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 \mathrm{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 \mathrm{cis } \left(\frac{\pi}{2}\right)</math>. |
| + | ==Trigonometric function cis== | ||
| + | {{main|cis}} | ||
| + | The trigonometric function <math>\cis x</math> is also defined as <math>e^{ix}</math> or <math>\sin x+i(\cos x)</math>. | ||
| + | |||
| + | ==Series== | ||
| + | When <math>i</math> is used in an exponential series, it repeats at every fifth term: | ||
| + | #<math>i^1=\sqrt{-1}</math> | ||
| + | #<math>i^2=\sqrt{-1}\cdot\sqrt{-1}=-1</math> | ||
| + | #<math>i^3=-1\cdot i=-i</math> | ||
| + | #<math>i^4=-i\cdot i=-i^2=-(-1)=1</math> | ||
| + | #<math>i^5=1\cdot i=i</math> | ||
| + | This has many useful properties. | ||
==Problems== | ==Problems== | ||
=== Introductory === | === Introductory === | ||
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* [[Complex numbers]] | * [[Complex numbers]] | ||
* [[Geometry]] | * [[Geometry]] | ||
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[[Category:Constants]] | [[Category:Constants]] | ||
Revision as of 14:44, 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 
.
Trigonometric function cis
- Main article: cis
The trigonometric function $\cis x$ (Error compiling LaTeX. Unknown error_msg) is also defined as 
or 
.
Series
When 
is used in an exponential series, it repeats at every fifth term:
This has many useful properties.
Problems
Introductory
- Find the sum of
(Source)
(
