Circular definition of rationals.

Question

If we define rational numbers as

A rational number is any number that can be fraction $\frac pq$ of two integers $p$ and $q$, with the denominator $q$ not equal to zero.

But integers themselves are rational numbers of the form $\frac p1$ where $p$ is an integer. So the definition becomes circular. How do we avoid this?

Answer 1

The standard way consists of:

• To define integers in a way that doesn't use rational numbers.
• To define a rational number as an equivalence class $\bigl[(a,b)\bigr]$ of elements of $\mathbb Z\times(\mathbb Z\setminus\{0\})$, where the equivalence relation is$$(a,b)\sim(c,d)\text{ iff }ad=bc.$$

This approach also avoids another problem in what you wrote: it does not assume that there is a general concept of number.