Square root of 2
The square root of 2, or the (1/2)th power of 2, written in mathematics as √ or 21⁄2, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it is 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 the Pythagorean theorem. It was probably the first number known to be irrational.
As a good rational approximation for the square root of two, with a reasonably small denominator, the fraction 99/ (≈ 1.4142857) is sometimes used.
The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of √ in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits, and is the closest possible three-place sexagesimal representation of √:
Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC) 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,
This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of √. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
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, the Pythagoreans 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. The square root of two is occasionally called Pythagoras' number or Pythagoras' constant, for example by Conway & Guy (1996).
Ancient Roman architecture
In ancient Roman architecture, Vitruvius describes the use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato. The system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion was also used to design atria by giving them a length equal to a diagonal taken from a square which sides are equivalent to the intended atrium's width.
There are a number of algorithms for approximating √, 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, a0 > 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:
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 a0 = 1 the next approximations are
- 3/ = 1.5
- 17/ = 1.416...
- 577/ = 1.414215...
- 665857/ = 1.4142135623746...
The value of √ 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 √ was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010. For a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely. Such computations aim to check empirically whether such numbers are normal.
A simple rational approximation 99/ (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1/ (approx. +0.72×10−4). Since it is a convergent of the continued fraction representation of the square root of two, any better rational approximation has a denominator not less than 169, since 239/ (≈ 1.4142012) is the next convergent with an error of approx. −0.12×10−4.
The rational approximation of the square root of two, 665,857/, derived from the fourth step in the Babylonian method starting with a0 = 1, is too large by approx. 1.6×10−12: its square is 2.0000000000045…
Proofs of irrationality
A short proof of the irrationality of √ can be obtained from the rational root theorem, that is, 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) = x2 − 2, it follows that √ is either an integer or irrational. Because √ is not an integer (2 is not a perfect square), √ must therefore be irrational. This proof can be generalized to show that any square root of any natural number that is not the square of a 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, also known as an indirect proof, in that 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 √ is a rational number, meaning that there exists a pair of integers whose ratio is √.
- If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
- Then √ can be written as an irreducible fraction a/ such that a and b are coprime integers (having no common factor).
- It follows that a2/ = 2 and a2 = 2b2. ( (a/)n = an/ ) ( a2 and b2 are integers)
- Therefore, a2 is even because it is equal to 2b2. (2b2 is necessarily even because it is 2 times another whole number and multiples of 2 are even.)
- 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 = 2k.
- Substituting 2k from step 7 for a in the second equation of step 4: 2b2 = (2k)2 is equivalent to 2b2 = 4k2, which is equivalent to b2 = 2k2.
- 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.
- By steps 5 and 8 a and b are both even, which contradicts that a/ is irreducible as stated in step 3.
Because there is a contradiction, the assumption (1) that √ is a rational number must be false. This means that √ is not a rational number; i.e., √ is irrational.
This proof was hinted at 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 have agreed that this proof is an interpolation and not attributable to Euclid.
A simple proof is attributed by John Horton Conway to Stanley Tennenbaum when the latter was a student in the early 1950s and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of American Scientist. Given two squares with integer sides respectively a and b, one of which has twice the area of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle ((2b − a)2) must equal the sum of the two uncovered squares (2(a − b)2). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.
Another geometric reductio ad absurdum argument showing that √ is irrational appeared in 2000 in the American Mathematical Monthly. 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 algebraic proof of the previous section viewed geometrically in yet another way.
Let △ABC be a right isosceles triangle with hypotenuse length m and legs n as shown in Figure 2. By the Pythagorean theorem, m/ = √. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
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) = 2n − m.
Hence, there is 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 √ is irrational.
In a constructive approach, 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 positive integers a and b, because the valuation (i.e., highest power of 2 dividing a number) of 2b2 is odd, while the valuation of a2 is even, they must be distinct integers; thus |2b2 − a2| ≥ 1. Then
the latter inequality being true because it is assumed that a/ ≤ 3 − √ (otherwise the quantitative apartness can be trivially established). This gives a lower bound of 1/ for the difference |√ − a/|, yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop (1985, p. 18). This proof constructively exhibits a discrepancy between √ and any rational.
Proof by Diophantine equations
- Lemma: For the Diophantine equation in its primitive (simplest) form, integer solutions exist if and only if either or is odd, but never when both and are odd.
Proof: For the given equation, there are only six possible combinations of oddness and evenness for whole-number values of and that produce a whole-number value for . A simple enumeration of all six possibilities shows why four of these six are impossible. Of the two remaining possibilities, one can be proven to not contain any solutions using modular arithmetic, leaving the sole remaining possibility as the only one to contain solutions, if any.
|Both even||Even||Impossible. The given Diophantine equation is primitive and therefore contains no common factors throughout|
|Both odd||Odd||Impossible. The sum of two odd numbers does not produce an odd number.|
|Both even||Odd||Impossible. The sum of two even numbers does not produce an odd number.|
|One even, another odd||Even||Impossible. The sum of an even number and an odd number does not produce an even number.|
|One even, another odd||Odd||Possible.|
The fifth possibility (both and odd and even) can be shown to contain no solutions as follows.
Since is even, must be divisible by , hence
The square of any odd number is always . The square of any even number is always . Since both and are odd and is even:
which is impossible. Therefore, the fifth possibility is also ruled out, leaving the sixth to be the only possible combination to contain solutions, if any.
An extension of this lemma is the result that two identical whole-number squares can never be added to produce another whole-number square, even when the equation is not in its simplest form.
- Theorem: is irrational.
Proof: Assume is rational. Therefore,
- Squaring both sides,
But the lemma proves that the sum of two identical whole-number squares cannot produce another whole-number square.
Therefore, the assumption that is rational is contradicted.
is irrational. Q. E. D.
Properties of the square root of two
One-half of √, also the reciprocal of √, approximately 0.707106781186548, 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 √ is as follows:
This is related to the property of silver ratios.
if the square root symbol is interpreted suitably for the complex numbers i and −i.
√ is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1, x1 = c and xn+1 = cxn for n > 1, the limit of xn will be called as n → ∞ (if this limit exists) f(c). Then √ is the only number c > 1 for which f(c) = c2. Or symbolically:
√ appears in Viète's formula for π:
It is not known whether √ 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/ = sin π/ = 1/, along with the infinite product representations for the sine and cosine, leads to products such as
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos π/ gives
The Taylor series of √ with x = 1 and using the double factorial n!! gives
The convergence of this series can be accelerated with an Euler transform, producing
It is not known whether √ can be represented with a BBP-type formula. BBP-type formulas are known for π√ and √ln(1+√), however.
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 because of their use in approximating the ratio between the sides and diagonal of a square). The first convergents are: 1/, 3/, 7/, 17/, 41/, 99/, 239/, 577/. The convergent p/ differs from √ by almost exactly 1/ and then the next convergent is p + 2q/.
Nested square representations
The following nested square expressions converge to √:
The reciprocal of the square root of two (the square root of 1/) is a widely used constant.
In 1786, German physics professor Georg Lichtenberg found that any sheet of paper whose long edge is √ times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standarised paper sizes at the beginning of the 20 century, they used Lichtenberg's ratio to create the "A" series of paper sizes. Today, the (approximate) aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1:√.
Let shorter length and longer length of the sides of a sheet of paper, with
- as required by ISO 216.
Let be the analogue ratio of the halved sheet, then
- Square root of 3
- Square root of 5
- Silver ratio, 1 + √
- The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music.
- The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.
- The celestial latitude (declination) of the Sun during a planet's astronomical cross-quarter day points equals the tilt of the planet's axis divided by √.
- Gelfond–Schneider constant, 2√.
- Viète's formula
- Fowler and Robson, p. 368.
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine
High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
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Fowler and Robson, p. 376. Flannery, p. 32, 158.
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