poveli1e

2022-01-21

Which vectors define the complex number plane?

pripravyf

$1=\left(1,0\right)$ and $i=\left(0,1\right)$
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 $\left(1,0\right)$ and $\left(0,1\right)$ in ${\mathbb{R}}^{2}$.
In fact, if you start from a knowledge of the real numbers $\mathbb{R}$ and want to describe the complex numbers $\mathbb{C}$, then you can define them in terms of pairs of real numbers with arithmetic operations:
$\left(a,b\right)+\left(c,b\right)=\left(a+c,b+d\right)$ (this is just addition of vectors)
$\left(a,b\right)\cdot \left(c,b\right)=\left(ac-bd,ad+bc\right)$
The mapping $a\to \left(a,0\right)$ 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 $\left(a,0\right)\cdot \left(c,d\right)=\left(ac,ad\right)$
which is effectively scalar multiplication.

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