Division of complex numbers is achieved by rationalizing with the conjugate of the denominator. That is, we multiply the numerators and denominators by the complex conjugate of the denominator.

Before we proceed, you must remember this property of imaginary number:


With that in mind, let us try some examples.

Example 1

Solve $\frac{4+10i}{8-2i}$


By rationalizing the denominator





If $i^2=-1$, then




Splitting the fraction 



Example 2

Solve $\frac{4-2i}{6+8i}$


First, we rationalize the denominator




Given that $i^2=-1$, therefore




Splitting the fraction



Example 3

Solve $\frac{3+2i}{5-i}$





Recalled that $i^2=-1$




Splitting the fraction



Example 4

If $z_1=1-3i$, $z_2=-2+5i$ and $z_3=-3-4i$. Find

i. $\frac{Z_1}{Z_3}$

ii. $\frac{Z_1Z_2}{Z_1+Z-2}$


I. By substitution


Now, let's rationalize the denominator







ii. To solve this, we have to evaluate the numerator and denominator separately.

we start with the numerator

 we know that $z_1=1-3i$, $z_2=-2+5i$, hence


So, let's multiply these complex numbers





Now, to the denominator...


So, let add these complex numbers



Now, let evaluate together


By rationalizing the denominator




if $i^2=-1$, then

$\frac{-13+22 -26i-11i}{5}$



That is all for now. Got questions relating to this? tell me in the comment box.

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