Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums

In the setting of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers, let real and imaginary parts be as in Real and Imaginary Parts of a Complex Number §parts, complex conjugates as in The Complex Conjugate of a Complex Number §conjugate and moduli as in The Modulus of a Complex Number §modulus. Let $z,w\in\mathbb{C}$ and $\lambda\in\mathbb{R}$, let $A$ be a finite set, and let $f:A\to\mathbb{C}$.

¶ $\overline{z+w}=\overline{z}+\overline{w}$, $\overline{zw}=\overline{z}\,\overline{w}$ and $\overline{\overline{z}}=z$; and $\overline{z}=z$ if and only if $z\in\mathbb{R}$.

¶ $\operatorname{Re}(z+w)=\operatorname{Re}z+\operatorname{Re}w$, $\operatorname{Im}(z+w)=\operatorname{Im}z+\operatorname{Im}w$, $\operatorname{Re}(\lambda z)=\lambda\operatorname{Re}z$ and $\operatorname{Im}(\lambda z)=\lambda\operatorname{Im}z$; moreover $z+\overline{z}=2\operatorname{Re}z$ and $z-\overline{z}=(2\operatorname{Im}z)\,i$.

¶ $|z|\ge0$; $|z|=0$ if and only if $z=0$; $z\overline{z}=|z|^{2}=(\operatorname{Re}z)^{2}+(\operatorname{Im}z)^{2}$; and $|\overline{z}|=|-z|=|z|$.

¶ $|zw|=|z|\,|w|$.

¶ If $z\neq0$, then $z^{-1}=\overline{z}/|z|^{2}$ and $|z^{-1}|=1/|z|$.

¶ $|\operatorname{Re}z|\le|z|$, $|\operatorname{Im}z|\le|z|$ and $|z|\le|\operatorname{Re}z|+|\operatorname{Im}z|$.

¶ $|z+w|\le|z|+|w|$ and $\big||z|-|w|\big|\le|z-w|$.

¶ $\overline{\sum_{x\in A}f(x)}=\sum_{x\in A}\overline{f(x)}$, $\operatorname{Re}\sum_{x\in A}f(x)=\sum_{x\in A}\operatorname{Re}f(x)$, $\operatorname{Im}\sum_{x\in A}f(x)=\sum_{x\in A}\operatorname{Im}f(x)$, and $\Big|\sum_{x\in A}f(x)\Big|\le\sum_{x\in A}|f(x)|$.