Subtracting Complex Numbers- Step-by-Step Tutorial

What Are Complex Numbers?

A complex number has two parts: a real component and an imaginary component. The standard form looks like this:

a + bi

Where a is the real part, b is the coefficient of the imaginary unit i, and i = √(-1).

Examples:

• 3 + 4i
• -2 + 7i
• 5 - 3i
• -1 - 6i

Subtracting complex numbers is straightforward once you understand the structure. You handle the real parts and imaginary parts separately. That's it.

The Rule for Subtraction

When you subtract two complex numbers, you subtract the real parts from each other and the imaginary parts from each other:

(a + bi) - (c + di) = (a - c) + (b - d)i

Notice the second complex number gets distributed with a negative sign. This changes the signs of both its real and imaginary components before combining like terms.

Step-by-Step: How to Subtract Complex Numbers

Step 1: Write Both Numbers in Standard Form

Make sure both complex numbers follow the a + bi format. If you see something like 5 - 2i, that's already standard form.

Step 2: Remove the Parentheses with Distribution

Apply the negative sign to the second complex number:

(a + bi) - (c + di) becomes a + bi - c - di

Step 3: Group Real Parts Together

Combine the real numbers: a - c

Step 4: Group Imaginary Parts Together

Combine the imaginary coefficients: (b - d)i

Step 5: Write the Final Answer

Your result is (a - c) + (b - d)i

Examples That Actually Show How It Works

Example 1: Basic Subtraction

Subtract: (7 + 5i) - (3 + 2i)

Step 1: Remove parentheses → 7 + 5i - 3 - 2i

Step 2: Group real parts → 7 - 3 = 4

Step 3: Group imaginary parts → 5i - 2i = 3i

Answer: 4 + 3i

Example 2: Negative Numbers Involved

Subtract: (2 + 4i) - (-3 + 6i)

Step 1: Remove parentheses → 2 + 4i + 3 - 6i

Step 2: Group real parts → 2 + 3 = 5

Step 3: Group imaginary parts → 4i - 6i = -2i

Answer: 5 - 2i

Example 3: Both Numbers Have Negatives

Subtract: (-4 + 3i) - (-1 + 8i)

Step 1: Remove parentheses → -4 + 3i + 1 - 8i

Step 2: Group real parts → -4 + 1 = -3

Step 3: Group imaginary parts → 3i - 8i = -5i

Answer: -3 - 5i

Example 4: Subtracting From a Real Number

Subtract: 6 - (2 + 4i)

Step 1: Remove parentheses → 6 - 2 - 4i

Step 2: Group real parts → 6 - 2 = 4

Step 3: The imaginary part stays → -4i

Answer: 4 - 4i

Quick Reference: Subtraction vs Addition

⚠️ The order matters. (a + bi) - (c + di) gives a different result than (c + di) - (a + bi). Subtraction is not commutative.

Common Mistakes to Avoid

• Forgetting to distribute the negative sign — this is the most common error. The minus sign applies to both the real and imaginary parts of the second number.
• Mixing up signs in the result — always double-check your arithmetic on the real and imaginary parts separately.
• Dropping the i — your answer must include i unless the imaginary coefficient is zero.
• Confusing subtraction with multiplication — don't distribute the negative to only one term. That's multiplication, not subtraction.

Practice Problems

Try these on your own before checking the answers:

1. (8 + 6i) - (3 + 2i) = ?
2. (1 + 9i) - (5 + 3i) = ?
3. (-2 + 5i) - (4 - 2i) = ?
4. (7 - 3i) - (-1 + 2i) = ?

Answers:

1. 5 + 4i
2. -4 + 6i
3. -6 + 7i
4. 8 - 5i

Getting Started: Your Quick Checklist

1. Identify both complex numbers in a + bi format
2. Rewrite the problem with the negative sign distributed
3. Calculate the real part by subtracting the real numbers
4. Calculate the imaginary part by subtracting the imaginary coefficients
5. Write the answer as a new complex number

That's all there is to it. Separate the real from the imaginary, do the arithmetic, combine. No excuses for getting it wrong after this.