Last updated on July 4th, 2025
Division of complex numbers involves finding the quotient of two numbers in the form a + bi. This article covers the division of complex numbers with formulas and examples.
Complex numbers are numbers with two parts: real and imaginary. They are represented as a + bi, where a and b are the real and imaginary parts respectively. We use complex numbers to solve equations that cannot be solved only with real numbers.
Complex numbers can be divided by using a method that involves the conjugate of the denominator. Two complex numbers, z1 = a1 + iy1 and z2 = a2 + iy2, can be expressed as
z1/z2=a1 + iy1/a2 + iy2
To find the formula for dividing two complex numbers, let's consider the complex numbers as z1 = a + ib and z2 = c + id. So, it can be calculated as
z1 / z2=[ ac + bd/c2 + d2] + i[(bc - ad/c2 + d2)]
Follow these steps to divide two complex numbers.
Step 1: Make sure both numerator and denominator are complex numbers written correctly in standard form.
Step 2: Determine the denominator’s conjugate. Example, if the denominator is c + id, then the conjugate is c - id.
Step 3: The conjugate should be multiplied by both numerator and denominator.
Step 4: The denominator must be solved using the difference of squares formula.
Step 5: Separate the result into its real and imaginary parts, expressed as a + bi.
Let us try dividing 3 + 4i/2 + i
Step 1: Checking for the standard form, we conclude that both numerator and denominator are in the standard form.
Step 2: The denominator is 2 + i, so its conjugate is 2 - i.
Step 3: The conjugate must be multiplied with numerator and denominator.
So, (3 + 4i) x (2 - i)/(2 + i) x (2 - i)
Step 4: Expanding the numerator, we get: (3 + 4i) (2 - i) = 3(2) - 3(i) + 4i(2) - 4i(i)
= 6 - 3i + 8i - 4i2
As i2 = -1, 6 - 3i + 8i - 4i2 becomes,
= 6 - 3i + 8i + 4
= 10 + 5i
Applying the difference of squares formula to solve the denominator,
(2 + i) (2 - i) = 22 - i2
= 4 - (-1)
= 4 +1 = 5
Step 5: Divide the expanded numerator by the denominator
10 + 5i/5 = 10/5 + 5i/5 = 2 + i
Here, we will learn how to divide complex numbers in polar form. Let’s say we need to divide z1 by z2 where z1 = r1(cos Θ1 + i sin Θ1) and z2 = r2(cos Θ2 + i sin Θ2).
So, z1 / z2 = r1(cos Θ1 + i sin Θ1) / r2 (cos Θ2 + i sin Θ2)
= r1(cos Θ1 + i sin Θ1) / r2 (cos Θ2 + i sin Θ2) × (cos Θ2 - i sin Θ2) / (cos Θ2 - i sin Θ2)
= r1(cos Θ1 + i sin Θ1) (cos Θ2 + i sin Θ2) / r2 (cos2 Θ2 - (i)2 sin2Θ2)
= r1/r2[cos (Θ1 - Θ2) + i sin(Θ1 - Θ2)]
= r(cos Θ + i sin Θ)
Where, Θ = Θ1 - Θ2 and r = r1/r2
So, the formula for dividing complex number in polar form is:
z1/z2 = r(cos Θ + i sinΘ)
Complex numbers are used daily in fields like engineering and quantum mechanics. Here are a few examples where the division of complex numbers is applied:
It is easy for students to make mistakes while dividing complex numbers because the process involves conjugates and imaginary units. Learning about a few common mistakes can help us avoid them and get better at dividing complex numbers.
Find the value of 3 +2i / 1 + i
The value of 3 + 2i / 1 + i is 5/2 -½i
The division of complex number division formula is z1/z2= ac + bd/c2 + d2 + i(bc - ad/c2 + d2)Here a = 3, b = 2, c = 1, and d =1
So, 3 + 2i / 1 + i = (3 + 2i)(1 -i) / (1 + i)(1 - i)
Expanding the numerator;
(3 + 2i)(1 -i) = 3(1) - 3i + 2i - 2i2
As the value of i2 = -1
3(1) - 3i + 2i - 2i2 = 3 - i + 2
= 5 - i
Simplifying the denominator;
(1 + i)(1 - i) = 1 - i2
= 1 + 1 = 2
So, the value of 3 + 2i / 1 + i = 5 - i/2
= 5/2 - ½i
Solve: 8 (cos 60° + i sin 60°) / 4 (cos 30° + i sin 30°)
The value of 8 (cos 60° + i sin 60°) / 4 (cos 30° + i sin 30°) is √3 + i
To divide the complex number in polar form, we use the equation;
z1/z2 = = r1/r2[cos (Θ1 - Θ2) + i sin(Θ1 - Θ2)]
Where, r1 = 8
Θ1 = 60°
r2 = 4
Θ2 = 30°
Substituting the values in the equation r1/r2[cos (Θ1 - Θ2) + i sin(Θ1 - Θ2)],
= 8/4[cos (60° - 30°) + i sin(60° - 30°)]
= 2[cos(30°) + i sin(30°)]
The value of cos 30° = √3/2
The value of sin 30°= 1/2
That is 2[cos(30°) + i sin(30°)] = 2(√3/2 + i1/2)
= √3 + i
Find the value of 3 + 4i/1 + 2i
The value of 3 + 4i/1 + 2i = 11/5 - ⅖i
The division of complex number formula is z1/z2= ac + bd/c2 + d2 + i(bc - ad/c2 + d2)
Here a = 3, b = 4, c = 1, and d = 2
So, 3 + 4i/1 + 2i = (3 + 4i)(1 - 2i)/(1 + 2i)(1 - 2i)
Expanding the numerator;
(3 + 4i)(1 - 2i)= 3 - 6i + 4i - 8i2
As the value of i2 = -1
= 3 - 2i + 8
= 11 - 2i
Simplifying the denominator;
(1 + 2i)(1 - 2i) = 1 - (2i)2
= 1 + 4 = 5
So, the value of 3 + 4i/1 + 2i = 11 - 2i/5
= 11/5 - ⅖i
Find the value of 5 + i / 3 - i
The value of 5 + i/3 - i = 7/5 + ⅘i
The conjugate of the denominator, 3 - i, is 3 + i
Multiply both numerator and denominator with the conjugate, that is
5 + i / 3 - i = (5 + i)(3 +i) / (3 - i)(3+i)
Expanding the numerator: (5 + i)(3 +i)
= 15 + 5i + 3i + i2
= 15 + 8i - 1
= 14 + 8i
Simplifying the denominator: (3 - i)(3+i)
= 9 - i2
= 9 - (-1) = 10
So, the value of 5 + i / 3 - i = 14 + 8i / 10
= 14/10 + (4/5)i
= 7/5 + (4/5)i
Solve: 8 (cos 210° + i sin 210°) / 4 (cos 60° + i sin 60°)
The value of 8 (cos 210° + i sin 210°) / 4 (cos 60° + i sin 60°) is -√3 + i
To divide the complex number in polar form, we use the equation;
z1/z2 = = r1/r2[cos (Θ1 - Θ2) + i sin(Θ1 - Θ2)]
Where, r1 = 8
Θ1 = 210°
r2 = 4
Θ2 = 60°
Substituting the values in the equation r1/r2[cos (Θ1 - Θ2) + i sin(Θ1 - Θ2)],
= 8/4[cos (210° - 60°) + i sin(210° - 60°)]
= 2[cos(150°) + i sin(150°)]
The value of cos 150° = -√3/2
The value of sin 150°= 1/2
That is 2[cos(150°) + i sin(150°)] = 2(-√3/2 + i1/2)
= -√3 + i
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.