Adding & Subtracting Complex Numbers

Q: What is $i$?

The imaginary unit, defined by $i = \sqrt{-1}$, so $i^2 = -1$. It lets us work with square roots of negative numbers.

A complex number has a real part and an imaginary part, written $a + bi$ where $i = \sqrt{-1}$ . Adding and subtracting them is just like combining like terms.

Real parts combine with real parts, imaginary with imaginary. The $i$ acts like a variable you collect — nothing new to memorize beyond keeping the two parts separate.

Combine like parts

To add $(a + bi) + (c + di)$ , add the real parts and the imaginary parts separately: $(a + c) + (b + d)i$ .

Subtraction works the same after distributing the minus sign to both parts of the second number. Treat $i$ exactly like a variable term.

Watch the subtraction sign

The most common slip is forgetting to distribute the subtraction to the imaginary part. $(3 + 2i) - (5 - 7i)$ becomes $3 - 5 + (2 - (-7))i$ .

Keep the signs straight and the answer is another complex number in $a + bi$ form.

Worked examples

Example 1: adding

Add $(3 + 2i) + (5 - 7i)$ .

Combine real parts

3 + 5 = 8

Combine imaginary parts

2i - 7i = -5i

Write as a + bi

8 - 5i

Answer: $8 - 5i$

Example 2: subtracting

Subtract $(6 + i) - (2 + 4i)$ .

Distribute the minus

6 + i - 2 - 4i

Combine like parts

(6 - 2) + (1 - 4)i

Simplify

4 - 3i

Answer: $4 - 3i$

Example 3: subtracting a negative imaginary part

Subtract $(4 - 3i) - (1 - 8i)$ .

Distribute the minus to both parts

4 - 3i - 1 + 8i

Combine like parts

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

Simplify

3 + 5i

Answer: $3 + 5i$

Try one yourself

Common questions

What is

i

The imaginary unit, defined by $i = \sqrt{-1}$ , so $i^2 = -1$ . It lets us work with square roots of negative numbers.

Can I add a real number to a complex number?

Yes — a real number is just a complex number with imaginary part zero. $5 + (2 + 3i) = 7 + 3i$ .

Do I ever multiply when adding?

No. Addition and subtraction only combine like parts. Multiplication of complex numbers is a separate skill involving the FOIL method and $i^2 = -1$ .

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Adding & Subtracting Complex Numbers

Combine like parts

Watch the subtraction sign

Worked examples

Example 1: adding

Example 2: subtracting

Example 3: subtracting a negative imaginary part

Try one yourself

Common questions

Keep learning

Multiplying Complex Numbers & Conjugates

The Imaginary Unit i & Powers of i

The Quadratic Formula over the Complex Numbers