Square_root_of_2

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

$\backslash sqrt\{2\}$

but can also be written as

$2^\{\backslash frac\{1\}\{2\}\},$

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:

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 $\backslash tfrac\{99\}\{70\}$ for the square root of two is better than the quick approximation $\backslash tfrac\{22\}\{7\}$ for pi, probably the most widely known irrational number.

List of numbers - Irrational numbers ζ(3) - $\backslash sqrt\{2\}$ - φ - √3 - √5 - α - e - π - δ
Binary	1.0110101000001001111...
Decimal	1.4142135623730950488...
Hexadecimal	1.6A09E667F3BCC908B2F...
Continued fraction	$1\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{2\; +\; \backslash cfrac\{1\}\{\backslash ddots\}\}\}\}$

The silver ratio is

$1+\backslash sqrt\{2\}.\backslash ,$

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

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 $\backslash sqrt\{2\}$ in four sexagesimal figures, which is about six decimal figures: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

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

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. Henderson. 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 $\backslash sqrt\{2\}.$

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.Washingtonpost.com: The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity Other legends report that Hippasus was drowned by fanatical Pythagoreans,Hippasus of Metapontum (ca. 500 BC) - from Eric Weisstein\'s World of Scientific Biography or merely expelled from their circle.Washingtonpost.com: The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity

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 methodAlthough 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. 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:

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

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.^{[citation needed]}

Among mathematical constants with nonrepeating decimal expansions, only π has been calculated more accurately.Number of known digits

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.

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.
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. 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:

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 (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. 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 $\backslash sqrt\{2\}\; =\; \backslash frac\{\backslash sqrt\{2\}\backslash cdot\; n(\backslash sqrt\{2\}-1)\}\{n(\backslash sqrt\{2\}-1)\}\; =\; \backslash frac\{2n-\backslash sqrt\{2\}n\}\{\backslash sqrt\{2\}n-n\}\; =\; \backslash frac\{2n-m\}\{m-n\},\backslash text\{\; because\; \}\backslash sqrt\{2\}\backslash ,n\backslash ,=\backslash ,m.$
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. 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.Apostol (2000), p. 841 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) = 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

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

This number satisfies

$\backslash frac\{\backslash sqrt\{2\}\}\{2\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{2\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; =\; \backslash cos(45^\backslash circ)\; =\; \backslash sin(45^\backslash circ).$

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

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

This is a result of a property of silver means.

Another interesting property of the square root of two:

$\backslash sqrt\{2+\backslash sqrt\{2+\backslash sqrt\{2\}\backslash cdots\}\}\; =\; 2$

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:

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

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

$\backslash frac\{1\}\{\backslash sqrt\; 2\}\; =\; \backslash prod\_\{k=0\}^\backslash infty$

\left(1-\frac{1}{(4k+2)^2}\right) = \left(1-\frac{1}{4}\right) \left(1-\frac{1}{36}\right) \left(1-\frac{1}{100}\right) \cdots

and

$\backslash sqrt\{2\}\; =$

\prod_{k=0}^\infty \frac{(4k+2)^2}{(4k+1)(4k+3)} = \left(\frac{2 \cdot 2}{1 \cdot 3}\right) \left(\frac{6 \cdot 6}{5 \cdot 7}\right) \left(\frac{10 \cdot 10}{9 \cdot 11}\right) \left(\frac{14 \cdot 14}{13 \cdot 15}\right) \cdots

or equivalently,

$\backslash sqrt\{2\}\; =$

\prod_{k=0}^\infty \left(1+\frac{1}{4k+1}\right) \left(1-\frac{1}{4k+3}\right) = \left(1+\frac{1}{1}\right) \left(1-\frac{1}{3}\right) \left(1+\frac{1}{5}\right) \left(1-\frac{1}{7}\right) \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 gives

$\backslash sqrt\{2\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^\{k+1\}\; \backslash frac\{(2k-3)!!\}\{(2k)!!\}\; =$

1 + \frac{1}{2} - \frac{1}{2\cdot4} + \frac{1\cdot3}{2\cdot4\cdot6} - \frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8} + \cdots.

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

$\backslash sqrt\{2\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(2k+1)!\}\{(k!)^2\; 2^\{3k+1\}\}\; =\; \backslash frac\{1\}\{2\}\; +\backslash frac\{3\}\{8\}\; +$

\frac{15}{64} + \frac{35}{256} + \frac{315}{4096} + \frac{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. [1]

Continued fraction representation

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

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

Paper size

The square root of two is the aspect ratio of paper sizes under ISO 216. 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 $x\; \backslash sqrt\{2\}$, its half has sides $x$ and $x\; \backslash sqrt\{2\}/2$, the latter being the same as $x/\backslash sqrt\{2\}$. Therefore, the proportion between the long side ($x$) and the short side ($x/\backslash sqrt(2)$) is again $\backslash sqrt(x)$.

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.
• 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 $\backslash sqrt\{2\}$.
• Henderson, David W., Square Roots in the Sulbasutra
• Eric W. Weisstein, Pythagoras\'s Constant at MathWorld.

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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