Which vectors define the complex number plane?

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Which vectors define the complex number plane?

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Answer 1 · 2017-07-21T13:27:05

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

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 $1$ $1$ and $i$ $i$ , $1$ $1$ being the real unit and $i$ $i$ the imaginary unit.

We can consider these as vectors $(1, 0)$ $(1, 0)$ and $(0, 1)$ $(0, 1)$ 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.

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Which vectors define the complex number plane?

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