A Simple Proof by Contradiction

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 r is an irrational number and s is a rational number, then r+s is irrational.

Proof: For a proof by contradiction, assume that the hypotheses are true (i.e., that r is irrational and s is  rational) but that the conclusion is false, i.e., r+s is not irrational.

That means r+s is rational, and so by the definition of rational numbers, r + s = a/b for integers a, b.

Then r = (a/b) - s.  There are two cases for s:

(i) s is irrational: this contradicts the hypothesis that s is rational.

(ii) s is rational: then s = c/d for integers c, d. But then

    \[r = (a/b) - s = (a/b) - (c/d) = (ad - bc)/bd\]

meaning r is rational.  But that contradicts the hypothesis that r is irrational.

Since we get a contra

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