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Square root of 2
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The square root of 2, also known as Pythagoras' constant,
is the positive real number that, when multiplied by itself, gives the number 2.
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 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 is often denoted by
or v2
but can also be written in exponent notation as
On basic calculators with no square-root function, the quick approximation for the square root may be used.
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Encyclopedia
The square root of 2, also known as Pythagoras' constant,
is the positive real number that, when multiplied by itself, gives the number 2.
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 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 is often denoted by
or v2
but can also be written in exponent notation as
On basic calculators with no square-root function, the quick approximation for the square root may be used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10 000.
History
The Babylonian clay tablet YBC 7289 (c. 18001600 BCE) gives an approximation of in four sexagesimal figures, which is about six decimal figures:
Another early close approximation of this number is given in ancient Indian mathematical texts, the Sulbasutras (c. 800200 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,
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
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. Other legends report that Hippasus was drowned by some Pythagoreans, or merely expelled from their circle.
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 methodFowler 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, a0>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 v2 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 v2 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 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 Gauss's lemma, 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) = x2 − 2, it follows that v2 is either an integer or irrational. Since v2 is not an integer (2 is not a perfect square), v2 must therefore be irrational.
See quadratic irrational 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 v2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = v2.
- Then v2 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 a2 / b2 = 2 and a2 = 2 b2. ((a / b)n = an / bn)
- Therefore a2 is even because it is equal to 2 b2. (2 b2 is necessarily even because it's divisible by 2that is, (2 b2)/2 = b2 and numbers divisible by two are even by definition.)
- 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).
- Because a is even, there exists an integer k that fulfills: a = 2k.
- 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.
- 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 (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 v2 is a rational number must be false. The opposite is proven: v2 is irrational.
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 v2 is a rational number. Then there are integers a and b such that a is coprime to b and v2 = a / b. In other words, v2 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 say which divides b and which does not divide into a otherwise the fraction would not be irreducible.
- The square of a can be factorized as the product of the primes into which a is factorized 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.
Another proof
The following reductio ad absurdum argument showing the irrationality of v2 is less well-known. It uses the additional information v2 > 1.
- Assume that v2 is a rational number. This would mean that there exist integers m and n with n ? 0 such that m/n = v2.
- Then v2 can also be written as an irreducible fraction m/n with positive integers, because v2 > 0.
- Then
- Since v2 > 1, it follows that m > n, which in turn implies that m > 2n m.
- So the fraction m/n for v2, 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 v2 is rational must be false.
Geometric proof
Another reductio ad absurdum showing that v2 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.
Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = v2. 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 v2 is irrational.
Analytic proof
- Lemma 1: let and such that for all and
-
-
Then is irrational.
Proof: suppose with .
For sufficiently big
-
then
-
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but is an integer, absurd, then is irrational.
Proof: let and
-
-
for all .
By induction,
-
for all . For ,
-
and if is true for then is true for . In fact
-
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By lemma 1 applications is irrational.
Properties of the square root of two
One-half of v2, 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
This number satisfies
One interesting property of the square root of two is as follows:
This is a result of a property of silver means.
Another interesting property of the square root of two:
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
The square root of two is also the only real number whose infinite tetrate is equal to its square.
Series and product representations
The identity cos(p/4) = sin(p/4) = 1/v2, along with the infinite product representations for the sine and cosine, leads to products such as
and
or equivalently,
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos(p/4) gives
The Taylor series of v(1+x) with x = 1 gives
The convergence of this series can be accelerated with an Euler transform, producing
It is not known whether v2 can be represented with a BBP-type formula. BBP-type formulas are known for pv2 and v2 ln(1+v2), 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 due to their use in approximating the ratio between the sides and diagonal of a square).
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 and , its half has sides and , the latter being the same as . Therefore, the proportion between the long side and the short side is again .
See also
External links
-
- , a collection of proofs
- , enthusiast site with realtime computation
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