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4.5.1: Imaginary Numbers

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    Imaginary Numbers

    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 x2 + 4 = 0.

    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" : \(\ i=\sqrt{-1}\)

    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 \(\ \sqrt{-16}\) can be simplified by simplifying the radical into \(\ \sqrt{16} \cdot \sqrt{-1}\), yielding: \(\ 4 \sqrt{-1}\) or \(\ 4 i\).

    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 \(\ a+b i\), where \(\ a\) is a real number, \(\ b\) is a real number, and \(\ i\) is the imaginary constant \(\ \sqrt{-1}\).

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    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 \(\ \sqrt{-5}\).


    \(\ \begin{array}{l}
    \sqrt{-5}=\sqrt{(-1) \cdot(5)} \\
    =\sqrt{-1} \sqrt{5} \\
    =i \sqrt{5}

    Example 3

    Simplify \(\ \sqrt{-72}\).


    \(\ \begin{array}{l}
    \sqrt{-72}=\sqrt{(-1) \cdot(72)} \\
    =\sqrt{-1} \sqrt{72} \\
    =i \sqrt{72}

    But, we’re not done yet! Since 72=36⋅2

    \(\ \begin{array}{l}
    i \sqrt{72}=i \sqrt{36} \sqrt{2} \\
    =i(6) \sqrt{2} \\
    =6 i \sqrt{2}

    Example 4

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

    1. What is \(\ i^2\)?
    2. What is \(\ i^3\)?
    3. What is \(\ i^4\)?


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

      \(\ \therefore i^{2}=-1\)

    2. \(\ i^3\) is the same thing as \(\ i^{2} \cdot i\), which is \(\ -1 \cdot i\) or \(\ -i\)

      \(\ \therefore i^{3}=-i\)

    3. \(\ i^{4}=i^{2} \cdot i^{2}\) which is \(\ -1 \cdot-1\)

      \(\ \therefore i^{4}=1\)

    Example 5

    Simplify the following radical: \(\ \sqrt{108-140}\).


    \(\ \sqrt{-32}\): Subtract within the parenthesis

    \(\ \sqrt{32 \cdot-1}\): Rewrite \(\ −32\) as \(\ -1 \cdot 32\)

    \(\ \sqrt{32} \cdot \sqrt{-1}\): Rewrite as a product of radicals

    \(\ \sqrt{32} \cdot i\): Substitute \(\ \sqrt{-1} \rightarrow i\)

    \(\ \sqrt{16 \cdot 2} \cdot i\): Factor \(\ 32\)

    \(\ 4 i \sqrt{2}\): Simplify \(\ \sqrt{16}\)

    Example 6

    Multiply the imaginary numbers: \(\ 4 i \cdot 3 i\).


    \(\ 4 \cdot 3 \cdot i \cdot i\): Using the commutative law for multiplication

    \(\ 12 \cdot i^{2}\): Simplify

    \(\ 12 \cdot-1\): Recall \(\ i^{2}=-1\)

    \(\ −12\)

    Example 7

    Multiply the imaginary numbers: \(\ \sqrt{4 i^{2}} \cdot \sqrt{12} i\).


    \(\ \sqrt{4} \cdot \sqrt{i^{2}} \cdot \sqrt{4} \cdot \sqrt{3} \cdot i\): Factor

    \(\ 2 \cdot i \cdot 2 \cdot \sqrt{3} \cdot i\): Simplify the roots

    \(\ 4 \sqrt{3} \cdot i^{2}\): Collect terms and simplify

    \(\ 4 \sqrt{3} \cdot-1\): Recall \(\ i^{2}=-1\)

    \(\ -4 \sqrt{3}\)



    1. \(\ \sqrt{-49}\)
    2. \(\ \sqrt{-81}\)
    3. \(\ \sqrt{-324}\)
    4. \(\ \sqrt{-121}\)
    5. \(\ -\sqrt{-16}\)
    6. \(\ -\sqrt{-1}\)
    7. \(\ \sqrt{-1.21}\)


    1. \(\ i^{8}\)
    2. \(\ i^{12}\)
    3. \(\ i^{3}\)
    4. \(\ 24 i^{20}\)
    5. \(\ i^{225}\)
    6. \(\ i^{1024}\)


    1. \(\ i^{4} \cdot i^{11}\)
    2. \(\ 5 i^{6} \cdot 5 i^{8}\)
    3. \(\ 3 \sqrt{-75} \cdot 5 \sqrt{-3}\)
    4. \(\ 2 \sqrt{-12} \cdot 6 \sqrt{-27}\)
    5. \(\ -4 \sqrt{-10} \cdot 5 \sqrt{-3} \cdot 6 \sqrt{-18}\)

    Review (Answers)

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


    Term Definition
    \(\ i\) \(\ i\) 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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