Dividing Complex Numbers
Dividing complex numbers is a little more complicated than addition, subtraction, and multiplication of complex numbers because it is difficult to divide a number by an imaginary number. For dividing complex numbers, we need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary part of the denominator so that we end up with a real number in the denominator.
In this article, we will learn about the division of complex numbers, dividing complex numbers in polar form, the division of imaginary numbers, and dividing complex fractions.
1.  What is Dividing Complex Numbers? 
2.  Steps for Dividing Complex Numbers 
3.  Division of Complex Numbers in Polar Form 
4.  FAQs on Dividing Complex Numbers 
What is Dividing Complex Numbers?
Dividing complex numbers is mathematically similar to the division of two real numbers. If \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\) are the two complex numbers, then dividing complex numbers \(z_1\) and \(z_2\) is mathematically written as:
\[\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}\]
Dividing Complex Numbers Formula
The division of two complex numbers \(z_1=a+ib\) and \(z_2=c+id\) is given by the quotient \(\dfrac{a+ib}{c+id}\). This is calculated by using the division of complex numbers formula:
\[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bcad}{c^2+d^2}\right)\end{aligned}\]
Steps for Dividing Complex Numbers
Now, we know what dividing complex numbers is, let us discuss the steps for dividing complex numbers. To divide the two complex numbers, follow the given steps:
 First, calculate the conjugate of the complex number that is at the denominator of the fraction.
 Multiply the conjugate with the numerator and the denominator of the complex fraction.
 Apply the algebraic identity (a+b)(ab)=a^{2 } b^{2} in the denominator and substitute i^{2} = 1.
 Apply the distributive property in the numerator and simplify.
 Separate the real part and the imaginary part of the resultant complex number.
\[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{a+ib}{c+id}\\&=\dfrac{a+ib}{c+id}\times\dfrac{cid}{cid}\\&=\dfrac{(a+ib)(cid)}{c^2(id)^2}\\&=\dfrac{aciad+ibci^2bd}{c^2(1)d^2}\\&=\dfrac{aciad+ibc+bd}{c^2+d^2}\\&=\dfrac{(ac+bd)+i(bcad)}{c^2+d^2}\\&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bcad}{c^2+d^2}\right)\end{aligned}\]
Division of Complex Numbers in Polar Form
Let us divide the complex number \(z_{1}=r_1\left(\cos\theta_1+i\sin\theta_1\right)\) by the complex number \(z_{2}=r_2\left(\cos\theta_2+i\sin\theta_2\right)\). Dividing complex numbers in polar form is calculated as:
\[\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2i\sin\theta_2}{\cos\theta_2i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1\theta_2)+i\sin(\theta_1\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]
Where \(\theta=\theta_1\theta_2\) and \(r=\dfrac{r_1}{r_2}\).
Thus, the division of complex numbers \(z_{1}=r_1\left(\cos\theta_1+i\sin\theta_1\right)\) and \(z_{2}=r_2\left(\cos\theta_2+i\sin\theta_2\right)\) in polar form is given by the quotient \(\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\).
This is calculated by the formula:
\[\begin{aligned}\dfrac{z_1}{z_2}&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}\]
Important Notes on Dividing Complex Numbers
 To divide a complex number a+ib by c+id, multiply the numerator and denominator of the fraction a+ib/c+id by c−id and simplify.
 The conjugate of the complex z = a+ib is a−ib.
 The modulus of the complex number z = a+ib is z = √(a^{2 }+ b^{2})
Related Topics on Dividing Complex Numbers
Dividing Complex Numbers Examples

Example 1: Express the complex number (5+√2i)/(1−√2i) in the form of a+ib using the dividing complex numbers formula.
Solution: Let a = 5, b = √2, c = 1, and d = √2.
\[\begin{aligned}\dfrac{5+\sqrt{2}i}{1\sqrt{2}i}=\dfrac{a+ib}{c+id}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bcad}{c^2+d^2}\right)\\&=\dfrac{(5\times 1)+(\sqrt{2}\times(\sqrt{2}))}{1^2+(\sqrt{2})^2}+\\&i\left(\dfrac{(\sqrt{2} \times 1)(5\times(\sqrt{2}))}{1^2+(\sqrt{2})^2}\right)\\&=\dfrac{52}{1+2}+i\left(\dfrac{\sqrt{2}+5\sqrt{2}}{1+2}\right)\\&=\dfrac{3+6\sqrt{2}i}{1+2}\\&=1+2\sqrt{2}i\end{aligned}\]
Answer: (5+√2i)/(1−√2i) = 1 + i2√2

Example 2: Find the resultant complex number by dividing 3+4i by 82i by dividing the complex numbers.
Solution: We will simplify the complex number (3+4i)/(82i). The conjugate of the denominator 82i is 8+2i. Multiply the numerator and denominator of (3+4i)/(82i) by (8+2i).
\[\begin{align}\dfrac{3+4i}{82i}&=\dfrac{3+4i}{82i}\times\dfrac{8+2i}{8+2i}\\&=\dfrac{24+6i+32i+8i^2}{64+16i16i4i^2}\\&=\dfrac{16+38i}{68}\\&=\dfrac{4}{17}+\dfrac{19}{34}i\end{align}\]
Answer: 3+4i by 82i = \(\dfrac{4}{17}+\dfrac{19}{34}i\)
FAQs on Dividing Complex Numbers
What is Dividing Complex Numbers in Algebra?
Dividing complex numbers is mathematically similar to the division of two real numbers. If \(z_1=x_1+iy_1\) and \(z_2=x_2+iy_2\) are the two complex numbers, then dividing complex numbers \(z_1\) and \(z_2\) is mathematically written as:
\[\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}\]
How to Simplify Dividing Complex Numbers?
To divide a complex number a+ib by c+id, multiply the numerator and denominator of the fraction (a+ib)/(c+id) by c−id and simplify.
How to do Division of Complex Numbers?
The division of complex numbers \(z_1=a+ib\) and \(z_2=c+id\) is calculated by using the dividing complex numbers formula:
\[\dfrac{z_1}{z_2}=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bcad}{c^2+d^2}\right)\]
How Do you Write a Quotient when Dividing Complex Number?
Let the quotient be \(\dfrac{a+ib}{c+id}\). This can be written as \(\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bcad}{c^2+d^2}\right)\).
How Do you Write the Division of Complex Numbers by a Real Number?
Divide the real part and the imaginary part of the complex number by that real number separately.
What is the Quotient when Dividing Complex Numbers 4+8i by 1+3i?
The quotient \(\dfrac{4+8i}{1+3i}\) is given as \(\dfrac{14}{5}i\dfrac{2}{5}\).
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