The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.

1. Section 6-8
Imaginary Numbers
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2. Warmup
Simplify the following:
2 2
() ( )
1. 2 3 2. −3 2
3. 6 • 15 4. -100
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3. Warmup
Simplify the following:
2 2
() ( )
1. 2 3 2. −3 2
12
3. 6 • 15 4. -100
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4. Warmup
Simplify the following:
2 2
() ( )
1. 2 3 2. −3 2
12 18
3. 6 • 15 4. -100
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5. Warmup
Simplify the following:
2 2
() ( )
1. 2 3 2. −3 2
12 18
3. 6 • 15 4. -100
≈ 9.49
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6. Warmup
Simplify the following:
2 2
() ( )
1. 2 3 2. −3 2
12 18
3. 6 • 15 4. -100
≈ 9.49 ???
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7. Definition
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8. Definition
When k > 0, the two solutions to x2 = k are
k and − k
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9. Big Question
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10. What
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11. is
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12. −k ?
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13. Friday, February 13, 2009

14. What is −k ?
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15. Imaginary Number
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16. Imaginary Number
i = −1
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17. Theorem
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18. Theorem
If k > 0,
−k = −1 k = i k
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19. Example 1
Solve.
x2 = -100
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20. Example 1
Solve.
x2 = -100
2
x = ± −100
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21. Example 1
Solve.
x2 = -100
2
x = ± −100
x = ± −1 100
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22. Example 1
Solve.
x2 = -100
2
x = ± −100
x = ± −1 100
x=±
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23. Example 1
Solve.
x2 = -100
2
x = ± −100
x = ± −1 100
x = ±i
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24. Example 1
Solve.
x2 = -100
2
x = ± −100
x = ± −1 100
x = ± i 10
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25. Example 1
Solve.
x2 = -100
2
x = ± −100
x = ± −1 100
x = ± i 10
x = ±10i
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26. Example 2
Show that i 7 is a square root of -7.
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27. Example 2
Show that i 7 is a square root of -7.
2
(i 7 )
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28. Example 2
Show that i 7 is a square root of -7.
2
(i 7 )
2
( )( )
2
7
=i
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29. Example 2
Show that i 7 is a square root of -7.
2
(i 7 )
2
( )( 7 )
2
=i
= ( −1) ( 7 )
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30. Example 2
Show that i 7 is a square root of -7.
2
(i 7 )
2
( )( 7 )
2
=i
= ( −1) ( 7 )
= −7
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31. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
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32. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i
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33. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i -
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34. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i - 7i
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35. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i - 7i
= -5i
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36. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i - 7i = 24i2
= -5i
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37. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i - 7i = 24i2
= -5i = 24(-1)
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38. Example 3
Simplify.
b. (6i)(4i)
a. −4 − −49
= 2i - 7i = 24i2
= -5i = 24(-1)
= -24
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39. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
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40. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
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41. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
= −16 2 + −2
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42. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
= −16 2 + −2
= 4i 2 + i 2
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43. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
= −16 2 + −2
= 4i 2 + i 2
= 5i 2
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44. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
5i
= −16 2 + −2 =
9i
= 4i 2 + i 2
= 5i 2
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45. Example 3
Simplify.
−25
d.
c. −32 + −2
−81
2
= −16g + −2
5i
= −16 2 + −2 =
9i
= 4i 2 + i 2
5
= 5i 2 =
9
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46. Example 4
Simplify.
−36 −64
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47. Example 4
Simplify.
−36 −64
= 6ig8i
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48. Example 4
Simplify.
−36 −64
= 6ig8i
2
= 48i
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49. Example 4
Simplify.
−36 −64
= 6ig8i
2
= 48i
= −48
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50. Example 4
Simplify.
−36 −64
= 6ig8i
2
= 48i
= −48
**NOTE**
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51. Example 4
Simplify.
−36 −64
= 6ig8i
2
= 48i
= −48
**NOTE**
Do NOT combine radicals that have negatives inside!
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52. Example 4
Simplify.
−36 −64
= 6ig8i
2
= 48i
= −48
**NOTE**
Do NOT combine radicals that have negatives inside!
−36 −64 ≠ 2304 = 48
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53. Homework
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54. Homework
p. 391 #1 - 29
“They can because they think they can.” - Virgil
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