Square root of 2
The square root of 2, or the th power of 2, written in mathematics as or, 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 from 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 is sometimes used.
The sequence in the OEIS consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places:
History
The Babylonian clay tablet YBC 7289 gives an approximation of in four sexagesimal figures,, 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 as follows: Increase the length 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.
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.Computation algorithms
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, ; 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, the better approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits. Starting with the next approximations are
- = 1.5
- = 1.416...
- = 1.414215...
- = 1.4142135623746...
Such computations aim to check empirically whether such numbers are normal.
Rational approximations
A simple rational approximation is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than . 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 is the next convergent with an error of approx..The rational approximation of the square root of two,, derived from the fourth step in the Babylonian method starting with, is too large by approx. : its square is …
Record progression
This is a table of recent records in calculating digits of.| Date | Name | Number of digits |
| June 28, 2016 | Ron Watkins | 10 trillion |
| April 3, 2016 | Ron Watkins | 5 trillion |
| February 9, 2012 | Alexander Yee | 2 trillion |
| March 22, 2010 | Shigeru Kondo | 1 trillion |
Proofs of irrationality
A short proof of the irrationality of can be obtained from the rational root theorem, that is, if is a monic polynomial with integer coefficients, then any rational root of is necessarily an integer. Applying this to the polynomial, it follows that is either an integer or irrational. Because is not an integer, 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.See quadratic irrational or infinite descent 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, 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 such that and are coprime integers which additionally means that at least one of or must be odd.
- It follows that and.
- Therefore, is even because it is equal to.
- It follows that must be even.
- Because is even, there exists an integer that fulfills:.
- Substituting from step 7 for in the second equation of step 4: is equivalent to, which is equivalent to.
- Because is divisible by two and therefore even, and because, it follows that is also even which means that is even.
- By steps 5 and 8 and are both even, which contradicts that is irreducible as stated in step 3.
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.
Proof by Unique Factorization
As with the Proof by Infinite Descent, we obtain. But now the factor 2 appears an odd number of times on the right, but an even number of times on the left, a contradiction.Geometric proof
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 must equal the sum of the two uncovered squares. 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 be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem,. Suppose and are integers. Let be a ratio given in its lowest terms.
Draw the arcs and with centre. Join. It follows that, and the and coincide. Therefore, the triangles and are congruent by SAS.
Because is a right angle and is half a right angle, is also a right isosceles triangle. Hence implies. By symmetry,, and is also a right isosceles triangle. It also follows that.
Hence, there is an even smaller right isosceles triangle, with hypotenuse length and legs. These values are integers even smaller than and and in the same ratio, contradicting the hypothesis that is in lowest terms. Therefore, and cannot be both integers, hence is irrational.
Constructive proof
In a constructive approach, one distinguishes between on the one hand not being rational, and on the other hand being irrational, the latter being a stronger property. Given positive integers and, because the valuation of is odd, while the valuation of is even, they must be distinct integers; thus. Thenthe latter inequality being true because it is assumed that . This gives a lower bound of for the difference, yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop. This proof constructively exhibits a discrepancy between and any rational.
Proof by Diophantine equations
- Lemma: For the Diophantine equation in its primitive form, integer solutions exist if and only if either or is odd, but never when both and are odd.
| x, y | z | |
| 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. |
| Both odd | Even | Possible. |
| One even, another odd | Odd | Possible. |
The fifth possibility 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.
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, 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 coordinatesThis number satisfies
One interesting property of is as follows:
since
This is related to the property of silver ratios.
can also be expressed in terms of the copies of the imaginary unit using only the square root and arithmetic operations:
if the square root symbol is interpreted suitably for the complex numbers and.
is also the only real number other than 1 whose infinite tetrate is equal to its square. In other words: if for, and for, the limit of will be called as . Then is the only number for which. Or symbolically:
appears in Viète's formula for :
for square roots and only one minus sign.
Similar in appearance but with a finite number of terms, appears in various trigonometric constants:
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, along with the infinite product representations for the sine and cosine, leads to products such asand
or equivalently,
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for gives
The Taylor series of with and using the double factorial 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 and, however.
The number can be represented by an infinite series of Egyptian fractions, with denominators defined by 2nth terms of a Fibonacci-like recurrence relation a=34a-a, a=0, a=6.
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. The first convergents are:. The convergent differs from by almost exactly and then the next convergent is.
Nested square representations
The following nested square expressions converge to :Derived constants
The reciprocal of the square root of two is a widely used constant.Paper size
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 ratio as the original sheet. When Germany standardised paper sizes at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes. Today, the aspect ratio of paper sizes under ISO 216 is 1:.Proof:
Let shorter length and longer length of the sides of a sheet of paper, with
Let be the analogue ratio of the halved sheet, then