Are there numbers other than real numbers?

The German mathematician Leopold Kronecker once said: “The whole numbers are made by God, everything else is man-made!” Indeed, mathematicians have invented new sets over the centuries that expanded the existing number sets (also called sets of numbers).

For example, the natural numbers (N) are contained in the integers (Z), which in turn are contained in the set of quotients of integers, called the rational numbers (Q). These are in turn in the (many more elements containing) field of the real numbers (R). You probably already knew this.

I use the field name here. That is a set with addition and multiplication, which satisfies many beautiful properties (such as the distrubitivity). Q and R are fields. There is another (field) extension of R, which is widely used, and that is C, the field of complex numbers. It’s an interesting collection that you’ll find more about on the internet and in third grade math textbooks.

Then you can actually go further, but it just gets harder. For example, sets that expand C are the quaternions H and the octonions O, but they are not fields. By adding elements to C we lose the beautiful properties that Q, R and C did have as a field: with the quaternions there is no commutativity (a b ≠ b a) and with the octions no associativity ( a (b c) ≠ (a b) c ).

N, Z, Q and R have numbers that we can order: there is a “<" that says when one number is greater or less than another. From C onwards we also no longer have this property: we usually cannot tell whether one of two complex numbers is greater than the other (unless they are both in R, of course).

There are many other ways to expand numbers, for example by placing different kinds of “infinite” on the number axis (cardinal numbers, ordinal numbers), or for example by adding “infinitely small” elements, which are just larger than 0, but smaller than any real number (infinitesimals → surreal numbers). I’ll leave out the technical details, but an interesting starting point is Wikipedia: complex number. You read about the most important extension of R and the block on the right gives you a pack of links to other number sets!

Lots of fun!