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169 LearnersLast updated on 13 September 2025
Division of Complex Numbers
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.
What are Complex Numbers?
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.
How to Divide Two Complex 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)
What is the Formula for Complex Number Division?
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)]
Steps for Complex Numbers Division
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. For 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) × (2 – i)] / [(2 + i) × (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
Division of Complex Numbers in Polar Form
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 sin2 Θ2)]
= (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 sin Θ)
Real-world Applications of the Division of Complex Numbers
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:
- In electrical engineering, complex numbers are divided to analyze the alternating current (AC) circuits. For example, in electrical engineering, division of complex numbers is used in Ohm’s Law (V = IZ) to compute voltage or current when impedance Z is complex.
- In quantum mechanics, the division of complex numbers is used to understand wave function transformations and probability calculations.
- It is used to calculate ratios in the design of antennas, microwave circuits, and fiber optic communication systems. Here, engineers often work with signal reflection, impedance matching, and wave propagation, all of which involve complex numbers.
Common Mistakes and How to Avoid Them in the Division of Complex Numbers
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.
Mistake 1
Not Multiplying by the Conjugate
Failing to multiply by the conjugate leaves an imaginary denominator in the non-standard form.
Mistake 2
Using Wrong Conjugate
Using the wrong conjugate while multiplying it with the numerator and denominator will not properly eliminate the imaginary part. This will lead to errors in the final result. Therefore, always double-check the conjugate before the multiplication. For example, the conjugate of c + i d is c – i d.
Mistake 3
Forgetting to Substitute the Value of i2
Students forget to substitute the value of i2 while working on problems. The value of i2 is –1. It is important to substitute the value correctly in equations containing i2 to get the correct answer.
Mistake 4
Not Writing the Correct Sign
The value with any signs should keep under strict consideration. Any misplace of signs can lead to miscalculation.
Mistake 5
Confusion Between Real and Imaginary Parts
To avoid confusion between the real and imaginary part, we should always write the answer in the standard form, which is a + bi, where "a" is real and "b" is imaginary.
Solved Examples of Division of Complex Numbers
Problem 1
Find the value of 3 +2i / 1 + i
The value of (3 + 2i) / (1 + i) is 5/2 – (1/2) i
Explanation
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 – (1/2) i
Problem 2
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
Explanation
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 + i (1/2)]
= √3 + i
Problem 3
Find the value of 3 + 4i/1 + 2i
The value of (3 + 4i) / (1 + 2i) = 11/5 – (2/5)i
Explanation
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 – i (2/5)
= 11/5 – (2/5)i
Problem 4
Find the value of 5 + i / 3 - i
The value of (5 + i) / (3 - i) = 7/5 + (4/5)i
Explanation
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
Problem 5
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
Explanation
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 z1 / z2 = (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 + i (1/2)]
= – √3 + i
FAQs on Division of Complex Numbers
1.What are complex numbers?
2.Is 7 a complex number?
3.What is the formula for dividing a complex number?
4.Is 2i a complex number?
5.What is the result when a complex number is divided by itself?
6.How can children in India use numbers in everyday life to understand Division of Complex Numbers?
7.What are some fun ways kids in India can practice Division of Complex Numbers with numbers?
8.What role do numbers and Division of Complex Numbers play in helping children in India develop problem-solving skills?
9.How can families in India create number-rich environments to improve Division of Complex Numbers skills?
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Hiralee Lalitkumar Makwana
About the Author
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.
Fun Fact
: She loves to read number jokes and games.
