a) Es ist

${\displaystyle {}(4-7{\mathrm {i} })(5+3{\mathrm {i} })=20+21-35{\mathrm {i} }+12{\mathrm {i} }=41-23{\mathrm {i} }\,.}$

b) Das inverse Element zu ${\displaystyle {}z}$ ist ${\displaystyle {}{\frac {\overline {z}}{z{\overline {z}}}}}$, also ist

${\displaystyle {}z^{-1}={\frac {a-b{\mathrm {i} }}{a^{2}+b^{2}}}={\frac {3-4{\mathrm {i} }}{3^{2}+4^{2}}}={\frac {3}{25}}-{\frac {4}{25}}{\mathrm {i} }\,.}$

${\displaystyle {}z}$

${\displaystyle {}\vert {z}\vert ={\sqrt {25}}=5}$

${\displaystyle {}z^{-1}}$

${\displaystyle {}{\frac {1}{5}}}$