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Imaginary Numbers

Module by: Kenny M. Felder. E-mail the author

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

Exercise 1

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

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 x 2 =–1? Obviously it wouldn’t be a number that we are familiar with (such as 5, - 3 4 - 3 4 , or π π). So, let’s just give it a new name: ii, because it’s imaginary. What would it be like?

Definition of i

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

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

Exercise 2

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

Exercise 3

What is ( –i ) 2 (–i ) 2 ?

Exercise 4

What is ( 3 i ) 2 (3i ) 2 ?

Exercise 5

What is ( –3 i ) 2 (–3i ) 2 ?

Exercise 6

What is 2i22i2 size 12{ left ( sqrt {2} `i right ) rSup { size 8{2} } } {}?

Exercise 7

What is 2i22i2 size 12{ left ( sqrt {2i} right ) rSup { size 8{2} } } {}?

Exercise 8

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

Exercise 9

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

Exercise 10

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

Exercise 11

Fill in the following table.

Table 1
i1i1  
i2i2  
i3i3  
i4i4  
i5i5  
i6i6  
i7i7  
i8i8  
i9i9  
i10i10  
i11i11  
i12i12  

Exercise 12

Fill in the following table.

Table 2
i 100 i 100  
i 101 i 101  
i 102 i 102  
i 103 i 103  
i 104 i 104  

Now let’s have some more fun!

Exercise 13

( 3 + 4 i ) 2 = (3+4i ) 2 =

Exercise 14

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

Exercise 15

1i2=1i2 size 12{ left ( { {1} over {i} } right ) rSup { size 8{2} } } {}=

Exercise 16

Simplify the fraction 1i1i size 12{ { {1} over {i} } } {}.

Hint:

Multiply the top and bottom by i i.

Exercise 17

Square your answer to #16. Did you get the same answer you got to #15? Why or why not?

Exercise 18

Simplify the fraction 13+2i13+2i size 12{ { {1} over {3+2i} } } {}.

Hint:

Multiply the top and bottom by 3 – 2 i 3–2i.

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