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notes on complex planes
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Polar & Exponential FormMost people are familiar with complex numbers in the form z = a + bi, howeve
alternate forms that are useful at times. In this section we’ll look at both of those
couple of nice facts that arise from them.
Geometric Interpretation
Before we get into the alternate forms we should first take a very brief look at a
interpretation of complex numbers since this will lead us into our first alternate f
Consider the complex number z = a + bi. We can think of this complex numbe
point (a, b) in the standard Cartesian coordinate system or as the vector that st
and ends at the point (a, b). An example of this is shown in the figure below.
In this interpretation we call the x-axis the real axis and the y-axis the imagina
call the xy-plane in this interpretation the complex plane.
Note as well that we can now get a geometric interpretation of the modulus. Fr
above, we can see that |z| = √a2 + b2 is nothing more than the length of the
using to represent the complex number z = a + bi. This interpretation also tell
inequality |z1 | < |z2 | means that z1 is closer to the origin (in the complex plane
Polar Form
Let’s now take a look at the first alternate form for a complex number. If we thin
complex number z = a + bi as the point (a, b) in the xy-plane we also know t
represent this point by the polar coordinates (r, θ), where r is the distance of th
origin and θ is the angle, in radians, from the positive x-axis to the ray connecti
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