Difference between revisions of "Imaginary unit"
(→Introductory) |
(→Intermediate) |
||
Line 22: | Line 22: | ||
===Intermediate=== | ===Intermediate=== | ||
+ | [[1984 AIME Problems/Problem 8]] | ||
+ | |||
===Olympiad=== | ===Olympiad=== | ||
Revision as of 09:55, 27 October 2007
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
.