# Which vectors define the complex number plane?

##### 1 Answer

#### 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

We can consider these as vectors

In fact, if you start from a knowledge of the real numbers

#(a, b) + (c, d) = (a+c, b+d)" "# (this is just addition of vectors)

#(a, b) * (c, d) = (ac-bd, ad+bc)#

The mapping

Note that:

#(a, 0) * (c, d) = (ac, ad)#

which is effectively scalar multiplication.