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

Let's find

First pull out the

#### Powers of i

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

Step 1: Write out

Step 2: Write out

Step 3: Write out

Step 4: Write out

Step 5: Write out

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

You should see that the powers of

Now, let's find the following powers of i.

i32

32 is divisible by 4, so

i50

i7

Finally, let's simplify the following 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 to find 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*}

#### Example 2

Simplify \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

Simplify \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

Simplify \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

Simplify \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.