Complex number division

You complex numbers are those that have an imaginary part, and among which we can also perform operations.

There are specific ways to solve each of them. In the case of complex number division we use the concept of the conjugate of a complex number.

Conjugated of a complex number:

Consider a complex number written in algebraic form $\dpi{120} \boldsymbol{z=a +bi}$ , then, the conjugate of $\dpi{120} \boldsymbol{z}$ is represented by $\dpi{120} \boldsymbol{\bar{z}}$ and is given by:

$\dpi{120} \boldsymbol{\bar{z}=a -bi}$

That is, to get the conjugate, we just need to change the sign of the imaginary part of the complex number.

That said, let's learn how to divide complex numbers.

complex number division

To divide a complex number $\dpi{120} \boldsymbol{z_1}$ by a complex number $\dpi{120} \boldsymbol{z_2}$ , we must write the division in the form of fraction:

$\dpi{120} \boldsymbol{z_1:z_2=\frac{z_1}{z_2}}$

Since multiplying and dividing a fraction by the same number does not change the final result, then we divide and multiply the fraction by the conjugate of the denominator.

$\dpi{120} \boldsymbol{\frac{z_1}{z_2}\cdot \frac{\bar{z_2}}{\bar{z_2}}}$

We then substitute the terms and multiply the fractions.

Example: if $\dpi{120} \boldsymbol{z_1=2 -3i}$ and $\dpi{120} \boldsymbol{z_2=4 +2i}$ , what is the value of $\dpi{120} \boldsymbol{z_1:z_2}$ ?

$\dpi{120} \boldsymbol{\frac{z_1}{z_2}\cdot \frac{\bar{z_2}}{\bar{z_2}}}$

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$\dpi{120} \boldsymbol{\frac{(2-3i)}{(4+2i)}\cdot \frac{(4-2i)}{(4-2i)}}$

$\dpi{120} \boldsymbol{\frac{8-4i-12i+6i^2}{16-8i+8i-4i^2}}$

$\dpi{120} \boldsymbol{\frac{8-16i+6i^2}{16-4i^2}}$

Remembering that $\dpi{120} \boldsymbol{i^2 = -1}$ , we have:

$\dpi{120} \boldsymbol{\frac{8-16i+6\cdot (-1)}{16-4\cdot (-1)}}$

$\dpi{120} \boldsymbol{\frac{8-16i-6}{16+4}}$

$\dpi{120} \boldsymbol{\frac{2-16i}{20}}$

We can simplify this result:

$\dpi{120} \boldsymbol{\frac{2-16i}{20}= \frac{1}{10}-\frac{4}{5}i}$

Complex number division formula

Generally speaking, for and $\dpi{120} \boldsymbol{z_1=a +bi}$ and $\dpi{120} \boldsymbol{z_2=c +di}$ , you can check a formula for dividing complex numbers:

$\dpi{120} \boldsymbol{z_1:z_2=\frac{z_1}{z_2} = \frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+ d^2}i}$

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