Chapter 1, Problem 71RE

To determine

To calculate: The complex number $\frac{4}{1-2i}$ in standard form.

The complex number $\frac{4}{1-2i}$ in the standard form is $\frac{4}{5}+\frac{8}{5}i$.

Calculation:

Consider the complex number,

$\frac{4}{1-2i}$

Multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of $\left(1-2i\right)\text{ is }\left(1+2i\right)$.

$\begin{array}{c}\frac{4}{\left(1-2i\right)}\times \frac{\left(1+2i\right)}{\left(1+2i\right)}=\frac{4+8i}{1-{\left(4i\right)}^2}\\ =\frac{4+8i}{1-\left(-4\right)}\\ =\frac{4}{5}+\frac{8}{5}i\end{array}$

The complex number $\frac{4}{1-2i}$ in standard form is $\frac{4}{5}+\frac{8}{5}i$.