Skill 7 of 12

Add, subtract, and multiply complex numbers

Add, subtract, and multiply complex numbers and write the result in the form $a+bi$ .

Common Core standards: N-CN.2
Builds on: The imaginary unit and square roots of negative numbers

Common mistakes

What students most often get wrong with this skill:

Combining real and imaginary parts as like terms
Forgetting to replace $i^2$ with $-1$ after multiplying
Distributing the minus sign to only the real part when subtracting

A worked example

From the lesson, with every step explained.

Example 1: Subtracting

Problem: Compute $(3+2i)-(1-5i)$ .

Plan: Remove the parentheses, applying the minus sign to both parts of the second number. Then combine real parts and imaginary parts.

1) Distribute the minus sign to every term in the second parentheses. This is the right move because subtraction means adding the opposite of the whole number.

3+2i-1+5i

2) Group the real parts.

3-1=2

3) Group the imaginary parts.

2i+5i=7i

4) Write in standard form.

2+7i

Answer: $2+7i$

Check: Add the answer to the number we subtracted. The result should be the first number.

$(2+7i)+(1-5i)=3+2i$

This matches the original first number, so the answer is correct.

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