# Lesson 14

More Arithmetic with Complex Numbers

• Let’s practice adding, subtracting, and multiplying complex numbers.

### 14.1: Which One Doesn’t Belong: Complex Expressions

Which one doesn’t belong?

A. $$i^2$$

B. $$(1 + i) + (1 - i)$$

C. $$(1 + i)^2$$

D. $$(1 + i)(1 - i)$$

### 14.2: Powers of $i$

1. Write each power of $$i$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. If $$a$$ or $$b$$ is zero, you can ignore that part of the number. For example, $$0+3i$$ can simply be expressed as $$3i$$.

$$i^0$$

$$i^1$$

$$i^2$$

$$i^3$$

$$i^4$$

$$i^5$$

$$i^6$$

$$i^7$$

$$i^8$$

2. What is $$i^{100}$$? Explain your reasoning.
3. What is $$i^{38}$$? Explain your reasoning.

1. Write each power of $$1+i$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. If $$a$$ or $$b$$ is zero, you can ignore that part of the number. For example, $$0+3i$$ can simply be expressed as $$3i$$.
1. $$(1+i)^0$$
2. $$(1+i)^1$$
3. $$(1+i)^2$$
4. $$(1+i)^3$$
5. $$(1+i)^4$$
6. $$(1+i)^5$$
7. $$(1+i)^6$$
8. $$(1+i)^7$$
9. $$(1+i)^8$$
2. Compare and contrast the powers of $$1+i$$ with the powers of $$i$$. What is the same? What is different?

### 14.3: Add 'Em Up (or Subtract or Multiply)

For each row, your partner and you will each rewrite an expression so it has the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. You and your partner should get the same answer. If you disagree, work to reach agreement.

partner A partner B
$$(7 + 9i) + (3 - 4i)$$ $$5i(1 - 2i)$$
$$2i(3 + 4i)$$ $$(1 + 2i) - (9 - 4i)$$
$$(4 - 3i)(4 + 3i)$$ $$(5 + i) + (20 - i)$$
$$(2i)^4$$ $$(3 + i \sqrt{7})(3 - i \sqrt{7})$$
$$(1 + i \sqrt{5}) - (\text- 7 - i \sqrt{5})$$ $$(\text- 2i)(\text- \sqrt{5} + 4i)$$
$$\left( \frac12 i \right) \left( \frac13 i \right) \left( \frac34 i \right)$$ $$\left( \frac12 i \right)^3$$

### Summary

Suppose we want to write the product $$(3+5i)(7-2i)$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. For example, we might want to compare our solution with a partner’s, and having answers in the same form makes that easier. Using the distributive property,

\displaystyle \begin{align} (3+5i)(7-2i) &= 21 - 6i +35i - 10i^2 \\ &= 21 + 29i - 10(\text-1) \\ &= 21+ 29i +10 \\ &= 31 +29i \end{align}

Keeping track of the negative signs is especially important since it is easy to mix up the fact that $$i^2=\text-1$$ with the fact that $$\text-i^2=\text-(\text-1) = 1$$.

Next, suppose we want to write the difference $$(\text-6+3i) - (2-4i)$$ as a single complex number in the form $$a+bi$$. Distributing the negative and combining like terms, we get:

\displaystyle \begin{align} (\text-6+3i) - (2-4i) &= \text-6 +3i - 2 - (\text-4i) \\ &= \text-8 +3i +4i \\ &= \text-8 +7i \end{align}

Again, it is important to be precise with negative signs. It is a common mistake to just subtract $$4i$$ rather than subtracting $$\text-4i$$.

### 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.