Why is the square root of 2 an irrational number?

Answer 1

By way of contradiction, suppose sqrt(2) is rational. Then there exist integers n and d such that sqrt(2) = n/d. We can also assume (without loss of generality) that n and d have no common factors, like writing the fraction in lowest terms. Multiplying both sides by d, and then squaring both sides, gives 2d2 = n2. Every integer can be written as a power of 2 times an odd number, so write n = 2ia and d = 2jb, where a and b are odd. Plugging into the previous equation gives 22j+1b2 = 22ia2. Since a2 and b2 must be odd, they must be equal, and hence 2j+1 = 2i. This is impossible; an odd integer cannot equal an even integer. Therefore the original assumption must be false, namely sqrt(2) is an irrational number.