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Homework: Imaginary Numbers

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

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

We began our in-class assignment by talking about why x 3 = –1 x 3 =–1 does have a solution, whereas x 2 = –1 x 2 =–1 does not. Let’s talk about the same thing graphically.

Exercise 1

On the graph below, do a quick sketch of y = x 3 y= x 3 .

ppp

  • a. Draw, on your graph, all the points on the curve where y = 1 y=1. How many are there?
  • b. Draw, on your graph, all the points on the curve where y = 0 y=0. How many are there?
  • c. Draw, on your graph, all the points on the curve where y = –1 y=–1. How many are there?

Exercise 2

On the graph below, do a quick sketch of y = x 2 y= x 2 .

ppp

  • a. Draw, on your graph, all the points on the curve where y = 1 y=1. How many are there?
  • b. Draw, on your graph, all the points on the curve where y = 0 y=0. How many are there?
  • c. Draw, on your graph, all the points on the curve where y = -1 y=-1. How many are there?

Exercise 3

Based on your sketch in exercise #2…

  • a. If a a is some number such that a > 0 a>0, how many solutions are there to the equation x 2 = a x 2 =a?
  • b. If a a is some number such that a = 0 a=0, how many solutions are there to the equation x 2 = a x 2 =a?
  • c. If a a is some number such that a < 0 a<0, how many solutions are there to the equation x 2 = a x 2 =a?
  • d. If ii is defined by the equation i 2 = –1 i 2 =–1, where the heck is it on the graph?

OK, let’s get a bit more practice with ii.

Exercise 4

In class, we made a table of powers of ii, and found that there was a repeating pattern. Make that table again quickly below, to see the pattern.

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

Exercise 5

Now let’s walk that table backward. Assuming the pattern keeps up as you back up, fill in the following table. (Start at the bottom.)

Table 2
i-4i-4  
i-3i-3  
i-2i-2  
i-1i-1  
i0i0  

Exercise 6

Did it work? Let’s figure it out. What should i 0 i 0 be, according to our general rules of exponents?

Exercise 7

What should i –1 i –1 be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

Exercise 8

What should i –2 i –2 be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

Exercise 9

What should i –3 i –3 be, according to our general rules of exponents? Can you simplify it to look like the answer in your table?

Exercise 10

Simplify the fraction i43ii43i size 12{ { {i} over {4 - 3i} } } {}.

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