Chapter 4.3, Problem 39ES

In 35-39 find the mistakes in the “proofs” that the sum of any two rational numbers is a rational number.

39. “Proof: Suppose r and s are rational numbers. If $r+s$ is rational, then by definition of rational $r+s=a/b$ for some integers a and b with $b\ne 0$ . Also since r and s are rational, $r=i/j$ and $s=m/n$ for some integers i,j, m, and n with $j\ne 0$ and $n\ne 0$ . It follows that

$r+s=\frac{i}{j}+\frac{m}{n}=\frac{a}{b}$ ,

which is a quotient of two integers with a nonzero denominator. Hence it is a rational number. This is what was to be shown.”