A common construction is this:

First define natural numbers as sysops did (or you can use peanos axioms).

Then add the negative numbers (I actuallly don't remember how this is done, ig it's usually hand waved as trivial). The negative and natural numbers together make up the integers.

The rationals are introduced as a pair of numbers (a, b) where a is an integer and b is a positive integer. (a, b) is considered the same rational as (c, d) if a * d = b * c.

The reals are finally introduced as sets of rationals with a certain property, namely that if p is in S, then all smaller rationals must also be in S. Edit: there are a few more properties, see https://en.wikipedia.org/wiki/Dedekind_cut