>>11412234>>11412329That's a good question. Here is the main formal definition of a/b.
First we take the integers for granted, including the operations + and * (times).
Then we look at the set of pairs of numbers (a,b) where a,b are integers and b!=0. We define an equivalence relation on this set of pairs ~ by requiring that
(a,b)~(c,d) if and only if ad=bc.
You can verify that this is indeed an equivalence relation. (use the fact that in the integers, ab=0 implies that a or b =0).
Now look at the equivalence classes of the equivalence relation. We write [(a,b)] for the equivalence class of (a,b), and it consists of (a,b), (2a,2b), (3a,3b) and infinitely many other elements.
With this, given integers a and b, with b nonzero, we define a/b to be [(a,b)] and the operations +, * as [(a,b)]+[(c,d)] = [(ad + bc, bd)],
[(a,b)]*[(c,d)] = [(ac,bd)]. You have to verify that these operations are well defined, meaning that taking different representatives of the same equivalence class gives you the same result, but that's easy. These operations define the field of rational numbers Q.
