'')^''n'' = ''a''^''n'' / ''b''^''n''  ) # Therefore ''a''² is even because it is equal to 2 ''b''². (2 ''b''² is necessarily even because it is 2 times another whole number and even numbers are multiples of 2.) # It follows that ''a'' must be even (as squares of odd integers are never even). # Because ''a'' is even, there exists an integer ''k'' that fulfills: ''a'' = 2''k''. # Substituting 2''k'' from (6) for ''a'' in the second 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 divisible by two and therefore even, and because 2''k''² = ''b''², it follows that ''b''² is also even which means that ''b'' is even. # By (5) and (8) ''a'' and ''b'' are both even, which contradicts that ''a'' / ''b'' is irreducible as stated in (2). ::''Q.E.D.''

Because there is a contradiction, the assumption (1) that √2 is a rational number must be false. By the law of excluded middle, the opposite is proven: √2 is irrational.

This proof was hinted by Aristotle, in his ''Analytica Priora'', §I.23. It appeared first as a full proof in Euclid's ''Elements'', as proposition 117 of Book X. However, since the early 19th century historians agree that this proof is an interpolation and not attributable to Euclid.

Proof by unique factorization

An alternative proof uses the same approach with the fundamental theorem of arithmetic which says every integer greater than 1 has a unique factorization into powers of primes.

# Assume that √2 is a rational number. Then there are integers ''a'' and ''b'' such that ''a'' is coprime to ''b'' and √2 = a / b. In other words, √2 can be written as an irreducible fraction. # The value of ''b'' cannot be 1 as there is no integer ''a'' the square of which is 2. # There must be a prime ''p'' which divides ''b'' and which does not divide ''a'' otherwise the fraction would not be irreducible. # The square of ''a'' can be factored as the product of the primes into which ''a'' is factored but with each power doubled. # Therefore by unique factorization the prime ''p'' which divides ''b'', and also its square, cannot divide the square of ''a''. # Therefore the square of an irreducible fraction cannot be reduced to an integer. # Therefore the square root of 2 cannot be a rational number.

This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational. The article quadratic irrational gives a proof of the same result but not using the fundamental theorem of arithmetic.

Proof by infinite descent, not involving factoring

The following reductio ad absurdum argument showing the irrationality of √2 is less well-known. It uses the additional information 2 > √2 > 1 so that 1 > √2 − 1 > 0. # Assume that √2 is a rational number. This would mean that there exist positive integers ''m'' and ''n'' with ''n'' ≠ 0 such that ''m''/''n'' = √2. Then ''m'' = ''n''√2 and ''m''√2 = 2''n''. # We may assume that ''n'' is the smallest integer so that ''n''√2 is an integer. That is, that the fraction ''m''/''n'' is in lowest terms. # Then $\backslash sqrt\{2\}\; =\; \backslash frac\{m\}\{n\}=\backslash frac\{m(\backslash sqrt\{2\}-1)\}\{n(\backslash sqrt\{2\}-1)\}=\backslash frac\{2n-m\}\{m-n\}$ # Because 1 > √2 − 1 > 0, it follows that ''n'' > ''n''(√2 − 1) = ''m'' − ''n'' > 0. # 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.

Geometric proof

Another reductio ad absurdum showing that √2 is irrational is less well-known. 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. It is essentially the previous proof viewed geometrically.

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.

Because ∠''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.

Analytic proof

Lemma 1: let $\backslash alpha\; \backslash in\; \backslash mathbb\{R\}^+$ and $p\_1,p\_2,\backslash dots,\; q\_1,\; q\_2,\; \backslash ldots\; \backslash in\; \backslash mathbb\{N\}$ such that $\backslash left|\backslash alpha\; q\_n\; -\; p\_n\; \backslash right|\; \backslash neq\; 0$ for all $n\; \backslash in\; \backslash mathbb\{N\}$ and

: $\backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; p\_n\; =\; \backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; q\_n\; =\; \backslash infty\backslash ,$

: $\backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; \backslash left|\; \backslash alpha\; q\_n\; -\; p\_n\; \backslash right|\; =\; 0.\backslash ,$

Then ''α'' is irrational.

''Proof:'' suppose ''α'' = ''a''/''b'' with ''a'', ''b'' ∈ N⁺.

For sufficiently big ''n'',

:

then

:

:

but ''aq''_''n'' − ''bp''_''n'' is an integer, absurd, then ''α'' is irrational.

√2 is irrational.

''Proof:'' let ''p''₁ = ''q''₁ = 1 and

: $p\_\{n+1\}\; =\; p\_n^2\; +\; 2\; q\_n^2\; \backslash ,$

: $q\_\{n+1\}\; =\; 2\; p\_n\; q\_n\; \backslash ,\backslash !$

for all ''n'' ∈ N.

By induction,

:

for all ''n'' ∈ N. For ''n'' = 1,

:

and if is true for ''n'' then is true for ''n'' + 1. In fact

:

:

:

By lemma 1 applications √2 is irrational.

Constructive proof

In a constructive proof, one distinguishes between on the one hand not being rational, and on the other hand being irrational (i.e., being quantifiably apart from every rational), the latter being a stronger property. Given integers ''a'' and ''b'', because the valuation of 2''b''² is odd, while the valuation of ''a''² is even, they must be distinct integers, so that, applying the law of trichotomy in the context of an effectively computable predicate over N, we obtain $|2\; b^2\; -\; a^2|\; \backslash geq\; 1$. An easy calculation then yields a lower bound of $\backslash frac\{1\}\{3b^2\}$ for the difference $|\backslash sqrt2\; -\; a\; /\; b|$, yielding a direct proof of irrationality not relying on the law of excluded middle, see Errett Bishop (1985, p. 18).

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

:$\backslash left(\backslash frac\{\backslash sqrt\{2\}\}\{2\},\; \backslash frac\{\backslash sqrt\{2\}\}\{2\}\backslash right).$

This number satisfies

:$\backslash tfrac12\backslash sqrt\{2\}\; =\; \backslash sqrt\{\backslash tfrac12\}\; =\; \backslash frac1\backslash sqrt\{2\}\; =\; \backslash cos(45^\backslash circ)\; =\; \backslash sin(45^\backslash circ).$

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

:$\backslash !\backslash \; \{1\; \backslash over\; \{\backslash sqrt\{2\}\; -\; 1\}\}\; =\; \backslash sqrt\{2\}\; +\; 1$

since $(\backslash sqrt\{2\}+1)(\backslash sqrt\{2\}-1)=2-1=1.$ This is related to the 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:

:$\backslash frac\{\backslash sqrt\{i\}+i\; \backslash sqrt\{i\}\}\{i\}\backslash text\{\; and\; \}\backslash frac\{\backslash sqrt\{-i\}-i\; \backslash sqrt\{-i\}\}\{-i\}.$

The square root of two is also the only real number other than 1 whose infinite tetrate is equal to its square.

:$\backslash sqrt\{2\}^\; \{\backslash sqrt\{2\}^\; \{\backslash sqrt\{2\}^\; \{\backslash \; \backslash cdot^\; \{\backslash cdot^\; \backslash cdot\}\}\}\}\; =\; 2$

The square root of two can also be used to approximate ''π'':

: $2^m\backslash sqrt\{2-\backslash sqrt\{2+\backslash sqrt\{2+\backslash cdots+\backslash sqrt\{2\}\}\}\}\; \backslash to\; \backslash pi\backslash text\{\; as\; \}m\; \backslash to\; \backslash infty\backslash ,$

for ''m'' square roots and only one minus sign.

It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.

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

:$\backslash frac\{1\}\{\backslash sqrt\; 2\}\; =\; \backslash prod\_\{k=0\}^\backslash infty\; \backslash left(1-\backslash frac\{1\}\{(4k+2)^2\}\backslash right)\; =\; \backslash left(1-\backslash frac\{1\}\{4\}\backslash right)\; \backslash left(1-\backslash frac\{1\}\{36\}\backslash right)\; \backslash left(1-\backslash frac\{1\}\{100\}\backslash right)\; \backslash cdots$

and

:$\backslash sqrt\{2\}\; =\; \backslash prod\_\{k=0\}^\backslash infty\; \backslash frac\{(4k+2)^2\}\{(4k+1)(4k+3)\}\; =\; \backslash left(\backslash frac\{2\; \backslash cdot\; 2\}\{1\; \backslash cdot\; 3\}\backslash right)\; \backslash left(\backslash frac\{6\; \backslash cdot\; 6\}\{5\; \backslash cdot\; 7\}\backslash right)\; \backslash left(\backslash frac\{10\; \backslash cdot\; 10\}\{9\; \backslash cdot\; 11\}\backslash right)\; \backslash left(\backslash frac\{14\; \backslash cdot\; 14\}\{13\; \backslash cdot\; 15\}\backslash right)\; \backslash cdots$

or equivalently,

:$\backslash sqrt\{2\}\; =\; \backslash prod\_\{k=0\}^\backslash infty\; \backslash left(1+\backslash frac\{1\}\{4k+1\}\backslash right)\; \backslash left(1-\backslash frac\{1\}\{4k+3\}\backslash right)\; =\; \backslash left(1+\backslash frac\{1\}\{1\}\backslash right)\; \backslash left(1-\backslash frac\{1\}\{3\}\backslash right)\; \backslash left(1+\backslash frac\{1\}\{5\}\backslash right)\; \backslash left(1-\backslash frac\{1\}\{7\}\backslash right)\; \backslash cdots.$

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

:$\backslash frac\{1\}\{\backslash sqrt\{2\}\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^k\; \backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash right)^\{2k\}\}\{(2k)!\}.$

The Taylor series of √(1 + ''x'') with ''x'' = 1 and using the double factorial ''n''!! gives

:$\backslash sqrt\{2\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^\{k+1\}\; \backslash frac\{(2k-3)!!\}\{(2k)!!\}\; =\; 1\; +\; \backslash frac\{1\}\{2\}\; -\; \backslash frac\{1\}\{2\backslash cdot4\}\; +\; \backslash frac\{1\backslash cdot3\}\{2\backslash cdot4\backslash cdot6\}\; -\; \backslash frac\{1\backslash cdot3\backslash cdot5\}\{2\backslash cdot4\backslash cdot6\backslash cdot8\}\; +\; \backslash cdots.$

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

:$\backslash sqrt\{2\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(2k+1)!\}\{(k!)^2\; 2^\{3k+1\}\}\; =\; \backslash frac\{1\}\{2\}\; +\backslash frac\{3\}\{8\}\; +\; \backslash frac\{15\}\{64\}\; +\; \backslash frac\{35\}\{256\}\; +\; \backslash frac\{315\}\{4096\}\; +\; \backslash frac\{693\}\{16384\}\; +\; \backslash cdots.$

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.

Continued fraction representation

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

:$\backslash !\backslash \; \backslash sqrt\{2\}\; =\; 1\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash ddots\}\}\}\}.$

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 because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408. The convergent p/q differs from the square root of 2 by almost exactly $\backslash scriptstyle\{\backslash frac\{1\}\{q^2\backslash sqrt\{2\}\}\}$ and then the next convergent is (''p'' + 2''q'')/(''p'' + ''q'').

Paper size

The square root of two is the aspect ratio of paper sizes under ISO 216 (A4, A0, etc.). This ratio guarantees that cutting a sheet in half along a line parallel to its short side results in two sheets having the same ratio.

See also

Pythagoras' constant

Square root of 3

Square root of 5

Silver ratio, 1 + √2

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, which in turn means that the ratio of ''areas'' between two successive apertures is 2.

Viète's formula

Notes

References

.

Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1--32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.

. . . .

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

Square Root of 2 Search Engine 2 billion searchable digits of √2, and

Category:Algebraic numbers Category:Mathematical constants Category:Irrational numbers Category:Articles containing proofs

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