<meta http-equiv="refresh" content="1; url=/nojavascript/"> Defining Complex Numbers | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra II with Trigonometry Concepts Go to the latest version.

# 5.8: Defining Complex Numbers

Difficulty Level: At Grade Created by: CK-12
%
Best Score
Practice Complex Numbers
Best Score
%

The coldest possible temperature, known as absolute zero is almost –460 degrees Fahrenheit. What is the square root of this number?

### Watch This

First, watch this video.

Then, watch this video.

### Guidance

Before this concept, all numbers have been real numbers. 2, -5, $\sqrt{11}$ , and $\frac{1}{3}$ are all examples of real numbers. Look at #1 from the Review Queue. With what we have previously learned, we cannot find $\sqrt{-25}$ because you cannot take the square root of a negative number. There is no real number that, when multiplied by itself, equals -25. Let’s simplify $\sqrt{-25}$ .

$\sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}$

In order to take the square root of a negative number we are going to assign $\sqrt{-1}$ a variable, $i$ . $i$ represents an imaginary number . Now, we can use $i$ to take the square root of a negative number.

$\sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}=5i$

All complex numbers have the form $a + bi$ , where $a$ and $b$ are real numbers. $a$ is the real part of the complex number and $b$ is the imaginary part . If $b = 0$ , then $a$ is left and the number is a real number. If $a = 0$ , then the number is only $bi$ and called a pure imaginary number . If $b \ne 0$ and $a \ne 0$ , the number will be an imaginary number.

#### Example A

Find $\sqrt{-162}$ .

Solution: First pull out the $i$ . Then, simplify $\sqrt{162}$ .

$\sqrt{-162}=\sqrt{-1} \cdot \sqrt{162}=i\sqrt{162}=i\sqrt{81 \cdot 2}=9i\sqrt{2}$

#### Investigation: Powers of i

In addition to now being able to take the square root of a negative number, $i$ also has some interesting properties. Try to find $i^2,i^3,$ and $i^4$ .

1. Write out $i^2$ and simplify. $i^2=i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2=-1$

2. Write out $i^3$ and simplify. $i^3=i^2 \cdot i = -1 \cdot i = -i$

3. Write out $i^4$ and simplify. $i^4=i^2 \cdot i^2 = -1 \cdot -1 = 1$

4. Write out $i^5$ and simplify. $i^5=i^4 \cdot i = 1 \cdot i = i$

5. Write out $i^6$ and simplify. $i^6=i^4 \cdot i^2 = 1 \cdot -1 = -1$

6. Do you see a pattern? Describe it and try to find $i^{19}$ .

You should see that the powers of $i$ repeat every 4 powers. So, all the powers that are divisible by 4 will be equal to 1. To find $i^{19}$ , divide 19 by 4 and determine the remainder. That will tell you what power it is the same as.

$i^{19}=i^{16} \cdot i^3 = 1 \cdot i^3=-i$

#### Example B

Find:

a) $i^{32}$

b) $i^{50}$

c) $i^7$

Solution:

a) 32 is divisible by 4, so $i^{32}=1$ .

b) $50 \div 4=12$ , with a remainder of 2. Therefore, $i^{50}=i^2=-1$ .

c) $7 \div 4=1$ , with a remainder of 3. Therefore, $i^7=i^3=-i$

#### Example C

Simplify the complex expressions.

a) $(6-4i)+(5+8i)$

b) $9-(4+i)+(2-7i)$

Solution: To add or subtract complex numbers, you need to combine like terms. Be careful with negatives and properly distributing them. Your answer should always be in standard form , which is $a + bi$ .

a) $(6-4i)+(5+8i)={\color{red}6}{\color{blue}-4i}+{\color{red}5}+{\color{blue}8i}={\color{red}11}+{\color{blue}4i}$

b) $9-(4+i)+(2-7i)={\color{red}9-4}{\color{blue}-i}+{\color{red}2}{\color{blue}-7i}={\color{red}7}{\color{blue}-8i}$

Intro Problem Revisit We're looking for $\sqrt{-460}$ .

First we need to pull out the $i$ . Then, we need to simplify $\sqrt{460}$ .

$\sqrt{-460}=\sqrt{-1} \cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4 \cdot 115}=2i\sqrt{115}$

### Guided Practice

Simplify.

1. $\sqrt{-49}$

2. $\sqrt{-125}$

3. $i^{210}$

4. $(8-3i)-(12-i)$

1. Rewrite $\sqrt{-49}$ in terms of $i$ and simplify the radical.

$\sqrt{-49}=i\sqrt{49}=7i$

2. Rewrite $\sqrt{-125}$ in terms of $i$ and simplify the radical.

$\sqrt{-125}=i\sqrt{125}=i\sqrt{25 \cdot 5}=5i\sqrt{5}$

3. $210 \div 4=52$ , with a remainder of 2. Therefore, $i^{210}=i^2=-1$ .

4. Distribute the negative and combine like terms.

$(8-3i)-(12-i)=8-3i-12+i=-4-2i$

### Vocabulary

Imaginary Numbers
Any number with an $i$ associated with it. Imaginary numbers have the form $a + bi$ or $bi$ .
Complex Numbers
All real and imaginary numbers. Complex numbers have the standard form $a + bi$ , where $a$ or $b$ can be zero. $a$ is the real part and $bi$ is the imaginary part .
Pure Imaginary Numbers
An imaginary number without a real part, only $bi$ .

### Practice

Simplify each expression and write in standard form.

1. $\sqrt{-9}$
2. $\sqrt{-242}$
3. $6\sqrt{-45}$
4. $-12i\sqrt{98}$
5. $\sqrt{-32} \cdot \sqrt{-27}$
6. $7i\sqrt{-126}$
7. $i^8$
8. $16i^{22}$
9. $-9i^{65}$
10. $i^{365}$
11. $2i^{91}$
12. $\sqrt{-\frac{16}{80}}$
13. $(11-5i)+(6-7i)$
14. $(14+2i)-(20+9i)$
15. $(8-i)-(3+4i)+15i$
16. $-10i-(1-4i)$
17. $(0.2+1.5i)-(-0.6+i)$
18. $6+(18-i)-(2+12i)$
19. $-i+(19+22i)-(8-14i)$
20. $18-(4+6i)+(17-9i)+24i$

## Date Created:

Mar 12, 2013

Jul 16, 2014
Files can only be attached to the latest version of Modality