Square root of two

The square root of two is the positive real number which, when multiplied by itself, gives a product of two. It was the first known irrational number.

A common geometrical case of the square root of two is that it is the length of a diagonal across a square with sides of one unit of length. This follows from Pythagoras' theorem.

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

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

1. Assume that √2 is a rational number, meaning that there exists an integer a and b so that a / b = √2.
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.
3. It follows that a² / b² = 2 and a² = 2 b².
4. Therefore a² is even because it is equal to 2 b² which is obviously even.
5. It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)
6. Because a is even, there exists a k that fulfills: a = 2k.
7. We insert the last equation of (3) in (6): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k².
8. Because 2k² is even it follows that b² is also even which means that b is even because only even numbers have even squares.
9. 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 and has sufficient charm that it is worth including here. It is an example of proof by infinite descent.

It proceeds by observing that if √2=m/n then √2=(2n−m)/(m−n), so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction if n and m are positive integers, so the assumption that √2 is rational must be false. It is possible to construct from an isosceles right triangle whose leg and hypotenuse have respective lengths n and m, by a classic straightedge-and-compass construction, a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m−n and 2n−m. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.