Saturday, November 05, 2011

Where (in the complex plane) are the solutions of an equation?

First of all: a polynomial of degree N has exactly N roots (in the complex plane) [ref]
Second: since for every root there is also its complex-conjugate [ref], we can only have some combinations...

DegreeRootsPossible Combinations
11Real Root
22Real Roots, or 2 Complex Roots
33Real Roots, or 1 Real and 2 Complex Roots
44Real Roots, or 2 Real and 2 Complex Roots, or 4 Complex Roots
etc etc!

Eventually, complex numbers are defined like this.

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