Appendix C. Complex Numbers

Each complex number z is a point in the complex plane, which is spanned by the real axis and the imaginary axes:

image with no caption

Two coordinate systems are commonly used for a (two-dimensional) plane: Cartesian and polar coordinates. For every complex number there exist two equivalent representations:

Complex Numbers

Here

Complex Numbers

is the “imaginary unit.”

We can transform between those representations as follows:

    Real part
    Imaginary part
    Magnitude
    Phase

Basic Operations

Complex numbers are added and multiplied component by component while taking into account that i2 = –1. If z1 = x1 + iy1 and z2 = x2 + iy2, then

z1 + z2 = (x1 + x2) + i(y1 + y2)
z1z2 = (x1x2y1y2) + i(x1y2 + x2y1)

Each complex number z has a “complex conjugate,” denoted , which is the same as except that the sign ...

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