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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

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    $\begingroup$ It depends on the basis field. It is 1-dimensional with respect to itself $\mathbb{C}$. But it is 2-dimensional wrt $\mathbb{R}$ $\endgroup$ May 14, 2019 at 20:43
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    $\begingroup$ What distinguishes a complex number from a typical $2D$ real vector is the fact that you can multiply complex numbers to get another complex number: $(a+bi)(c+di) = (ac-bd)+(ab+bc)i$ $\endgroup$
    – Henry
    May 14, 2019 at 20:44
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    $\begingroup$ As a perspective on what people have said, sometimes we can regard mathematical objects like the complex numbers as things in themselves which have an inherent unity. Other times we can decompose them into pieces in an interesting way. Sometimes we can do both, and quite often that shows that something interesting is going on. $\endgroup$ May 14, 2019 at 20:56
  • $\begingroup$ The complex numbers are infinite dimensional over $\Bbb Q$. Yes, this is confusing. Yes, this is on purpose. Yes, this is to demonstrate to you that context is everything. $\endgroup$
    – Asaf Karagila
    May 14, 2019 at 23:04

2 Answers 2

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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 $\mathbb{R}$.

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  • $\begingroup$ (+1) Just what I was about to write. $\endgroup$ May 14, 2019 at 20:45
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    $\begingroup$ Yes, but if yo think of it as a vector space you have to be careful, because it is not an inner product space in the usual sense. That is, the "dot product" between tow complex "vectors" is not the usual complex multiplcation of the two numbers represented. $\endgroup$ May 14, 2019 at 20:48
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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.

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