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Imaginary Numbers

An imaginary number is the square root of a negative number. How is that imaginary?  It's something that can happen when you're doing math. For instance, what if you have the equation:

$x^2=-16$

You'll square root both sides.

$\eqalign{x^2&=-16\\\sqrt{x^2}&=\sqrt{-16}\\x&=\sqrt{-16}}$

$x$ equals the square root of -16 is an answer that you could get.  But, it's not a mathematically possible number.  Why not?  What number, times itself, is equal to a negative number?

$\eqalign{4\times4&=16\\-4 \times -4&=16}$

Think about it. The  only way to get a negative number is to multiple a negative times a positive -- and that's not a perfect square. So, the square root of a negative number is not possible in real math.  It's not a real number.

But it can happen. Math deals with this with imaginary numbers.  Imaginary numbers are written as $i$.

$i=\sqrt{-1}$

Officially, the definition of $i$ is that $i^2=-1$.

Because squares and square roots are inverse operations $i^2=-1$ and $i=\sqrt{-1}$ are equivalent statements.

The most interesting thing about $i$ is that you can work with it in ways that can make it turn back into a real number.  

Critically, know that working with exponents of $i$ results in a pattern:

$\eqalign{i&=\sqrt{-1}\\i^2&=\sqrt{-1}\times \sqrt{-1}=-1\\i^3&=\sqrt{-1}\times \sqrt{-1}\times \sqrt{-1}=-1\sqrt{-1}\\i^4&= \sqrt{-1}\times \sqrt{-1}\times \sqrt{-1}\times\sqrt{-1}=-1 \times -1=1}$

This pattern continues:

$\eqalign{i&=\sqrt{-1}\\i^2&=-1\\i^3&=-1\sqrt{-1}\\i^4&=1\\i^5&=\sqrt{-1}\\i^6&=-1\\i^7&=-1\sqrt{-1}\\i^8&= 1\\i^9&=\sqrt{-1}\\i^10&=-1\\i^11&=-1\sqrt{-1}\\i^12&=1}$

You can figure out the value of $i$ to any power by dividing the exponent by 4. 

If the remainder is 0, $i^x=1$

If the remainder is 1, $i^x=\sqrt{-1}$

If the remainder is 2, $i^x=-1$

If the remainder is 3, $i^x=-1\sqrt{-1}$

Let's try it. 

Example:

$i^{88}=$

$88\div4=11$  

There is no remainder, so $i^{88}=1$

 

How can you use $i$?  First thing of it as a radical.  So anything you need to do with a radical (for instance, rationalize it out of the denominator of a fraction) you need to do with $i$.  But, you can also use it in problems like the one we started with.  Rather than stopping at an "impossible" answer, you can use $i$ to finish the problem:

$\eqalign{x^2&=-16\\\sqrt{x^2}&=\sqrt{-16}\\x&=\sqrt{-16}\\x&=4i}$

Practice Problems:

  • Imaginary Numbers

    Write the following numbers without using $i$.  

    1. $i$
    2. $i^2$
    3. $i^3$
    4. $i^4$
    5. $i^5$
    6. $4i^3$
    7. $6i^9$
    8. $8i^{12}$
    9. $2i^2$
    10. $5i^3$
    11. $7i^6$
    12. $11i^9$
    13. $21i^{14}$
    14. $56i^{16}$
    15. $7i^{20}$
    16. $9i^{33}$
    17. $3i^{32}$
    18. $4i^{30}$
    19. $(5i^2)(6i^2)$
    20. $(7i^3)(8i^7)$

    Answer Key:

Skill:

Common Core Grade Level/Subject

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