Question

Which vectors define the complex number plane?

Answer 1

`1 = (1, 0)` and `i = (0, 1)`

Explanation:

The complex number plane is usually considered as a two dimensional vector space over the reals. The two coordinates represent the real and imaginary parts of the complex numbers.

As such, the standard orthonormal basis consists of the number `1` and `i`, `1` being the real unit and `i` the imaginary unit.

We can consider these as vectors `(1, 0)` and `(0, 1)` in `RR^2`.

In fact, if you start from a knowledge of the real numbers `RR` and want to describe the complex numbers `CC`, then you can define them in terms of pairs of real numbers with arithmetic operations:

`(a, b) + (c, d) = (a+c, b+d)" "` (this is just addition of vectors)

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

The mapping `a -> (a, 0)` embeds the real numbers in the complex numbers, allowing us to consider real numbers as just complex numbers with a zero imaginary part.

Note that:

`(a, 0) * (c, d) = (ac, ad)`

which is effectively scalar multiplication.