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Imaginary Numbers Homework -- Sample Test: Complex Numbers

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

Summary: This module provides a sample test on complex numbers.

Exercise 1

Fill in the following table.

Table 1
i -1 i -1  
i 0 i 0  
i 1 i 1  
i 2 i 2  
i 3 i 3  
i 4 i 4  
i 5 i 5  
i 6 i 6  
i 7 i 7  

Simplify.

Exercise 2

( i ) 85 = (i ) 85 =

Exercise 3

( 5 i ) 2 = (5i ) 2 =

Exercise 4

( n i ) 103 = (ni ) 103 =

Exercise 5

  • a. 2020 size 12{ sqrt { - "20"} } {} =
  • b. Other than your answer to part (a), is there any other number that you can squareto get –20? If so, what is it?

Exercise 6

( 3 w z i ) 2 = (3wzi ) 2 =

Exercise 7

  • a. Complex conjugate of 4 + i = 4+i=
  • b. What do you get when you multiply 4 + i 4+i by its complex conjugate?

If the following are simplified to the form a + b i a+bi, what are a a and b b in each case?

Exercise 8

–i–i

  • a. a = a=
  • b. b = b=

Exercise 9

nini size 12{ { {n} over {i} } } {}

  • a. a = a=
  • b. b = b=

Exercise 10

4x16ix2i3i4x16ix2i3i size 12{ { {4x} over {1 - 6 ital "ix"} } - { {2i} over {3 - i} } } {}

  • a. a = a=
  • b. b = b=

Exercise 11

If 2 x + 3 x i + 2 y = 28 + 9 i 2x+3xi+2y=28+9i, what are xx and yy?

Exercise 12

Make up a quadratic equation (using all real numbers) that has two non-real roots, and solve it.

Exercise 13

  • a. Find the two complex numbers (of course in the form z = a + b i ) z=a+bi) that fill the condition z 2 = –2 i z 2 =–2i.
  • b. Check one of your answers to part (a), by squaring it to make sure you get –2 i –2i.

Extra credit:

Complex numbers cannot be graphed on a number line. But they can be graphed on a 2-dimensional graph: you graph the point x + i y x+iy at ( xx, yy).

  1. If you graph the point 5 + 12 i 5+12i, how far is that point from the origin (0,0)?
  2. If you graph the point x + i y x+iy, how far is that point from the origin (0,0)?
  3. What do you get if you multiply the point x + i y x+iy by its complex conjugate? How does this relate to your answer to part (b)?

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