Question

Find the multiplicative inverse of $3+4i$ .

Answer 1

Find the multiplicative inverse of complex number :

Let $z=3+4i$

The multiplicative inverse of a complex number is its reciprocal.

So, $\frac{1}{z}=\frac{1}{3+4i}$

$$\begin{gathered} \Rightarrow z^{-1}=\frac{1}{3+4i}\times \frac{3-4i}{3-4i} \\ \Rightarrow z^{-1}=\frac{3-4i}{(3-4i)(3+4i)} \\ \Rightarrow z^{-1}=\frac{3-4i}{3^2-{\left(4i\right)}^2} \\ \Rightarrow z^{-1}=\frac{3-4i}{9+16}\{i=\sqrt{-1}\} \\ \Rightarrow z^{-1}=\frac{3}{25}-\frac{4i}{25} \end{gathered}$$

Therefore, the multiplicative inverse of $3+4i$ will be $\frac{3}{25}-\frac{4i}{25}$.