Lesson 12

Arithmetic with Complex Numbers

• Let’s work with complex numbers.

12.1: Math Talk: Telescoping Sums

Find the value of these expressions mentally.

$$2 - 2 + 20 - 20 + 200 - 200$$

$$100 - 50 + 10 - 10 + 50 - 100$$

$$3 + 2 + 1 + 0 - 1 - 2 - 3$$

$$1 + 2 + 4 + 8 + 16 + 32 - 16 - 8 - 4 - 2 - 1$$

1. This diagram represents $$(2 + 3i) + (\text- 8 - 8i)$$.
1. How do you see $$2 + 3i$$ represented?
2. How do you see $$\text- 8 - 8i$$ represented?
3. What complex number does $$A$$ represent?
4. Add “like terms” in the expression $$(2 + 3i) + (\text- 8 - 8i)$$. What do you get?
2. Write these sums and differences in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.
1. $$(\text- 3 + 2i) + (4 - 5i)$$ (Check your work by drawing a diagram.)
2. $$(\text- 37 - 45i) + (11 + 81i)$$
3. $$(\text- 3 + 2i) - (4 - 5i)$$
4. $$(\text- 37 - 45i) - (11 + 81i)$$

12.3: Multiplication on the Complex Plane

1. Draw points to represent 2, 22, 23, and 24 on the real number line.
1. Write $$2i$$, $$(2i)^2$$, $$(2i)^3$$, and $$(2i)^4$$ in the form $$a+bi$$.
2. Plot $$2i$$, $$(2i)^2$$, $$(2i)^3$$, and $$(2i)^4$$ on the complex plane.

1. If $$a$$ and $$b$$ are positive numbers, is it true that $$\sqrt{ab}=\sqrt{a}\sqrt{b}$$? Explain how you know.
2. If $$a$$ and $$b$$ are negative numbers, is it true that $$\sqrt{ab}=\sqrt{a}\sqrt{b}$$? Explain how you know.

Summary

When we add a real number with an imaginary number, we get a complex number. We usually write complex numbers as:

$$\displaystyle a + bi$$

where $$a$$ and $$b$$ are real numbers. We say that $$a$$ is the real part and $$bi$$ is the imaginary part.

To add (or subtract) two complex numbers, we add (or subtract) the real parts and add (or subtract) the imaginary parts. For example:

$$\displaystyle (2 + 3i )+(4 + 5i) = (2 + 4) + (3i + 5i) = 6 + 8i$$

$$\displaystyle (2 + 3i )-(4 + 5i) = (2 - 4) + (3i - 5i) = \text-2 - 2i$$

In general:

$$\displaystyle (a+bi) + (c+di) = (a+c) + (b+d)i$$

and:

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

When we raise an imaginary number to a power, we can use the fact that $$i^2=\text-1$$ to write the result in the form $$a+bi$$. For example, $$(4i)^3=4i \boldcdot 4i \boldcdot 4i$$. We can group the $$i$$ factors together to see how to rewrite this.

\begin{align*} 4i \boldcdot 4i \boldcdot 4i &= (4 \boldcdot 4 \boldcdot 4) \boldcdot (i \boldcdot i \boldcdot i) \\ &= 64 \boldcdot (i^2 \boldcdot i) \\ &= 64 \boldcdot \text-1 \boldcdot i \\ &= \text-64i \end{align*}

So in this example, $$a$$ is 0 and $$b$$ is -64.

Glossary Entries

• complex number

A number in the complex plane. It can be written as $$a + bi$$, where $$a$$ and $$b$$ are real numbers and $$i^2 = \text-1$$.

• imaginary number

A number on the imaginary number line. It can be written as $$bi$$, where $$b$$ is a real number and $$i^2 = \text-1$$.

• real number

A number on the number line.