How do you prove this question?
Q: Prove that if x is irrational, then 1/x is irrational?
I am thinking Proof by contradiction but I am not sure. Even where to begin?
Hint: If we were to assume that 1/x WAS rational, then it could be written as a ratio of integers p/q.
If x is irrational, and 1/x is rational, then we are saying that the rational number q can be written as a product of an irrational number x, and a rationa number. Is this valid?
Proof by contradiction works when we assume the opposite of what we want to proove is true. We then derive an erroneous result... a contradiction.
Our problem is... if x is irrational, prove that 1/x is irrational. The first thing we do is assume the opposite. Assume that x is irrational, but conversely, 1/x is rational. From this we will try to derive an impossibility.
A property of a rational number is that it can be written as a ratio of integers. ANY rational number can be written in the form , where p and q are both integers.
So if we assume that 1/x is rational then:
Now multiply both sides by xq
What this statement says is that the integer q can be written as a product of the integer p, and the irrational number q. THIS is invalid, because multiplying an integer by an irrational number gives an irrational number. Hence q can't be an integer, and our initial assumption MUST be wrong.
Hence 1/x is not rational. And any number that is not rational is irrational.
So the next step is to observe that 1/x = p/q, such that p and q are integers. And this leads the contradiction q = px, which cannot be.