Imaginary unit: Difference between revisions
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The '''imaginary unit''', <math>i=\sqrt{-1}</math>, 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 <math>1 \text{cis } \left(\frac{\pi}{2}\right)</math>. | The '''imaginary unit''', <math>i=\sqrt{-1}</math>, 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 <math>1 \text{cis } \left(\frac{\pi}{2}\right)</math>. Any [[complex number]] can be expressed as <math>a+bi</math> for some real numbers <math>a</math> and <math>b</math>. | ||
==Trigonometric function cis== | ==Trigonometric function cis== | ||
Revision as of 13:49, 26 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 
.
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
- Find the sum of
(Source)
(
