Imaginary unit
The imaginary unit, , is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as
. Any complex number can be expressed as
for some real numbers
and
.
Contents
Trigonometric function cis
- Main article: cis
The trigonometric function is also defined as
or
.
Series
When is used in an exponential series, it repeats at every four terms:
This has many useful properties.
Use in factorization
is often very helpful in factorization. For example, consider the difference of squares:
. With
, it is possible to factor the otherwise-unfactorisable
into
.
Problems
Introductory
Intermediate
- The equation
has complex roots with argument
between
and
in the complex plane. Determine the degree measure of
. (Source)