Properties of Complex Conjugation and Modulus

Let $\mathbb{C}$ be the field of complex numbers with imaginary unit $i$, let $\mathbb{R}$ be the set of real numbers, and let $z,w\in\mathbb{C}$. Sums, products and inverses are those of the field $\mathbb{C}$, and $z-w$ abbreviates $z+(-w)$; for a real number $x$, $2x$ abbreviates $x+x$, and $x^{2}$ abbreviates $x\cdot x$. Real and imaginary parts are as in that definition, $\overline{z}$ denotes the complex conjugate of $z$, and $|z|$ its modulus. Real numbers are elements of $\mathbb{C}$ by condition 1 of The Complex Numbers, and by claim 1 of Canonical Form and Arithmetic of Complex Numbers their identities and inverses in $\mathbb{C}$ are the real ones. Then the following hold.

1. (Conjugation is an involutive field automorphism)

and $\overline{z}=z$ holds if and only if $z\in\mathbb{R}$.

2. (Real and imaginary parts from the conjugate)

3. (Modulus and conjugate) $z\overline{z}=|z|^{2}$ and $|\overline{z}|=|z|$. Moreover $|z|=0$ if and only if $z=0$.

4. (Multiplicativity) $|zw|=|z|\,|w|$, the product on the right being that of real numbers.

5. (Inverse) If $z\neq0$, then the multiplicative inverse of $z$ in $\mathbb{C}$ is $z^{-1}=\bigl(|z|^{-2}\bigr)\,\overline{z}$, where $|z|^{-2}$ denotes the real number $1/|z|^{2}$.

6. (Component bounds) $\operatorname{Re}z\le|z|$, $-\operatorname{Re}z\le|z|$, $\operatorname{Im}z\le|z|$ and $-\operatorname{Im}z\le|z|$, these being inequalities between real numbers.

7. (Triangle inequality) $|z+w|\le|z|+|w|$.

8. (Modulus of a real number) For every real number $a$, viewed as an element of $\mathbb{C}$, the modulus $|a|$ equals $a$ if $0\le a$, and equals $-a$ otherwise.

9. (The modulus metric) The function $d_{\mathbb{C}}$ assigning to each pair $(z,w)$ of complex numbers the real number $d_{\mathbb{C}}(z,w)=|z-w|$ is a metric on $\mathbb{C}$.