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>. | ||
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* [[Complex numbers]] | * [[Complex numbers]] | ||
* [[Geometry]] | * [[Geometry]] | ||
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Revision as of 13:36, 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 
.
Problems
Introductory
- Find the sum of
Solutions
Introductory
- Let's begin by computing powers of
.
- We can now stop because we have come back to our original term. This means that the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences
have a sum of zero (k is a natural number). Since 
, the original series sums to the first two terms of the powers of i, which equals 
.
.
have a sum of zero (k is a natural number). Since 
, the original series sums to the first two terms of the powers of i, which equals 
.
