Complex Number Division- Techniques and Practice Problems
What Is Complex Number Division?
Complex number division is the process of dividing numbers that have both a real part and an imaginary part. It sounds intimidating, but once you see the pattern, it becomes mechanical.
The trick is multiplying by the conjugate. That's it. Everything else is just algebra.
A Quick Refresher on Complex Numbers
A complex number looks like this:
a + bi
Where:
- a is the real part
- b is the imaginary coefficient
- i is √(-1)
Examples: 3 + 4i, -2 + 7i, 5 - 3i
Why Can't You Just Divide Normally?
With real numbers, division is straightforward. But with complex numbers, you're working with two components at once.
Try dividing (3 + 4i) by (1 + 2i) using basic division. You'll get stuck. The result needs to end up in the form a + bi, and simple division doesn't get you there.
This is why we use the conjugate method.
The Conjugate Method: Step by Step
To divide two complex numbers, follow these steps:
Step 1: Identify the Conjugate
The conjugate of a + bi is a - bi. Just flip the sign of the imaginary part.
Step 2: Multiply Numerator and Denominator by the Conjugate
For (3 + 4i) ÷ (1 + 2i), multiply both by the conjugate of (1 + 2i), which is (1 - 2i).
Step 3: Apply FOIL to the Numerator
Multiply out (3 + 4i)(1 - 2i). Use FOIL like normal.
Step 4: Apply FOIL to the Denominator
Multiply out (1 + 2i)(1 - 2i). Here's where the magic happens — this always simplifies to a real number.
Step 5: Simplify and Separate Real from Imaginary
Combine like terms. Your answer should end up as a + bi.
Worked Examples
Example 1: (3 + 4i) ÷ (1 + 2i)
Step 1: Multiply by the conjugate: (1 - 2i)
Step 2: Numerator: (3 + 4i)(1 - 2i)
- F: 3 × 1 = 3
- O: 3 × (-2i) = -6i
- I: 4i × 1 = 4i
- L: 4i × (-2i) = -8i²
Combine: 3 - 6i + 4i - 8i² = 3 - 2i - 8i²
Remember that i² = -1, so -8i² = -8(-1) = 8
Numerator simplifies to: 3 - 2i + 8 = 11 - 2i
Step 3: Denominator: (1 + 2i)(1 - 2i)
- 1 × 1 = 1
- 1 × (-2i) = -2i
- 2i × 1 = 2i
- 2i × (-2i) = -4i²
Combine: 1 - 2i + 2i - 4i² = 1 - 4i²
Since i² = -1: 1 - 4(-1) = 1 + 4 = 5
Step 4: Final answer: (11 - 2i) ÷ 5 = 11/5 - (2/5)i
Or: 2.2 - 0.4i
Example 2: (7 - 3i) ÷ (2 + i)
Conjugate of denominator: 2 - i
Numerator: (7 - 3i)(2 - i)
- 7 × 2 = 14
- 7 × (-i) = -7i
- -3i × 2 = -6i
- -3i × (-i) = 3i²
14 - 7i - 6i + 3i² = 14 - 13i + 3(-1) = 14 - 13i - 3 = 11 - 13i
Denominator: (2 + i)(2 - i) = 4 - 2i + 2i - i² = 4 - (-1) = 5
Answer: (11 - 13i) ÷ 5 = 11/5 - (13/5)i
Practice Problems
Try these on your own before checking the answers below:
- (5 + 10i) ÷ (1 + i)
- (8 - 2i) ÷ (3 - i)
- (4 + 6i) ÷ (2 - 3i)
- (1 + i) ÷ (1 - i)
Answers:
- 7.5 + 2.5i (or 15/2 + 5/2i)
- 26/10 + 2/10i (or 13/5 + 1/5i)
- -1/13 + 24/13i
- i (This one equals exactly i — no real part!)
Common Mistakes to Avoid
- Forgetting to multiply both numerator AND denominator by the conjugate. You must do both.
- Messing up i² = -1. This shows up twice in every problem. Get it wrong and everything falls apart.
- Not simplifying the final answer. Combine like terms. Write it cleanly.
- Using the wrong conjugate. The conjugate is always a - bi. Not a + bi.
Quick Reference Table
| Problem Type | Method | Result Form |
|---|---|---|
| Divide by (a + bi) | Multiply by (a - bi) | x + yi |
| Divide by (a - bi) | Multiply by (a + bi) | x + yi |
| Divide by pure imaginary (bi) | Multiply numerator and denominator by i | x + yi |
| Divide real by complex | Still use conjugate method | x + yi |
Getting Started: Your First 3 Problems
Master the basics with these three problems:
- (2 + 3i) ÷ (1 + i) → Answer: 2.5 + 0.5i
- (6 - 4i) ÷ (2 - i) → Answer: 16/5 + 2/5i
- (5i) ÷ (1 + 2i) → Answer: 2 - i
Work through each one step by step. Don't skip steps. Once you've done 10 problems without looking at examples, you'll have it locked in.