NOTE: The ``existence'' of complex numbers wasn't generally accepted until people got used to a geometric interpretation of them.
Finding the polar form of a complex number is exactly the same problem as finding polar coordinates of a point in rectangular coordinates. The only hard part is figuring out what is.
If we write complex numbers in rectangular form, their sum is easy to compute:
For example, the power of a singular complex number in polar form is easy to compute; just power the and multiply the angle.
|EXAM 1: Wednesday 7:00-7:50pm in Pepper Canyon 109 (!)
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Given a point in the plane, we can also view it as or in polar form as . Polar form is great since it's good for multiplication, powering, and for extracting roots:
Theorem 4.3.6 (De Moivre's) For any integer we have
Since we know how to raise a complex number in polar form to the th power, we can find all numbers with a given power, hence find the th roots of a complex number.
An application of De Moivre is to computing and in terms of and . For example,
William Stein 2006-03-15