# 4.4: Imaginary Numbers

Difficulty Level: At Grade Created by: CK-12
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Have you ever had an imaginary pet? Many people have, particularly as young children.

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

Silly thought, what does it have to do with imaginary numbers?

### Watch This

'Note:' For a very detailed explanation of i and the complex numbers, watch the video below, then visit: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

Embedded Video:

### Guidance

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 number 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*} can be simplified by simplifying the radical into \begin{align*}\sqrt{16} \cdot \sqrt{-1}\end{align*}, yielding: \begin{align*}4 \sqrt{-1}\end{align*} or \begin{align*}4i\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 i's with real numbers, you get complex numbers:

The definition of complex numbers: Complex numbers are of the form \begin{align*}a + bi\end{align*}, where \begin{align*}a\end{align*} is a real number, \begin{align*}b\end{align*} is a constant, and \begin{align*}i\end{align*} is the imaginary constant \begin{align*}\sqrt{-1}\end{align*}.

#### Example A

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

Solutions

\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 B

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

Solution

\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 C

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

a) What is i2?
b) What is i3?
c) What is i4?

Solutions

\begin{align*}i^2\end{align*} is the same as \begin{align*}(\sqrt{-1})^2\end{align*}. When you square a square root, they cancel and you are left with the number originally inside the radical, in this case \begin{align*}-1\end{align*}

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

\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*}

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

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

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

Concept question wrap up

Do you see the application of the crazy analogy from the introduction?

Two imaginary pets 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!

### Vocabulary

The imaginary constant i is the square root of -1.

A complex number is the sum of a real number and an imaginary number, written in the form: \begin{align*}a + bi\end{align*}.

Simplifying the radical involves factoring the term(s) under the root symbol, so that perfect squares may be "pulled out", and simplified.

### Guided Practice

1)\begin{align*}\sqrt{-9}\end{align*}

2)\begin{align*}\sqrt{-12}\end{align*}

3)\begin{align*}\sqrt{-17}\end{align*}

4)\begin{align*}\sqrt{108-140}\end{align*}

Multiply the imaginary numbers

5) \begin{align*}4i \cdot 3i\end{align*}

6 \begin{align*}\sqrt{16}i \cdot 3\end{align*}

7) \begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}

Solutions

1) To simplify the radical \begin{align*}\sqrt{-9}\end{align*}

\begin{align*}\sqrt{9 \cdot -1}\end{align*} : Rewrite \begin{align*}-9\end{align*} as \begin{align*}-1 \cdot 9\end{align*}
\begin{align*}\sqrt{9} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
\begin{align*}\sqrt{9} \cdot i\end{align*} : Substitute \begin{align*}\sqrt{-1} \to i\end{align*}
\begin{align*}3i\end{align*} : Simplify \begin{align*}\sqrt9\end{align*}

2) To simplify the radical \begin{align*}\sqrt{-12}\end{align*}

\begin{align*}\sqrt{12 \cdot -1}\end{align*} : Rewrite \begin{align*}-12\end{align*} as \begin{align*}-1 \cdot 12\end{align*}
\begin{align*}\sqrt{12} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
\begin{align*}\sqrt{12} \cdot i\end{align*} : Substitute \begin{align*}\sqrt{-1} \to i\end{align*}
\begin{align*}\sqrt{3 \cdot 4} \cdot i\end{align*} : Factor 12
\begin{align*}2\sqrt3 \cdot i\end{align*} : Simplify \begin{align*}\sqrt4\end{align*}

3) To simplify the radical\begin{align*}\sqrt{-17}\end{align*}

\begin{align*}\sqrt{17 \cdot -1}\end{align*} : Rewrite \begin{align*}-17\end{align*} as \begin{align*}-1 \cdot 17\end{align*}
\begin{align*}\sqrt{17} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
\begin{align*}\sqrt{17} \cdot i\end{align*} : Substitute \begin{align*}\sqrt{-1} \to i\end{align*}
\begin{align*}\sqrt{17}i\end{align*} : Simplify

4) To simplify the 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\sqrt2\end{align*} : Simplify \begin{align*}\sqrt{16}\end{align*}

5) To multiply \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*}

6) To multiply \begin{align*}\sqrt{16}i \cdot 3\end{align*}

\begin{align*}4i \cdot 3\end{align*} : Simplify \begin{align*}\sqrt{16}\end{align*}
\begin{align*}12i\end{align*}

7) To multiply \begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}

\begin{align*}\sqrt4 \cdot \sqrt{i^2} \cdot \sqrt4 \cdot \sqrt3 \cdot i\end{align*} : Factor
\begin{align*}2 \cdot i \cdot 2 \cdot \sqrt3 \cdot i\end{align*} : Simplify the roots
\begin{align*}4\sqrt3 \cdot i^2\end{align*} : Collect terms and simplify
\begin{align*}4\sqrt3 \cdot -1\end{align*} : Recall \begin{align*}i^2 = -1\end{align*}
\begin{align*}-4\sqrt3\end{align*}

### Practice

Simplify:

1. \begin{align*}\sqrt{-49}\end{align*}
2. \begin{align*}\sqrt{-81}\end{align*}
3. \begin{align*}\sqrt{-324}\end{align*}
4. \begin{align*}\sqrt{-121}\end{align*}
5. \begin{align*}-\sqrt{-16}\end{align*}
6. \begin{align*}-\sqrt{-1}\end{align*}
7. \begin{align*}\sqrt{-1.21}\end{align*}

Simplify:

1. \begin{align*}i^8\end{align*}
2. \begin{align*}i^{12}\end{align*}
3. \begin{align*}i^3\end{align*}
4. \begin{align*}24i^{20}\end{align*}
5. \begin{align*}i^{225}\end{align*}
6. \begin{align*}i^{1024}\end{align*}

Multiply:

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

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### Vocabulary Language: English

TermDefinition
$i$ $i$ is an imaginary number. $i=\sqrt{-1}$.
complex number A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
i $i$ is an imaginary number. $i=\sqrt{-1}$.
Imaginary Number An imaginary number is a number that can be written as the product of a real number and $i$.
Imaginary Numbers An imaginary number is a number that can be written as the product of a real number and $i$.

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