Puzzles
Complex squares
For which complex numbers, \(z\), are \(\mathrm{Re}(z^2)\) and \(\mathrm{Im}(z^2)\) both positive?
Show answer & extension
Hide answer & extension
Any complex number can be written in the form \(z=re^{i\theta}\).
This gives that \(z^2=r^2e^{2i\theta}\), which will have positive real and complex parts when \(0+2\pi n < 2\theta < \frac{\pi}{2}+2\pi n\).
This will occur when \(0 < \theta < \frac{\pi}{4}\) and \(\pi < \theta < \frac{5\pi}{4}\).
A complex number \(z\) falls in these regions when \(|\mathrm{Re}(z)|>|\mathrm{Im}(z)|\) and \(\mathrm{sign}(\mathrm{Re}(z))=\mathrm{sign}(\mathrm{Im}(z))\).
Extension
For which complex numbers, \(z\), are \(\mathrm{Re}(z^3)\) and \(\mathrm{Re}(z^3)\) both positive?
![](javascript:showlimage("argand.png"))