The Polar Form of a Complex Number

The unit circle

The fundamental trigonometric identity (i.e the Pythagorean theorem) is


From this we can see that the complex numbers


are points on the circle of radius one centered at the origin.

Think of the point tex2html_wrap_inline11 moving counterclockwise around the circle as the real number tex2html_wrap_inline13 moves from left to right. Similarly, the point moves clockwise if tex2html_wrap_inline13 decreases. And whether tex2html_wrap_inline13 increases or decreases, the point returns to the same position on the circle whenever tex2html_wrap_inline13 changes by tex2html_wrap_inline21 or by tex2html_wrap_inline23 or by tex2html_wrap_inline25 where k is any integer.

Exercise: Verify that


Exercise: Prove de Moivre's formula


Now picture a fixed complex number on the unit circle


Consider multiples of z by a real, positive number r.


As r grows from 1, our point moves out along the ray whose tail is at the origin and which passes through the point z. As r shrinks from 1 toward zero, our point moves inward along the same ray toward the origin. The modulus of the point is r. We call the angle tex2html_wrap_inline32 which this ray makes with the x-axis, the argument of the number z. All the numbers rz have the same argument. We write


Just as a point in the plane is completely determined by its polar coordinates tex2html_wrap_inline40 , a complex number is completely determined by its modulus and its argument.

Notice that the argument is not defined when r=0 and in any case is only determined up to an integer multiple of tex2html_wrap_inline44 .

Why not just use polar coordinates? What's new about this way of thinking about points in the plane?

We now have a geometric interpretation of multiplication!

[Algebra] [Complex Variables]
[Geometry] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra]

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Author: Michael O'Neill

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