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.