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, , and are all examples of real numbers. With what we have previously learned, we cannot find 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 .

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

All **complex numbers** have the form , where and are real numbers. is the **real part** of the complex number and is the **imaginary part**. If , then is left and the number is a **real number.** If , then the number is only and called a **pure imaginary number**. If and , the number will be an imaginary number.

Let's find .

First pull out the . Then, simplify .

#### Powers of i

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

Step 1: Write out and simplify.

Step 2: Write out and simplify.

Step 3: Write out and simplify.

Step 4: Write out and simplify.

Step 5: Write out and simplify.

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

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

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

32 is divisible by 4, so .

, with a remainder of 2. Therefore, .

, with a remainder of 3. Therefore,

Finally, let's simplify the following complex expressions.

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 .

### Examples

#### Example 1

Earlier, you were asked to find the square root of -460 degrees.

We're looking for .

First we need to pull out the . Then, we need to simplify .

#### Example 2

Simplify .

Rewrite in terms of and simplify the radical.

#### Example 3

Simplify .

Rewrite in terms of and simplify the radical.

#### Example 4

Simplify .

, with a remainder of 2. Therefore, .

#### Example 5

Simplify .

Distribute the negative and combine like terms.

### Review

Simplify each expression and write in standard form.

### Answers for Review Problems

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