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 moving counterclockwise around the circle as the real number moves from left to right. Similarly, the point moves clockwise if decreases. And whether increases or decreases, the point returns to the same position on the circle whenever changes by or by or by 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 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 , 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 .
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!
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Author: Michael O'Neill