Use a proof by contradiction to show that the sum of an irrational number and a rational number is irrational (using the definitions of rational and irrational real numbers; cf p85 of the textbook, or your notes).
Theorem: If is an irrational number and is a rational number, then is irrational.
Proof: For a proof by contradiction, assume that the hypotheses are true (i.e., that is irrational and is rational) but that the conclusion is false, i.e., is not irrational.
That means is rational, and so by the definition of rational numbers, for integers .
Then . There are two cases for :
(i) is irrational: this contradicts the hypothesis that s is rational.
(ii) is rational: then for integers . But then
meaning r is rational. But that contradicts the hypothesis that r is irrational.
Since we get a contra