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suppose that were rational:
Then we can write as a fractionwhere and are integers with no common factors.
Since , square both sides=>
So,
By the definition of even, this means is even. But then must be even if is even .
So for some integer .
If and , then
.
So . This means that is even, so must be even.
We now have a contradiction. and were chosen not to have any common factors.
But they are both even, i.e. they are both divisible by .
Because assuming that was rational led to a , it must be the case that is irrational.
