Theorem 2.4.1: No Square Roots in Q
Why these ads ...
There is no rational number
x such that
x2 = x * x = 2.
Context
Proof:
Suppose there was such an
x. Being a rational number, we can write it
as
- x = a / b (with no common divisors)
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,
- a = 2 c for some integer c.
But then we have that
- 4 c2 = 2 b2, or
2 c2 = b2
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.