(The rational numbers, that is.)

Suppose we know about integers but not about rational numbers. That means we can solve equations like x+5=3, but not necessarily ones like 3x=7. (Of course we can solve some such equations; 3x=6, for instance.) As when constructing the integers, we would like to build on our knowledge of the integers to make a more general sort of number for which we can solve these equations.

We use exactly the same idea as when constructing the integers: consider pairs (a,b) where $b\neq{}0.$ (The idea is that this means a/b.) Say that (a,b)=(c,d) exactly when ad=bc. (That is: say that they are "equivalent" when this holds, and then work with equivalence classes of pairs instead of just with pairs.)

The rules for doing arithmetic with these are exactly the ones you learned in school for doing arithmetic with fractions. For instance, (a,b)+(c,d)=(ad+bc,bd).

As before, it turns out that the arithmetic operations are well defined; if you apply them to different elements of the same equivalence classes, the result is in the same equivalence class.

When constructing the integers we found that (a,a) "is zero"; here we find that (a,a) "is 1".

When constructing the integers we found that (b,a) "is the negative of" (a,b); here we find that (b,a) "is the reciprocal of" (a,b).

When constructing the integers we found that (a,0) "is a"; here we find that (a,1) "is a".

Our collection of equivalence classes of pairs of integers acts just as we would like the rational numbers to act.

So, now we can solve any linear equation ax+b=c. Want to solve more complicated equations? Well, then it's time to think about constructing the reals...

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