Difference between revisions of "Imaginary unit"
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=== Introductory === | === Introductory === | ||
*Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math> ([[Imaginary unit/Introductory|Source]]) | *Find the sum of <math>i^1+i^2+\ldots+i^{2006}</math> ([[Imaginary unit/Introductory|Source]]) | ||
+ | *Find the product of <math>i^1 \times i^2 \times \cdots \times i^{2006}</math>. ([[Imaginary unit/Introductory|Source]]) | ||
+ | |||
===Intermediate=== | ===Intermediate=== | ||
===Olympiad=== | ===Olympiad=== |
Revision as of 09:52, 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
.