What are complex numbers?

A lot of people find complex numbers mysterious and counter-intuitive.

Defining i to be "the square root of minus one" is about as sane as defining it to be the square root of the colour blue.

The first people who thought about it followed that approach, and were rightly scared stiff and confused by it. Even Euler made trivial mistakes. They called them "Imaginary Numbers" because they seemed to be useful for calculating, but no-one really believed that they existed.

 
Argand came up with the right way of thinking about it:

Take all the tuples (x,y). (Where x and y are just integers). Define on them addition and multiplication rules.

Oh look! There's a big system of these things, and embedded within it is a sub-system which works exactly the same as the integers and their rules.

Since they're exactly the same for all practical purposes, we may as well forget about the difference and say that we'll write (a,0) as a and (0,b) as ib, and (a,b) as a+ib

And although none of the pairs in the  sub-system square to be (-1,0), also known as -1, there are two other things that do! 

So now we know that both (0,1) and (0, -1), otherwise known as 1i and -1i, or i and -i, are square roots of -1.

No slight of hand or magical thinking necessary, and we have the complex integers.


As it happens, we can do the same thing starting from the reals, to form something that embeds the reals. But the reals really are dark and mysterious and need to be brought about by a kind of magic.

According to me. Some people think they're as real as the complex numbers.