# Multiplying and Dividing Complex Numbers

Learn how to multiply and divide complex numbers in few simple steps using the following step-by-step guide.

## Step by step guide to Multiplying and Dividing Complex Numbers

• Multiplying complex numbers: $$\color{blue}{(a+bi)+(c+di)=(ac-bd)+(ad+bc)i}$$
• Dividing complex numbers: $$\color{blue}{\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2- d^2 }+\frac{bc+ad}{c^2- d^2}i }$$
• Imaginary number rule: $$\color{blue}{i^2=-1}$$

### Example 1:

Solve: $$\frac{4-2i}{2+i}=$$

Solution:

Use the rule for dividing complex numbers:
$$\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2- d^2 }+\frac{bc+ad}{c^2- d^2 } i→$$
$$\frac{4-2i}{2+i}×\frac{2-i}{2-i}=\frac{(4×(2)+(-2)(1)}{2^2+ (1)^2 }+\frac{(-2×(2)-(4)(1)}{2^2+(1)^2 } i=\frac{6-8i}{5}=\frac{6}{5}-\frac{8}{5} i$$

### Example 2:

Solve: $$(2-3i)(4-3i)$$

Solution:

Use the rule: $$(a+bi)+(c+di)=(ac-bd)+(ad+bc)i$$
$$(2.4-(-3)(-3))+(2(-3)+(-3).4)i=-1-18i$$

### Example 3:

Solve: $$(2-8i)(3-5i)$$

Solution:

Use the rule: $$(a+bi)+(c+di)=(ac-bd)+(ad+bc)i$$
$$(2.3-(-8)9-5))+(2(-5)+(-8).3)i=-34-34i$$

### Example 4:

Solve: $$\frac{2-3i}{2+i}=$$

Solution:

Use the rule for dividing complex numbers:
$$\frac{a+bi}{c+di}=\frac{a+bi}{c+di}×\frac{c-di}{c-di}=\frac{ac+bd}{c^2- d^2 }+\frac{bc+ad}{c^2- d^2 } i→$$
$$\frac{2-3i}{2+i}×\frac{2-i}{2-i}=\frac{(2×(2)+(-3)(1)}{2^2- (-1)^2 }+\frac{(-3×(2)+(-1)(2)}{2^2-(-1)^2 } i=\frac{1-8i}{5}=\frac{1}{5}-\frac{8}{5} i$$

## Exercises

### Simplify.

• $$\color{blue}{(4i)(– i)(2 – 5i)}$$
• $$\color{blue}{(2 – 8i)(3 – 5i)}$$
• $$\color{blue}{(–5 + 9i)(3 + 5i)}$$
• $$\color{blue}{(7 + 3i)(7+ 8i)}$$
• $$\color{blue}{(5 + 4i)^2}$$
• $$\color{blue}{2(3i) – (5i)(– 8 + 5i)}$$

• $$\color{blue}{8 – 20i }$$
• $$\color{blue}{–34 – 34i}$$
• $$\color{blue}{–60 + 2i}$$
• $$\color{blue}{25 + 77i}$$
• $$\color{blue}{9 + 40i}$$
• $$\color{blue}{25 + 46i}$$

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