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This module provides sample problems related to i and imaginary numbers.

Explain, using words and equations, why the equation x 2 = –1 has no answer, but x 3 = –1 does.

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OK, so now we are going to use our imaginations. (Didn’t think we were allowed to do that in math class, did you?) Suppose there were an answer to x 2 = –1 ? Obviously it wouldn’t be a number that we are familiar with (such as 5, - 3 4 , or π ). So, let’s just give it a new name: i , because it’s imaginary . What would it be like?

Definition of i

The definition of the imaginary number i is that it is the square root of –1 :

i = 1 size 12{ sqrt { - 1} } {} or, equivalently, i 2 = –1

Based on that definition, answer the following questions. In each case, don’t just guess—give a good mathematical reason why the answer should be what you say it is!

What is i ( –i ) ? (*Remember that –i means –1 × i .)

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What is 2 i 2 size 12{ left ( sqrt {2} `i right ) rSup { size 8{2} } } {} ?

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What is 2i 2 size 12{ left ( sqrt {2i} right ) rSup { size 8{2} } } {} ?

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What is 25 size 12{ sqrt { - "25"} } {} ?

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What is 3 size 12{ sqrt { - 3} } {} ?

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What is 8 size 12{ sqrt { - 8} } {} ?

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Fill in the following table.

i 1
i 2
i 3
i 4
i 5
i 6
i 7
i 8
i 9
i 10
i 11
i 12
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Fill in the following table.

i 100
i 101
i 102
i 103
i 104
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Now let’s have some more fun!

( 3 + 4 i ) ( 3 4 i ) =

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1 i 2 size 12{ left ( { {1} over {i} } right ) rSup { size 8{2} } } {} =

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Simplify the fraction 1 i size 12{ { {1} over {i} } } {} .

Multiply the top and bottom by i .
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Square your answer to #16. Did you get the same answer you got to #15? Why or why not?

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Simplify the fraction 1 3 + 2i size 12{ { {1} over {3+2i} } } {} .

Multiply the top and bottom by 3 2 i .
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Source:  OpenStax, Advanced algebra ii: activities and homework. OpenStax CNX. Sep 15, 2009 Download for free at http://cnx.org/content/col10686/1.5
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