# Complex number operations review

Review complex number addition, subtraction, and multiplication.
$(a_1+b_1i)+(a_2+b_2i)=(a_1+a_2)+(b_1+b_2)i$
Subtraction
$(a_1+b_1i)-(a_2+b_2i)=(a_1-a_2)+(b_1-b_2)i$
Multiplication
$(a_1+b_1i)\cdot(a_2+b_2i)=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i$

## Practice set 1: Adding and subtracting complex numbers

### Example 1: Adding complex numbers

When adding complex numbers, we simply add the real parts and add the imaginary parts. For example:
\begin{aligned} &\phantom{=}(\blueD 3+\greenD4i)+(\blueD6\greenD{-10}i) \\\\ &=(\blueD3+\blueD6)+(\greenD4\greenD{-10})i \\\\ &=\blueD9\greenD{-6}i \end{aligned}

### Example 2: Subtracting complex numbers

When subtracting complex numbers, we simply subtract the real parts and subtract the imaginary parts. For example:
\begin{aligned} &\phantom{=}(\blueD 3+\greenD4i)-(\blueD6\greenD{-10}i) \\\\ &=(\blueD3-\blueD6)+(\greenD4-(\greenD{-10}))i \\\\ &=\blueD{-3}+\greenD{14}i \end{aligned}
Want to try more problems like this? Check out this exercise.

## Practice set 2: Multiplying complex numbers

When multiplying complex numbers, we perform a multiplication similar to how we expand the parentheses in binomial products:
$(a+b)(c+d)=ac+ad+bc+bd$
Unlike regular binomial multiplication, with complex numbers we also consider the fact that $i^2=-1$.

### Example 1

\begin{aligned} &\phantom{=}\blueD 2\cdot(\blueD{-3}+\greenD{4}i) \\\\ &=\blueD2\cdot(\blueD{-3})+\blueD2\cdot\greenD4i \\\\ &=\blueD{-6}+\greenD8i \end{aligned}

### Example 2

\begin{aligned} &\phantom{=}\greenD3i\cdot(\blueD{1}\greenD{-5}i) \\\\ &=\greenD3i\cdot\blueD1+\greenD3i\cdot(\greenD{-5})i \\\\ &=\greenD3i-15i^2 \\\\ &=\greenD3i-15(-1) \\\\ &=\blueD{15}+\greenD3i \end{aligned}

### Example 3

\begin{aligned} &\phantom{=}(\blueD2+\greenD3i)\cdot(\blueD{1}\greenD{-5}i) \\\\ &=\blueD2\cdot\blueD1+\blueD2\cdot(\greenD{-5})i+\greenD3i\cdot\blueD1+\greenD3i\cdot(\greenD{-5})i \\\\ &=\blueD2\greenD{-10}i+\greenD3i-15i^2 \\\\ &=\blueD2\greenD{-7}i-15(-1) \\\\ &=\blueD{17}\greenD{-7}i \end{aligned}
Want to try more problems like this? Check out this basic exercise and this advanced exercise.