More Notation and Terminology

It's already been mentioned that the rectilinear coordinates of a complex number in the complex plane are called the real part and imaginary part, respectively.

We also have special notation and various names for the polar coordinates of a complex number :

The complex conjugate of is denoted (or ) and is defined by

where, of course, .

In general, you can always obtain the complex conjugate of any expression by simply replacing with . In the complex plane, this is a vertical flip about the real axis; i.e., complex conjugation replaces each point in the complex plane by its mirror image on the other side of the axis.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.