Have you ever had an imaginary pet? Some people have, particularly as young children.

Wouldn't you be surprised if you and your real friend left your two imaginary puppies alone together, and you came back to find a real puppy?

It's a silly thought, so what does it have to do with imaginary numbers?

### Imaginary Numbers

What is the square root of -1?

You may recall running into roots of negatives in algebra, when attempting to solve equations like \begin{align*}x^2 + 4 = 0\end{align*}

Since there are no real numbers that can be squared to equal -4, this equation has no real solution. Enter the imaginary constant: "i".

**The definition of "i"** : \begin{align*}i = \sqrt{-1}\end{align*}

The use of the word *imaginary* does not mean these numbers are useless. For a long period in the history of mathematics, it was thought that the square root of a negative number was in fact only within the mathematical imagination, without real-world significance hence, imaginary. That has changed. Mathematicians now consider the imaginary numbers as another set of numbers that have real significance, but do not fit on what is called the number line, and engineers, scientists, and others solve real world problems using combinations of real and imaginary numbers (called complex numbers) every day.

Imaginary values such as \begin{align*}\sqrt{-16}\end{align*}

The uses of *i* become more apparent when you begin working with increased powers of *i*, as you will see in the examples below.

#### Complex Numbers

When you combine imaginary numbers with real numbers, you get complex numbers:

**Complex numbers** are of the form \begin{align*}a + bi\end{align*}

### Examples

#### Example 1

Earlier, you were given an analogy about imaginary pets.

Do you see its application?

Two imaginary puppies creating a real puppy is an oddly effective metaphor for the behavior of the powers of *i*.

One *i* is imaginary, but two *i's* multiply to be a real number. In fact, *every* even power of *i* results in a real number!

#### Example 2

Simplify \begin{align*}\sqrt{-5}\end{align*}

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

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

\begin{align*}= i\sqrt{5}\end{align*}

#### Example 3

Simplify \begin{align*}\sqrt{-72}\end{align*}

\begin{align*}\sqrt{-72} = \sqrt{(-1)\cdot (72)}\end{align*}

\begin{align*}= \sqrt{-1}\sqrt{72}\end{align*}

\begin{align*}= i\sqrt{72}\end{align*}

But, we’re not done yet! Since \begin{align*}72 = 36 \cdot 2\end{align*}

\begin{align*}i\sqrt{72} = i\sqrt{36} \sqrt{2}\end{align*}

\begin{align*}= i(6)\sqrt{2}\end{align*}

\begin{align*}= 6i\sqrt{2}\end{align*}

#### Example 4

Strange things happen when the imaginary constant *i* is multiplied by itself different numbers of times.

- What is
*i*^{2}?

\begin{align*}i^2\end{align*}

\begin{align*}\therefore i^2 = -1\end{align*}

- What is
*i*^{3}?

\begin{align*}i^3\end{align*} is the same thing as \begin{align*}i^2 \cdot i\end{align*}, which is \begin{align*}-1 \cdot i\end{align*} or \begin{align*}-i\end{align*}

\begin{align*}\therefore i^3 = -i\end{align*}

- What is
*i*^{4}?

\begin{align*}i^4 = i^2 \cdot i^2\end{align*} which is \begin{align*}-1 \cdot -1\end{align*}

\begin{align*}\therefore i^4 = 1\end{align*}

#### Example 5

Simplify the following radical: \begin{align*}\sqrt{108-140}\end{align*}.

\begin{align*}\sqrt{-32}\end{align*} : Subtract within the parenthesis

\begin{align*}\sqrt{32 \cdot -1}\end{align*} : Rewrite \begin{align*}-32\end{align*} as \begin{align*}-1 \cdot 32\end{align*}

\begin{align*}\sqrt{32} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals

\begin{align*}\sqrt{32} \cdot i\end{align*} : Substitute \begin{align*}\sqrt{-1} \to i\end{align*}

\begin{align*}\sqrt{16 \cdot 2} \cdot i\end{align*} : Factor \begin{align*}32\end{align*}

\begin{align*}4i\sqrt{2}\end{align*} : Simplify \begin{align*}\sqrt{16}\end{align*}

#### Example 6

Multiply the imaginary numbers: \begin{align*}4i \cdot 3i\end{align*}.

\begin{align*}4 \cdot 3 \cdot i \cdot i\end{align*} : Using the commutative law for multiplication

\begin{align*}12 \cdot i^2\end{align*} : Simplify

\begin{align*}12 \cdot -1\end{align*} : Recall \begin{align*}i^2 = -1\end{align*}

\begin{align*}-12\end{align*}

#### Example 7

Multiply the imaginary numbers: \begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}.

\begin{align*}\sqrt{4} \cdot \sqrt{i^2} \cdot \sqrt{4} \cdot \sqrt{3} \cdot i\end{align*} : Factor

\begin{align*}2 \cdot i \cdot 2 \cdot \sqrt{3} \cdot i\end{align*} : Simplify the roots

\begin{align*}4\sqrt{3} \cdot i^2\end{align*} : Collect terms and simplify

\begin{align*}4\sqrt{3} \cdot -1\end{align*} : Recall \begin{align*}i^2 = -1\end{align*}

\begin{align*}-4\sqrt{3}\end{align*}

### Review

Simplify:

- \begin{align*}\sqrt{-49}\end{align*}
- \begin{align*}\sqrt{-81}\end{align*}
- \begin{align*}\sqrt{-324}\end{align*}
- \begin{align*}\sqrt{-121}\end{align*}
- \begin{align*}-\sqrt{-16}\end{align*}
- \begin{align*}-\sqrt{-1}\end{align*}
- \begin{align*}\sqrt{-1.21}\end{align*}

Simplify:

- \begin{align*}i^8\end{align*}
- \begin{align*}i^{12}\end{align*}
- \begin{align*}i^3\end{align*}
- \begin{align*}24i^{20}\end{align*}
- \begin{align*}i^{225}\end{align*}
- \begin{align*}i^{1024}\end{align*}

Multiply:

- \begin{align*}i^4 \cdot i^{11}\end{align*}
- \begin{align*}5i^6 \cdot 5i^8\end{align*}
- \begin{align*}3\sqrt{-75} \cdot 5\sqrt{-3}\end{align*}
- \begin{align*}2\sqrt{-12} \cdot 6\sqrt{-27}\end{align*}
- \begin{align*}-4 \sqrt{-10} \cdot 5 \sqrt{-3} \cdot 6\sqrt {-18}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.4.