Powers of Complex Numbers
Positive Whole Number Powers
As an application of the rule for multiplying together complex numbers in polar form, it is
a simple matter to multiply a complex number by itself any desired number of times.
Suppose that
Then,
and, by continuing this process,
This result is due to De Moivre, but other aspects of it will need to be discussed before we
may formalise what is called â€œDe Moivreâ€™s Theoremâ€.
Example
Negative Whole Number Powers
If n is a negative whole number, we shall suppose that
n = −m,
where m is a positive whole number.
Thus, if
then
In more detail,
giving
But −m = n, and so
showing that the result of the previous section remains true for negative whole number
powers.
Example
