Question

How do you rationalize imaginary number $?$

Answer 1

To rationalize imaginary numbers we need to multiply them by their conjugate:

Suppose ,$a+ib$is an imaginary number or a complex number, then we need to multiple it by its conjugate ,

that is., $a-ib$ to rationalize.

$\left(a+ib\right)\times \left(a-ib\right)=a^2-b^2$

Therefore, the product of an imaginary number and its conjugate is not an imaginary number.

Example:

Let $\frac{6-7i}{1-2i}$

$\Rightarrow$$\frac{6-7i}{1-2i}\times \frac{1+2i}{1+2i}$

$$\begin{gathered} =\frac{\left(6-7i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)} \\ =\frac{\left(6\right)\left(1\right)+\left(6\right)\left(2i\right)-\left(7i\right)\left(1\right)-\left(7i\right)\left(2i\right)}{1^2-{\left(2i\right)}^2} \\ =\frac{6+12i-7i-14i^2}{1^2-{\left(2i\right)}^2} \end{gathered}$$

$=\frac{6+5i+14}{1+4}$ ($i^2=-1$)

$=\frac{20+5i}{5}$

$\Rightarrow$$\frac{\mathbf{6}\mathbf{-}\mathbf{7}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{i}}\mathbf{=}$$\mathbf{4}\mathbf{+}\mathit{i}$