The Polar Form of a Complex Number

The unit circle

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

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From this we can see that the complex numbers

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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

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Exercise: Prove de Moivre's formula

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Now picture a fixed complex number on the unit circle

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

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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

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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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