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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