# Puzzles

## Complex squares

For which complex numbers, \(z\), are \(\mathrm{Re}(z^2)\) and \(\mathrm{Im}(z^2)\) both positive?

#### Show answer & extension

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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?