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The Complex Plane and Polar Coordinates

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Equations

(Eq1)     z = x + yi = r cosθ + ri sin θ

relationship between complex and polar coordinates

Nomenclature

x	real number
y	real number
i	√  −1   , the imaginary unit
z	complex number

Details

It is often useful to picture a complex number z = x + yi in the plane, with x along the horizontal axis and y along the vertical. The xy-plane is then called the complex plane. The following figure shows the locations of some complex numbers on the complex plane.



The triangle in the next figure:



shows that a complex number can be written using polar coodinates as follows:

(Eq1)

z = x + yi = r cosθ + ri sin θ