Polar Form

Let and be arbitrary complex numbers. Not that and , so:

Now we define 3 points on the complex plane: let be the origin , B be the point corresponding to , and be the point corresponding to . In triangle , let be the side length of , be the side length of , and be the length of a side of . A standard fact from geometry is that the length of any side of a triangle is less than or equal to the sum of the lengths of the other 2 sides of the triangle, so

By the Pythagorean theorem, we have , , and . Substituting these into the inequality above gives

Express in polar form

It follows that

So the polar form evaluates to: