Complex Numbers Division- Methods and Examples

What Is Division of Complex Numbers?

Complex numbers have a real part and an imaginary part. When you divide them, you're working with expressions like (a + bi) ÷ (c + di). The trick is getting rid of the imaginary number in the denominator.

That's it. That's the whole problem. The denominator has i in it, and you need to make it go away.

The Method: Multiply by the Conjugate

To divide complex numbers, you multiply both the numerator and denominator by the complex conjugate of the denominator.

The conjugate of c + di is c − di. When you multiply these together, the result is a real number.

Here's why this works:

(c + di)(c − di) = c² + d²

The imaginary parts cancel out. You're left with a real number in the denominator.

Why the Conjugate Works

Complex conjugates are like mirror images across the real number line. One has +di, the other has −di. Multiply them and the i² term becomes −1, which combines with the other terms to give you something real.

Step-by-Step Process

Here's how to divide any two complex numbers:

1. Write the division problem as a fraction
2. Find the conjugate of the denominator
3. Multiply both top and bottom by that conjugate
4. Simplify the numerator (expand and combine like terms)
5. Simplify the denominator (it becomes a real number)
6. Separate into real and imaginary parts
7. Write your final answer in a + bi form

Examples

Example 1: Simple Division

Divide: (6 + 2i) ÷ (1 + i)

Step 1: Write as a fraction

6 + 2i
─────────
1 + i

Step 2: Multiply by the conjugate of the denominator (1 − i)

(6 + 2i)(1 − i)
────────────────
(1 + i)(1 − i)

Step 3: Expand the numerator

(6 × 1) + (6 × −i) + (2i × 1) + (2i × −i)
= 6 − 6i + 2i − 2i²
= 6 − 4i − 2(−1)
= 6 − 4i + 2
= 8 − 4i

Step 4: Expand the denominator

(1 + i)(1 − i) = 1 − i² = 1 − (−1) = 2

Step 5: Divide each part

8 ÷ 2 = 4
−4i ÷ 2 = −2i

Answer: 4 − 2i

Example 2: With Fractions

Divide: (3 + 4i) ÷ (2 − i)

Step 1: Set up the fraction

3 + 4i
─────────
2 − i

Step 2: Conjugate of denominator is (2 + i)

(3 + 4i)(2 + i)
────────────────
(2 − i)(2 + i)

Step 3: Expand numerator

(3 × 2) + (3 × i) + (4i × 2) + (4i × i)
= 6 + 3i + 8i + 4i²
= 6 + 11i + 4(−1)
= 6 + 11i − 4
= 2 + 11i

Step 4: Expand denominator

(2 − i)(2 + i) = 4 − i² = 4 − (−1) = 5

Step 5: Divide

2 ÷ 5 = 2/5
11i ÷ 5 = 11/5 i

Answer: 2/5 + 11/5 i

Example 3: Denominator Is Pure Imaginary

Divide: (5 + 10i) ÷ (2i)

This one's simpler. Just divide each part by 2i.

5 ÷ 2i = 5/(2i)

Rationalize: multiply top and bottom by i

5i ÷ (2i × i) = 5i ÷ (2i²) = 5i ÷ (2 × −1) = 5i ÷ (−2) = −5/2 i

10i ÷ 2i = 10 ÷ 2 = 5

Answer: 5 − 5/2 i

Quick Reference: Division Methods

Common Mistakes to Avoid

• Forgetting the conjugate — you must multiply both top and bottom by the same conjugate
• Screwing up the signs when expanding — take your time with (a + bi)(c − di)
• i² = −1 — not 1. This is where most errors happen
• Not simplifying your final answer — combine like terms and reduce fractions
• Forgetting to rationalize when the denominator has i in it

Practice Problems

Try these on your own before checking the answers:

1. (8 + 4i) ÷ (2 + i)
2. (7 − 3i) ÷ (1 + 2i)
3. (12 + 6i) ÷ 6
4. (5i) ÷ (1 + i)

Answers

1. 4 − 4/5 i (multiply by 2 − i, simplify)
2. 1/5 − 17/5 i (multiply by 1 − 2i)
3. 2 + i (just divide each part)
4. 5/2 + 5/2 i (multiply top and bottom by 1 − i)

The Bottom Line

Complex number division comes down to one skill: multiplying by the conjugate. Once you understand that trick, every problem is just expansion, simplification, and separation into real and imaginary parts.

Most students lose points on the arithmetic, not the concept. So practice your i² calculations until they're automatic.