Sorry for the delay; physics camp has been killing me lately.

In the previous Constructing the Reals post, I derived the non-negative rationals from the natural numbers. Now we’ll finally get symmetry by adding negative number to get the signed rational numbers. The naive way to do so is to simply say that a signed rational number is a rational number with a sign. This does make multiplication easy, but it makes addition fairly difficult to define. So we take an alternative definition that makes addition, subtraction, and multiplication all easy; the only difficult part is division.

Define a signed rational number x as an ordered pair (|a|,|b|) of unsigned rational numbers, identifying the unsigned rational x with the signed rational (|x|, |0|). We define (|a|, |b|) = (|c|, |d|) if and only if |a| + |d| = |b| + |c|. Addition is simply pairwise: (|a|, |b|) + (|c|, |d|) = (|a| + |c|, |b| + |d|). Multiplication isn’t, however, but by analogy with algebra, we can easily guess that (|a|, |b|)(|c|, |d|) = (|a||c| + |b||d|, |a||d| + |b||c|). As it turns out, we can finally define subtraction as well: (|a|, |b|) – (|c|, |d|) = (|a|, |b|) + (|d|, |c|).

Division, however, is harder; we have to define it by saying that when |w/x| > |y/z|, the inverse of (|w/x|, |y/z|), written (|w/x|, |y/z|)^-1, is equal to (|zx/(wz-xy)|, |0|), where we define a – b for non-negative integers a, b to be the unique integer c such that a = b + c (although I should probably prove that such a c exists, that’s not the point). On the other hand, if |w/x|< |y/z|, the inverse is (|0|, |zx/(xy-wz)|). If |w/x|= |y/z|, of course, then (|w/x|, |y/z|) has no inverse (since it would be like the inverse of 0). But using inverses, we can define x/y = x y^-1. It can be shown that multiplication and addition under these definitions follow the rules you expect them to, as do addition and subtraction. In other words, they form what is called a field, which can be roughly defined as ‘anything with addition, subtraction, multiplication, and division that behave like they do for real numbers’. Other common fields are the complex numbers and the real numbers themselves, although there are obviously others.

Tomorrow, I’ll finally get to everyone’s favorite: the reals! And this time I mean it; I’ve got the post queued up in WordPress and everything. After that, I’ll show you a different method that both gets you to the reals faster (we get them all at once!) and extends them to both infinitely large and infinitely small numbers, and even numbers that are neither positive, negative nor zero. But that’ll have to wait, and with graduation and physics camp and everything, I don’t know when I’ll start on that series.