The square root of 2, √2, also known as Pythagoras s constant, is the positive real number which, when multiplied by itself, gives the product 2 (number). Its numerical value approximated to 65 decimal is: :1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799. A number called the silver ratio has the same expansion only starting with 2.4 rather than 1.4.

√2 was the first known irrational number. Geometrically, √2 is the length of a diagonal across a square with sides of one unit of length; this follows from Pythagoras theorem.

=History=

The first approximation of this number was given in ancient Indian mathematical texts, the Sulba Sutras (800 B.C. to 200 B.C.) as follows: Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth. That is,

1 + frac{1}{3} + frac{1}{3 cdot 4} - frac{1}{3 cdot4 cdot 34} = frac{577}{408} approx 1.414215686.

The discovery of the irrational numbers is usually attributed to Pythagoras or one of his followers, who produced a (most likely geometrical) proof of the irrationality of √2.

=Proof of irrationality=

One proof of the number s irrationality is the following proof by contradiction. The proposition is proved by assuming that the opposite of the proposition is true and showing that the proposition is false, which means that the proposition must be true.

# Assume that √2 is a rational number, meaning that there exist an integer a and an integer b such that a / b = √2. # Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) 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. # Therefore a 2 is even because it is equal to 2 b 2 which is obviously even. # It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.) # Because a is even, there exists a k that fulfills: a = 2 k . # We insert the last equation of (3) in (6): 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 even it follows that b 2 is also even which means that b is even because only even numbers have even squares. # By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

Since we have found 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.

=A different proof=

Another reductio ad absurdum showing that √2 is irrational is less well-known. It is an example of proof by infinite descent. It makes use of a classic ruler-and-compass 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 .

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.

Since ∠ EBF is a right angle and ∠ BEF is half a right angle, BEF is also a right isosceles triangle. Hence BF = m  −  n . By symmetry, DF = m  −  n , and FDC is also a right isosceles triangle. It also follows that FC = 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.

= Further reading =