# Complex Numbers

## a + bi, the sum of a real and an imaginary number.

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Defining Complex Numbers

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, \begin{align*}\sqrt{11}\end{align*} , and \begin{align*}\frac{1}{3}\end{align*} are all examples of real numbers. Look at #1 from the Review Queue. With what we have previously learned, we cannot find \begin{align*} \sqrt{-25}\end{align*} 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 \begin{align*} \sqrt{-25}\end{align*}.

\begin{align*} \sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}\end{align*}

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

\begin{align*} \sqrt{-25}= \sqrt{25 \cdot -1}=5 \sqrt{-1}=5i\end{align*}

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

#### Example A

Find \begin{align*}\sqrt{-162}\end{align*} .

Solution: First pull out the \begin{align*}i\end{align*}. Then, simplify \begin{align*}\sqrt{162}\end{align*} .

\begin{align*}\sqrt{-162}=\sqrt{-1} \cdot \sqrt{162}=i\sqrt{162}=i\sqrt{81 \cdot 2}=9i\sqrt{2}\end{align*}

#### Investigation: Powers of i

In addition to now being able to take the square root of a negative number, \begin{align*}i\end{align*} also has some interesting properties. Try to find \begin{align*}i^2,i^3,\end{align*} and \begin{align*}i^4\end{align*}.

1. Write out \begin{align*}i^2\end{align*} and simplify. \begin{align*}i^2=i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1}^2=-1\end{align*}

2. Write out \begin{align*}i^3\end{align*} and simplify. \begin{align*}i^3=i^2 \cdot i = -1 \cdot i = -i\end{align*}

3. Write out \begin{align*}i^4\end{align*} and simplify. \begin{align*}i^4=i^2 \cdot i^2 = -1 \cdot -1 = 1\end{align*}

4. Write out \begin{align*}i^5\end{align*} and simplify. \begin{align*}i^5=i^4 \cdot i = 1 \cdot i = i\end{align*}

5. Write out \begin{align*}i^6\end{align*} and simplify. \begin{align*}i^6=i^4 \cdot i^2 = 1 \cdot -1 = -1\end{align*}

6. Do you see a pattern? Describe it and try to find \begin{align*}i^{19}\end{align*}.

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

\begin{align*}i^{19}=i^{16} \cdot i^3 = 1 \cdot i^3=-i \end{align*}

#### Example B

Find:

a) \begin{align*}i^{32}\end{align*}

b) \begin{align*}i^{50}\end{align*}

c) \begin{align*}i^7\end{align*}

Solution:

a) 32 is divisible by 4, so \begin{align*}i^{32}=1\end{align*}.

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

c) \begin{align*}7 \div 4=1\end{align*}, with a remainder of 3. Therefore, \begin{align*}i^7=i^3=-i\end{align*}

#### Example C

Simplify the complex expressions.

a) \begin{align*}(6-4i)+(5+8i)\end{align*}

b) \begin{align*}9-(4+i)+(2-7i)\end{align*}

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 \begin{align*}a + bi\end{align*}.

a) \begin{align*}(6-4i)+(5+8i)={\color{red}6}{\color{blue}-4i}+{\color{red}5}+{\color{blue}8i}={\color{red}11}+{\color{blue}4i}\end{align*}

b) \begin{align*}9-(4+i)+(2-7i)={\color{red}9-4}{\color{blue}-i}+{\color{red}2}{\color{blue}-7i}={\color{red}7}{\color{blue}-8i}\end{align*}

Intro Problem Revisit We're looking for \begin{align*}\sqrt{-460}\end{align*} .

First we need to pull out the \begin{align*}i\end{align*}. Then, we need to simplify \begin{align*}\sqrt{460}\end{align*} .

\begin{align*}\sqrt{-460}=\sqrt{-1} \cdot \sqrt{460}=i\sqrt{460}=i\sqrt{4 \cdot 115}=2i\sqrt{115}\end{align*}

### Guided Practice

Simplify.

1. \begin{align*}\sqrt{-49}\end{align*}

2. \begin{align*}\sqrt{-125}\end{align*}

3. \begin{align*}i^{210}\end{align*}

4. \begin{align*}(8-3i)-(12-i)\end{align*}

1. Rewrite \begin{align*}\sqrt{-49}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.

\begin{align*}\sqrt{-49}=i\sqrt{49}=7i\end{align*}

2. Rewrite \begin{align*}\sqrt{-125}\end{align*} in terms of \begin{align*}i\end{align*} and simplify the radical.

\begin{align*}\sqrt{-125}=i\sqrt{125}=i\sqrt{25 \cdot 5}=5i\sqrt{5}\end{align*}

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

4. Distribute the negative and combine like terms.

\begin{align*}(8-3i)-(12-i)=8-3i-12+i=-4-2i\end{align*}

### Vocabulary

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

### Practice

Simplify each expression and write in standard form.

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

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