There is no rational number x such that x2 = x * x = 2.
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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
a2 = 2 b2
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.
Interactive Real Analysis, ver. 1.9.5 (c) 1994-2007, Bert G. Wachsmuth Page last modified: Mar 28, 2007
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