Question

Question #c9e1c

Answer 1

OK, because the problem statement is ambiguous, I'd rather try to prove a more general and--presumably--more useful result.

I shall prove that, if `a/b` is rational and `x` is irrational, then their sum `a/b + x` is irrational. (`a` and `b` are integers and `b!=0`)

I'll use a proof by contradiction: let's assume that the sum of the two numbers is rational. This means that the sum can be written as a ratio of two integers `c` and `d`, `d!=0`:

`a/b + x = c/d <=> x = c/d - a/b <=> x = (bc - ad)/(bd)`

Note that, because `a, b, c` and `d` are integers, then `bd` is an integer and `bc - ad` is also an integer. Therefore, `x` is the ratio of two integers, i.e. `x` is rational, which contradicts the hypothesis that `x` is irrational.

Thus, the assumption that `a/b + x` is a rational number led us to a contradiction. Hence, its opposite must be true, that is `a/b + x` is an irrational number.

This kind of reasoning can be applied to any particular cases.