The question I am working on is:
Prove that if x is irrational, then 1/x is irrational.
My proof differs from the one given in the answer key; but I still feel that mine is valid. Could someone possibly look over my proof to see if it is correct?
Proof by contraposition: If $1/x$ is rational, then x is a rational number.
Assuming that $x \ne 0$, then $1/x$ is by definition a rational number; taking the reciprocal of this, x is will be some number, other than zero, that can be written as $x/1$, which is a rational number by definition.
Since we have proven the contrapositive to be true, then the original statement must be true.
EDIT: I found this solution on the internet.
Proof: We prove the contrapositive: If 1=x is rational, then x is rational. So suppose 1=x is rational. Then there exist integers p; q, with q = 0, such that 1 6 =x = p=q. Then x = q=p is clearly rational, unless p = 0. However, the case that p = 0 can't occur, because if p = 0, then 1=x = p=q = 0. But 1=x is never zero.
My question is, what is the point in mentioning the case that $p=0$. Isn't it safe to assume that, once you reach the point when you take the reciprocal, $p$ can't equal zero??