Which vectors define the complex number plane?

1 Answer
Jul 21, 2017

#1 = (1, 0)# and #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# and #i#, #1# being the real unit and #i# the imaginary unit.

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