Question

Are the complex numbers the same as `RR xx RR` ?

Answer 1

A few thoughts...

Explanation:

If `A` and `B` are any two sets, then their cartesian product `A xx B` is the set of ordered pairs `(a, b)` where `a in A` and `b in B`.

As a particular example `RR^2 = RR xx RR` is the set of ordered pairs of real numbers.

We can define addition and multiplication of elements of `RR^2` as follows:

`(a, b) + (c, d) = (a+c, b+d)`

`(a, b) * (c, d) = (ac-bd, ad+bc)`

If we do so, then what we have effectively arrived at is the complex numbers. It has all the properties of a field and the real numbers embed in it via the map `x -> (x, 0)`.

So you could say that the complex numbers are a specific cartesian product `RR xx RR` equipped with specific definitions of addition and multiplication.