Question

Find the multiplicative inverse of each of the complex numbers given in the Exercises $11$ to $13$.
$4-3i$

Answer 1

$\Rightarrow$ Multiplicative inverse of $z=z^{-1}$

$\Rightarrow$ Multiplicative inverse of $z=\frac{1}{z}$

Putting $z=4-3i$

Multiplicative inverse of $4-3i=\frac{1}{4-3i}$

$=\frac{1}{4-3i}\times \frac{4+3i}{4+3i}$

$=\frac{4+3i}{(4-3i)(4+3i)}$

$=\frac{4+3i}{(4{)}^2-(3i{)}^2}$ $\left[\right(a+b\left)\right(a-b)=a^2-b^2]$

$=\frac{4+3i}{16-9i^2}$

$=\frac{4+3i}{16-9(-1)}$ $[i^2=-1]$

$=\frac{4+3i}{16+9}$

$=\frac{4+3i}{25}$

$=\frac{4}{25}+\frac{3}{25}i$