Are complex numbers two dimensional or one dimensional?

Question

Complex numbers are represented as:

z = x + yi

This gives the impression that complex numbers are a real component plus an imaginary component. However, when doing math with complex numbers, they are represented as 2-D vectors like in this picture:

Complex Vector

Are complex numbers one or two dimensional?

If they are one dimensional, then why do we do math with them and represent them as vectors?

If they are two dimensional, then why are they represented as one number: "x + yi" instead of a coordinate pair: "(x, yi)"

I know this has been asked a zillion times, but most of the answers I've found online don't explain the subject well. The best explanation I've found so far was here on Quora: https://www.quora.com/Are-complex-numbers-2-dimensional

Answer 1

You can consider $\mathbb{C}$ as a $\mathbb{C}$vector space in this case complex numbers is a $1$-dimensional vector space over $\mathbb{C}$.

You can consider $\mathbb{C}$ as a $\mathbb{R}$vector space in this case complex numbers is a $2$-dimensional vector space over $\mathbb{R}$.

Answer 2

Quoting from https://en.m.wikipedia.org/wiki/Complex_number#Construction_as_ordered_pairs:

William Rowan Hamilton introduced the approach to define the set C of complex numbers as the set R^2 of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed:

\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}

It is then just a matter of notation to express (a, b) as a + bi.