2.4. The Real Number System

Theorem 2.4.1: No Square Roots in Q

There is no rational number x such that x2 = x * x = 2.


Suppose there was such an x. Being a rational number, we can write it as Since x2 = x * x = 2 we have In other words, a2 is even, and therefore a must be even as well. (Can you prove this ?). Hence, But then we have that As before, this means that b is even.. But then both a and b are divisible by 2. That's a contradiction, because a and b were supposed to have no common divisors.

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