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

### Complex Numbers

Before this concept, all numbers have been real numbers. 2, -5,

In order to take the square root of a negative number we are going to assign **imaginary number**. Now, we can use

All **complex numbers** have the form **real part** of the complex number and **imaginary part**. If **real number.** If **pure imaginary number**. If

#### Solve the following problems

Find

First pull out the

#### Investigation: Powers of i

In addition to now being able to take the square root of a negative number,

1. Write out

2. Write out

3. Write out

4. Write out

5. Write out

6. Do you see a pattern? Describe it and try to find

You should see that the powers of

Find:

32 is divisible by 4, so

Simplify the complex expressions.

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

\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*}

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

\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*}

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*}.

### Examples

#### Example 1

Earlier, you were asked what is the square root of -460 degrees.

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*}

Simplify.

#### Example 2

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

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*}

#### Example 3

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

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*}

#### Example 4

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

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

#### Example 5

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

Distribute the negative and combine like terms.

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

### Review

Simplify each expression and write in standard form.

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

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.8.