We now have integers, and can solve any equation of the form $x + a = b$, with a and b being arbitrary natural numbers.  In fact, we can solve those equations with a and b being integers too.

The reason that we needed to create new numbers is that we wanted to solve equations like $x = 2$) but not all.

Consider $3\times 3 = 9 > 7$.  We want a number that’s in between the two, but as long as we stick with integers, there isn’t one.

So, remembering how we defined the integers, we try the same trick again: create an equivalance relation on pairs of integers, such that $(b,a)$.  That’s all assuming that a isn’t zero, of course.

These new numbers are called “rational numbers”, because they represent a ratio of two integers.

We need to define addition and multiplication of rationals.  We want them to be consistent with the definition we already have for integers, so $(a,1) \times (b,1) = (ab,1)$ (I’m abusing notation here, writing (a,1) when I mean “the equivalence class that contains (a,1).  It’s much easier, and mathematicians do this all the time.  It doesn’t cause any problems as long as you’re paying attention).

It’s not obvious how to extend this to more general cases, so let’s go back to the equation we used as motivation.  Say we have two numbers, x and y, defined as the solutions to $acx + acy = ac(x+y) = ad+bc$.

And that gives us a definition for addition: if x and y are rational numbers, with x containing the pair (b,a) and y the pair (d,c), then x+y is the rational containing $(ad+bc,ac)$.

We can do something similar for multiplication.  Again, I’ll leave the details to you.

Are there any more numbers that we need to invent?  Surely not, because there isn’t any more room to look for them.  We had all that space less than zero in the natural numbers, so we had to invent integers to fill it.  And we had all that space between integers to fill with rationals.  But in between the rationals, there are just more rationals: $\frac{1}{2}(x+y)$ is between x and y, and there are infinitely many more along with it.  If there are any more numbers, there’s nowhere for them to go.  We have all the numbers we need.

At least, that was the thinking about two and a half thousand years ago.  Then something shocking happened.  I’m sure you already know what that was, but it’s late and I’m tired, so I’ll leave that story for next time.