Must the opposite of a rational number be a rational number?

Answer 1

No. A rational number is one that may be represented as a simple multiple, or a division. Such as one times seven, or three times seven; OR one divided by seven, or three divided by seven. Or similar.

A rational number, if a fraction, when multiplied by the divisor of the fraction, will give a simple whole number. e.g. 7/5, when multiplied by 5 becomes the whole number 7.

An irrational number is one that cannot be formed by a simple ratio, or a simple division. e and pi are a couple of common Irrational Numbers. [irrational demonstrates where the name comes from.]

No number can be found which will convert them, by multiplication, into a whole number.

Answer 2

The answer depends on what you mean by "opposite". The additive inverse of any rational number must be rational. The multiplicative inverse of a rational number is rational only if the number is non-zero. If it is zero, its "opposite" is not defined.