Difference between revisions of "Imaginary unit"
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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 \ | + | 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>. |
==Trigonometric function cis== | ==Trigonometric function cis== | ||
{{main|cis}} | {{main|cis}} | ||
− | The trigonometric function <math>\ | + | The trigonometric function <math>\text{cis } x</math> is also defined as <math>e^{ix}</math> or <math>\sin x+i(\cos x)</math>. |
==Series== | ==Series== |
Revision as of 14:48, 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
.
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
- Find the sum of
(Source)