Corollary of De Moivre's Theorem

Let \(z,z_1,z_2\) be complex numbers and \(n\in\mathbb Z\). Quotients require \(z_2\ne0\); zero or negative powers require \(z\ne0\). Each argument identity additionally requires every complex number whose argument appears to be nonzero.

Then, regarding modulus, we have:

1. \(|z_1 z_2| = |z_1| |z_2|\)
2. \(\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}\)

3. \(|z^n| = |z|^n\)

Then, regarding argument, we have:

4. \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\) modulo \(2\pi\)
5. \(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\) modulo \(2\pi\)
6. \(\arg(z^n) = n \arg(z)\) modulo \(2\pi\)

Remarks
The second group of results concerning argument deserves some elaboration.

For example, consider \(z = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\).

\[z^{12} = \cos 2\pi + i \sin 2\pi = 1\]

We know that \(\arg(z^{12}) = \arg(1) = 0\), which is not \(2\pi\). Although \(2\pi \neq 0\), we say that

\[2\pi = 0 \pmod{2\pi}\]

This is to say that up to an addition or subtraction of integral multiple of \(2\pi\) they are the same angle. In general we have that \(\alpha = \beta \pmod{2\pi}\) if and only if there exists an integer \(k\) such that \(\alpha - \beta = 2k\pi\).