Loesung
$$\begin{matrix}
z_1 = 3 + 4 \cdot i &&&&&&z_2 = 7 - 13 \cdot i\cr
\end{matrix}$$
- $z_1 + z_2=3 + 4 \cdot i + 7 - 13 \cdot i = 10 -9 \dot i$
- $z_2 - z_1 =3 + 4 \cdot i - (7 - 13 \cdot i) = -4 +17 \dot i$
- $z_1 \cdot z_2 = (3 + 4\cdot i)\cdot(7 - 13\cdot i)=3\cdot 7 -(4\cdot(-13) +(3\cdot (-13) + 4\cdot7)i =73 - 11 i$
d.
$$
\frac{z_1}{z_2}
=\frac{a_1a_2+b_1b_2}{a_2^2+b_2^2}+\frac{(a_2b_1-a_1b_2)}{a_2^2+b_2^2}i
=\frac{a_1a_2+b_1b_2}{7^2+(-13)^2}+\frac{(a_2b_1-a_1b_2)}{7^2+(-13)^2}i
=\frac{3\cdot7+4(-13)}{49+169}+\frac{(7\cdot4-3\cdot(-13))}{49+169}i
$$
$$
\hskip 7cm
=\frac{21-52}{218}+\frac{28+39}{218}i
=\frac{-31}{218}+\frac{67}{218}i
$$
e. $z_2^2= z_2\cdot z_2 = ( 7 - 13 i)( 7 - 13 i)=49-169 - 91i -91i=-120-182i $