The square root of 2 is irrational

Written by Dylan Hendrickson last updated

, the unique positive real number whose square is 2, is not a rational number.

Proof

Suppose is rational. Then for some integers and ; without_loss_of_generality let be in lowest_terms, i.e. . We have

From the definition of ,

So is a multiple of . Since is prime, must be a multiple of 2; let . Then

So is a multiple of , and so is . But then , which contradicts the assumption that is in lowest terms! So there isn't any way to express as a fraction in lowest terms, and thus there isn't a way to express as a ratio of integers at all. That is, is irrational.