# Which vectors define the complex number plane?

Jul 21, 2017

$1 = \left(1 , 0\right)$ and $i = \left(0 , 1\right)$

#### 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 $\left(1 , 0\right)$ 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:

$\left(a , b\right) + \left(c , d\right) = \left(a + c , b + d\right) \text{ }$ (this is just addition of vectors)

$\left(a , b\right) \cdot \left(c , d\right) = \left(a c - b d , a d + b c\right)$

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:

$\left(a , 0\right) \cdot \left(c , d\right) = \left(a c , a d\right)$

which is effectively scalar multiplication.