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# Imaginary Numbers

## i = sqrt (-1)

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Imaginary Numbers

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: x2+4=0\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" : i=1\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 16\begin{align*}\sqrt{-16}\end{align*} can be simplified by simplifying the radical into 161\begin{align*}\sqrt{16} \cdot \sqrt{-1}\end{align*}, yielding: 41\begin{align*}4 \sqrt{-1}\end{align*} or 4i\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 a+bi\begin{align*}a + bi\end{align*}, where a\begin{align*}a\end{align*} is a real number, b\begin{align*}b\end{align*} is a constant, and i\begin{align*}i\end{align*} is the imaginary constant 1\begin{align*}\sqrt{-1}\end{align*}.

#### Example A

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

Solutions

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

=15\begin{align*}= \sqrt{-1}\sqrt{5}\end{align*}

=i5\begin{align*}= i\sqrt{5}\end{align*}

#### Example B

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

Solution

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

=172\begin{align*}= \sqrt{-1}\sqrt{72}\end{align*}

=i72\begin{align*}= i\sqrt{72}\end{align*}

But, we’re not done yet! Since 72=362\begin{align*}72 = 36 \cdot 2\end{align*}

i72=i362\begin{align*}i\sqrt{72} = i\sqrt{36} \sqrt{2}\end{align*}

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

=6i2\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

i2\begin{align*}i^2\end{align*} is the same as (1)2\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 1\begin{align*}-1\end{align*}

a) i2=1\begin{align*}\therefore i^2 = -1\end{align*}

i3\begin{align*}i^3\end{align*} is the same thing as i2i\begin{align*}i^2 \cdot i\end{align*}, which is 1i\begin{align*}-1 \cdot i\end{align*} or i\begin{align*}-i\end{align*}

b) i3=i\begin{align*}\therefore i^3 = -i\end{align*}

i4=i2i2\begin{align*}i^4 = i^2 \cdot i^2\end{align*} which is 11\begin{align*}-1 \cdot -1\end{align*}

c) i4=1\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: a+bi\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)9\begin{align*}\sqrt{-9}\end{align*}

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

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

4)108140\begin{align*}\sqrt{108-140}\end{align*}

Multiply the imaginary numbers

5) 4i3i\begin{align*}4i \cdot 3i\end{align*}

6 16i3\begin{align*}\sqrt{16}i \cdot 3\end{align*}

7) 4i212i\begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}

Solutions

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

91\begin{align*}\sqrt{9 \cdot -1}\end{align*} : Rewrite 9\begin{align*}-9\end{align*} as 19\begin{align*}-1 \cdot 9\end{align*}
91\begin{align*}\sqrt{9} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
9i\begin{align*}\sqrt{9} \cdot i\end{align*} : Substitute 1i\begin{align*}\sqrt{-1} \to i\end{align*}
3i\begin{align*}3i\end{align*} : Simplify 9\begin{align*}\sqrt9\end{align*}

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

121\begin{align*}\sqrt{12 \cdot -1}\end{align*} : Rewrite 12\begin{align*}-12\end{align*} as 112\begin{align*}-1 \cdot 12\end{align*}
121\begin{align*}\sqrt{12} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
12i\begin{align*}\sqrt{12} \cdot i\end{align*} : Substitute 1i\begin{align*}\sqrt{-1} \to i\end{align*}
34i\begin{align*}\sqrt{3 \cdot 4} \cdot i\end{align*} : Factor 12
23i\begin{align*}2\sqrt3 \cdot i\end{align*} : Simplify 4\begin{align*}\sqrt4\end{align*}

3) To simplify the radical17\begin{align*}\sqrt{-17}\end{align*}

171\begin{align*}\sqrt{17 \cdot -1}\end{align*} : Rewrite 17\begin{align*}-17\end{align*} as 117\begin{align*}-1 \cdot 17\end{align*}
171\begin{align*}\sqrt{17} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
17i\begin{align*}\sqrt{17} \cdot i\end{align*} : Substitute 1i\begin{align*}\sqrt{-1} \to i\end{align*}
17i\begin{align*}\sqrt{17}i\end{align*} : Simplify

4) To simplify the radical 108140\begin{align*}\sqrt{108-140}\end{align*}

32\begin{align*}\sqrt{-32}\end{align*} : Subtract within the parenthesis
321\begin{align*}\sqrt{32 \cdot -1}\end{align*} : Rewrite 32\begin{align*}-32\end{align*} as 132\begin{align*}-1 \cdot 32\end{align*}
321\begin{align*}\sqrt{32} \cdot \sqrt{-1}\end{align*} : Rewrite as a product of radicals
32i\begin{align*}\sqrt{32} \cdot i\end{align*} : Substitute 1i\begin{align*}\sqrt{-1} \to i\end{align*}
162i\begin{align*}\sqrt{16 \cdot 2} \cdot i\end{align*} : Factor 32\begin{align*}32\end{align*}
4i2\begin{align*}4i\sqrt2\end{align*} : Simplify 16\begin{align*}\sqrt{16}\end{align*}

5) To multiply 4i3i\begin{align*}4i \cdot 3i\end{align*}

43ii\begin{align*}4 \cdot 3 \cdot i \cdot i\end{align*} : Using the commutative law for multiplication
12i2\begin{align*}12 \cdot i^2\end{align*} : Simplify
121\begin{align*}12 \cdot -1\end{align*} : Recall i2=1\begin{align*}i^2 = -1\end{align*}
12\begin{align*}-12\end{align*}

6) To multiply 16i3\begin{align*}\sqrt{16}i \cdot 3\end{align*}

4i3\begin{align*}4i \cdot 3\end{align*} : Simplify 16\begin{align*}\sqrt{16}\end{align*}
12i\begin{align*}12i\end{align*}

7) To multiply 4i212i\begin{align*}\sqrt{4i^2} \cdot \sqrt{12}i\end{align*}

4i243i\begin{align*}\sqrt4 \cdot \sqrt{i^2} \cdot \sqrt4 \cdot \sqrt3 \cdot i\end{align*} : Factor
2i23i\begin{align*}2 \cdot i \cdot 2 \cdot \sqrt3 \cdot i\end{align*} : Simplify the roots
43i2\begin{align*}4\sqrt3 \cdot i^2\end{align*} : Collect terms and simplify
431\begin{align*}4\sqrt3 \cdot -1\end{align*} : Recall i2=1\begin{align*}i^2 = -1\end{align*}
43\begin{align*}-4\sqrt3\end{align*}

### Practice

Simplify:

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

Simplify:

1. i8\begin{align*}i^8\end{align*}
2. i12\begin{align*}i^{12}\end{align*}
3. i3\begin{align*}i^3\end{align*}
4. 24i20\begin{align*}24i^{20}\end{align*}
5. i225\begin{align*}i^{225}\end{align*}
6. i1024\begin{align*}i^{1024}\end{align*}

Multiply:

1. i4i11\begin{align*}i^4 \cdot i^{11}\end{align*}
2. 5i65i8\begin{align*}5i^6 \cdot 5i^8\end{align*}
3. 37553\begin{align*}3\sqrt{-75} \cdot 5\sqrt{-3}\end{align*}
4. 212627\begin{align*}2\sqrt{-12} \cdot 6\sqrt{-27}\end{align*}
5. 41053618\begin{align*}-4 \sqrt{-10} \cdot 5 \sqrt{-3} \cdot 6\sqrt {-18}\end{align*}

### Vocabulary Language: English

$i$

$i$

$i$ is an imaginary number. $i=\sqrt{-1}$.
complex number

complex number

A complex number is the sum of a real number and an imaginary number, written in the form $a + bi$.
i

i

$i$ is an imaginary number. $i=\sqrt{-1}$.
Imaginary Number

Imaginary Number

An imaginary number is a number that can be written as the product of a real number and $i$.
Imaginary Numbers

Imaginary Numbers

An imaginary number is a number that can be written as the product of a real number and $i$.