Member-only story

A Tricky Complex Numbers Proof

But easy when you know how!

Russell Lim
Nice Math Problems
5 min readJul 3, 2024

--

All images by author

Draw a square in the complex plane with vertices at 1, i, -1 and -i. Prove that for any two complex numbers u and v within the square, their product uv also lies within the square.

When I first saw this problem, I thought it was trivial. And it would be, if it were a circle…

When you multiply complex numbers, it’s usually easiest to use the polar form. That’s where the complex number is defined in terms of its modulus or distance from the origin (r) and the angle from the horizontal axis (θ). To multiply complex numbers in polar form, you just multiply the moduli (distances) and add the angles.

Because we know both our complex numbers have modulus less than 1, the product of the two moduli will also be less than 1. This guarantees that the product will be inside a circle of radius 1.

But it does not guarantee that the product is within the square!

So the problem is not as simple as I thought…

--

--

Nice Math Problems
Nice Math Problems

Published in Nice Math Problems

Follow this publication if you enjoy ✨nice✨ mathematics problems.

Russell Lim
Russell Lim

Written by Russell Lim

I teach high school mathematics in Melbourne, Australia. I like thinking about interesting problems and learning new things.

No responses yet