Properties of conjugate and norm

Complex numbers: Construction of complex numbers and complex arithmetic

Properties of conjugate and norm

Properties of complex conjugation

If $z$ and $w$ are two complex numbers, then:

$\begin{aligned}\overline{z+w}&=\overline{z}+\overline{w}\\ \\ \overline{z-w}&=\overline{z}-\overline{w} \\ \\ \overline{z\cdot w}&=\overline{z}\cdot \overline{w}\\ \\ \overline{\left(\frac{z}{w}\right)}&=\frac{\overline{z}}{\overline{w}}\\ \\ \overline{\overline{z}}&=z\\ \\ \text{Re}(z)&= \frac{z+\overline{z}}{2}\\ \\ \text{Im}(z)&= \frac{z-\overline{z}}{2i} \end{aligned}$

Properties of the norm

If $z$ and $w$ are two complex numbers, then:

$\begin{aligned} |z+w|\; &\le |z|+|w|\\ \\ |z\cdot w|&=|z|\cdot |w|\\ \\ \left|\frac{z}{w}\right|&=\frac{|z|}{|w|}\\ \\ |\overline{z}|&= |z|\end{aligned}$

Any complex number $\zeta$ is a solution of the following quadratic equation with reël coefficients $a=\text{Re}(\zeta)$ and $r^2=|\zeta|^2$ :

$z^2-2a\cdot z+r^2 = 0\tiny.$

Any complex number $\zeta$ is of course a solution of the following quadratic equation

$(z-\zeta)\cdot(z+\overline{\zeta})=0$ We can expand the left-hand side as follows:

$\begin{aligned} (z-\zeta)\cdot(z+\overline{\zeta}) &= z^2-(\zeta+\overline{\zeta})\cdot z-\zeta\cdot \overline{\zeta} \\ \\ &= z^2 + 2\cdot\text{Re}(\zeta)\cdot z+|\zeta|^2 \\ \\ &= z^2-2a\cdot z+r^2 \end{aligned}$ Alle coefficients of this second degree polynomial in $z$ are real numbers.