I have this homework problem that I can't seem to be able to figure out:
Prove: If n∈N is not the square of some other m∈N, then √n must be irrational.
I know that a number being irrational means that it cannot be written in the form ab:a,b∈N b≠0 (in this case, ordinarily it'd be a∈Z, b∈Z∖{0}) but how would I go about proving this? Would a proof by contradiction work here?
Thanks!!
Answer
Let n be a positive integer such that there is no m such that n=m2. Suppose √n is rational. Then there exists p and q with no common factor (beside 1) such that
√n=pq
Then
n=p2q2.
However, n is an positive integer and p and q have no common factors beside 1. So q=1. This gives that
n=p2
Contradiction since it was assumed that n≠m2 for any m.
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