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