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