For $z \in \mathbb{C}\setminus \{0\}$, is $|1/z| = 1/|z|$? I have been trying to prove that for $z \in \mathbb{C}$ and $z \neq 0$ that $|1/z| = 1/|z|$ but with no success.  Not all the properties of real numbers can be applied to complex numbers, of course.
Yes.
Note that $1/|z|$ is, by definition, the real multiplicative inverse of $|z|$, so by uniqueness of such an inverse, since
$$|1/z|\times |z|=|z/z|=1,$$
we have $|1/z|=1/|z|$.
The main problem that you have here is that you have been trying to re-invent the wheel, rather than actually consulting a Complex Analysis textbook.  The following results are based on Chapter 1 of &quot;An Introduction To Complex Function Theory&quot; [Bruce Palka].

For $z,w \in \mathbb{C}, |z| \times |w| = |z \times w|.$

For $z = (x + iy) \in \mathbb{C}$, let $\overline{z}$ denote $(x - iy).$

Then $\frac{1}{z} = \frac{\overline{z}}{z \times \overline{z}}.$

$z \times \overline{z} = |z|^2.$

$|\overline{z}| = |z|$.


The result that you are trying to prove is a direct consequence of the above intermediate results.