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Resource packet
• Simplifying complex
numbers
• Operations on
complex numbers
© 2013 The Enlightened Elephant
Name: _____________________________________________________
Date: ___________
Complex Numbers
Main Ideas
The imaginary
unit i
Notes
Definition:____________________________________________________
______________________________________________________________
______________________________________________________________
Characteristics:_______________________________________________
______________________________________________________________
______________________________________________________________
Examples
Complex number
NON-Examples
Definition:____________________________________________________
______________________________________________________________
______________________________________________________________
Characteristics:_______________________________________________
______________________________________________________________
______________________________________________________________
Examples
NON-Examples
Main Ideas
Simplify a
complex number
Notes
Steps
 Rewrite radicand as a product of __________ ___________
factors, -1 and another factor.
 Simplify the __________________ (recall: −1 = 𝑖).
 Write as a + bi
Examples:
Simplify  18  6 and write as a +bi



Simplify  28  7 and write as a +bi



Practice
Simplify and write as a +bi
1.
−12 − 8
3. −27 + 17
2. − −32 + 3
4. − −50 − 25
Date: ___________
Name: ___________KEY______________________________________
Complex Numbers
Main Ideas
The imaginary
unit i
Notes
“i” is the number whose square is -1
Definition:____________________________________________________
i  1
______________________________________________________________
2
______________________________________________________________
1  i
Characteristics:_______________________________________________
______________________________________________________________
______________________________________________________________
Examples
i
6
1
Complex number
NON-Examples

6
A number in the form of a +bi
Definition:____________________________________________________
______________________________________________________________
______________________________________________________________
a and b are real numbers.
Characteristics:_______________________________________________
a is the real part and bi is the imaginary part
______________________________________________________________
______________________________________________________________
Examples
3i
-5i
5 +6i
7-2i
-5- I
25
NON-Examples
None, all numbers are
included in the set of
complex numbers
Main Ideas
Simplify a
complex number
Notes
Steps
 Rewrite radicand as a product of perfect square factors, -1
and another factor.
 Simplify the radical (recall: −1 = 𝑖).
 Write as a + bi
Examples:
Simplify  18  6 and write as a +bi

9(1)(2)  6

3i 2  6

6  3i 2
Simplify  28  7 and write as a +bi
Practice

4(1)(7)  7

2i 7  7
  7  2i 7
Simplify and write as a +bi
1.
−12 − 8
−8 + 4𝑖 3
3. −27 + 17
17 + 3𝑖 3
2. − −32 + 3
3 + 4𝑖 2
4. − −50 − 25
−25 − 5𝑖 2
Name ________________________________________________________ Date__________ Per._____
Simplify Complex Numbers
Directions: Simplify and write as a +bi
1.
4
2.
9  2
3.
 16  1
4.
8 3
5.   7  8
6.
 27  4
7.   32  5
8.
 72  8
10.
 48  9
9.
 50  6
© 2013 The Enlightened Elephant
Name _______KEY_____________________________________________ Date__________ Per._____
Simplify Complex Numbers
Simplify and write as a +bi
1.
4
2.
2. 2  3i
1. 2i
3.
 16  1
4.
3. - 1  4i
5.   7  8
6.
 50  6
9. 6  5i 2
© 2013 The Enlightened Elephant
 27  4
6.  4  3i 3
8.
 72  8
8. 8  6i 2
7. - 5  4i 2
9.
8 3
4.  3  2i 2
5. 8  i 7
7.   32  5
9  2
10.
 48  9
10. 9  4i 3
Name: _____________________________________________________
Date: ___________
Operations on Complex Numbers
Main Ideas
Additive Inverse of
a complex number
Notes
An additive inverse is also called the __________________________
Definition:____________________________________________________
______________________________________________________________
______________________________________________________________
Examples
Adding Complex
Numbers
Steps
 Add the ___________ parts.
 Add the _________________ parts.
 Write the _________________ as a + bi.
Examples
Simplify and write as a+bi
1. (3 + 2i) + (-4 + 9i)
2. (-6 - i) + (3 + 8i)
3. (9 - 2i) + (2 + 10i)
4. (7 + 2i) + (3 - 6i)
Main Ideas
Subtracting
complex numbers
Notes
Steps
 Distribute the _____________________.
 Combine __________________(real parts and imaginary parts).
 Write as a + bi
Example:
Simplify (5  2i )  (3  4i ) and write as a +bi



Multiplying
Monomials
Steps
 Multiply the _______________ parts.
 Multiply the ________________parts.
 Simplify (recall: i2 = -1.
Examples
Multiply.
1. (2i)(-4i)
Multiplying
Binomials
2. (3i)(7i)
Steps
 Multiply using the _______________ method.
 ___________________ and write as a +bi.
Examples
Multiply.
1. (3 + 4i)(7 - 4i)
2. (8 - 9i)(4 -6i)
Date: ___________
Name: _____________________________________________________
Operations on Complex Numbers
Main Ideas
Additive Inverse of
a complex number
Notes
An additive inverse is also called the __________________________
opposite
Definition:____________________________________________________
Two numbers whose sum is zero are called additive
inverses.
______________________________________________________________
______________________________________________________________
Examples
6+2i and -6-2i
Adding Complex
Numbers
Steps
 Add the real parts.
 Add the imaginary parts.
 Write the answer as a + bi.
Examples
Simplify and write as a+bi
1. (3 + 2i) + (-4 + 9i)
-1+11i
3. (9 - 2i) + (2 + 10i)
11+8i
2. (-6 - i) + (3 + 8i)
-3+7i
4. (7 + 2i) + (3 - 6i)
10-4i
Main Ideas
Subtracting
complex numbers
Notes
Steps
 Distribute the negative.
 Combine like terms(real parts and imaginary parts).
 Write as a + bi
Example:
Simplify (5  2i )  (3  4i ) and write as a +bi
 5-2i-3+4i
 2+2i
 2+2i
Multiplying
Monomials
Steps
 Multiply the real parts.
 Multiply the imaginary parts.
 Simplify.
Examples
Multiply.
1. (2i)(-4i)
-8i2 = 8
Multiplying
Binomials
2. (3i)(7i)
21i2 = -21
Steps
 Multiply using the FOIL method.
 Simplify and write as a +bi.
Examples
Multiply.
1. (3 + 4i)(7 - 4i)
21-12i+28i-16i2
=37+16i
2. (8 - 9i)(4 -6i)
32-48i - 36i+54i2
=-22 - 84i
Name ________________________________________________________ Date__________ Per._____
Adding and Subtracting Complex Numbers
State the additive inverse.
1.  25i
2. 7  6i
Add.
1. (4  2i )  (6-2i )
2. (-3  i )  (5  7i )
3. (12  8i )  (-2i )
4. (-13  4i )  (5  18i )  (15  3i )
Subtract.
1. (6  8i )  ( 4-i)
2. (-1  3i )  ( 4  9i )
3. (2  6i )  (9  i )
4. (-18  7i )  (15  12i )
5. (12  10i )  (19  5i )  (6  2i )
6. (-256  83i )  (12  123i )
© 2013 The Enlightened Elephant
Name ________________________________________________________ Date__________ Per._____
Multiply Complex Numbers
Multiply.
1. (4i )(i )
2. (-6i )(3i )
3. (3i )(10i )(-i)
4. 5(2  i )
5. (2  i )(4  i )
6. (12  i )(2  4i )
7. (5  9i )(6  2i )
8. (12  i )(3  6i )
9. ( 3  i )( 2  4i )
© 2013 The Enlightened Elephant
10. (3  i )(4  2i )(5  3i )
key
Name ________________________________________________________ Date__________ Per._____
Adding and Subtracting Complex Numbers
State the additive inverse.
1.  25i
2. 7  6i
25i
 7  6i
Add.
1. (4  2i )  (6-2i )
10
3. (12  8i )  (-2i )
1210i
2. (-3  i )  (5  7i )
2  8i
4. (-13  4i )  (5  18i )  (15  3i )
7  25i
Subtract.
1. (6  8i )  ( 4-i)
2  9i
3. (2  6i )  (9  i )
11 5i
5. (12  10i )  (19  5i )  (6  2i )
13 13i
© 2013 The Enlightened Elephant
2. (-1  3i )  ( 4  9i )
 5 6i
4. (-18  7i )  (15  12i )
 33 5i
6. (-256  83i )  (12  123i )
 244 40i
key
Name ________________________________________________________ Date__________ Per._____
Multiply Complex Numbers
Multiply.
2. (-6i )(3i )
1. (4i )(i )
4
3. (3i )(10i )(-i)
30i
5. (2  i )(4  i )
9  2i
7. (5  9i )(6  2i )
 48  44i
9. ( 3  i )( 2  4i )
6  4i 3  i 2  4
18
4. 5(2  i )
10  5i
6. (12  i )(2  4i )
 28  46i
8. (12  i )(3  6i )
 42  69i
10. (3  i )(4  2i )(5  3i )
 64  52i
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers A
Add or Subtract. Write as a +bi.
1. (4  5i )  (3i )
2. (7  8i )  (4  i )
3. (6  5i)  (2  9i)
4. (2  3i)  (6  7i)
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers A
Add or Subtract. Write as a +bi.
1. (4  5i )  (3i )
2. (7  8i )  (4  i )
3. (6  5i)  (2  9i)
4. (2  3i)  (6  7i)
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers B
Add or Subtract. Write as a +bi.
1. (3  7i )  (2i )
2. (5  9i )  (2  i )
3. (9  2i )  (3  7i)
4. (3  2i)  (5  9i)
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers B
Add or Subtract. Write as a +bi.
1. (3  7i )  (2i )
2. (5  9i )  (2  i )
3. (9  2i )  (3  7i)
4. (3  2i)  (5  9i)
Name ___________KEY______________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers A
Add or Subtract. Write as a +bi.
1. (4  5i )  (3i )
2. (7  8i )  (4  i )
4-2i
3 + 7i
3. (6  5i)  (2  9i)
4. (2  3i)  (6  7i)
4 - 4i
-4 + 10i
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers A
Add or Subtract. Write as a +bi.
1. (4  5i )  (3i )
2. (7  8i )  (4  i )
3. (6  5i)  (2  9i)
4. (2  3i)  (6  7i)
Name ___________KEY______________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers B
Add or Subtract. Write as a +bi.
1. (3  7i )  (2i )
2. (5  9i )  (2  i )
3 - 5i
7 + 8i
3. (9  2i )  (3  7i)
4. (3  2i)  (5  9i)
6 – 5i
-2 +11i
Name _____________________________________________________Date____________Per. _______
Quiz: Add and Subtract Complex Numbers B
Add or Subtract. Write as a +bi.
1. (3  7i )  (2i )
2. (5  9i )  (2  i )
3. (9  2i )  (3  7i)
4. (3  2i)  (5  9i)
Name _____________________________________________________Date____________Per. _______
Quiz: Multiply Complex Numbers
Multiply. Write as a +bi.
1.
(3  5i )(2i )
3. (2  4i)(5  8i)
2.
(8i )(i )
4. (2  3i)(4  5i)
Name _____________________________________________________Date____________Per. _______
Quiz: Multiply Complex Numbers
Multiply. Write as a +bi.
1.
(3  5i )(2i )
3. (2  4i)(5  8i)
2.
(8i )(i )
4. (2  3i)(4  5i)
Name __________KEY_______________________________________Date____________Per. _______
Quiz: Multiply Complex Numbers
Multiply. Write as a +bi.
1.
(3  5i )(2i )
2.
(8i )(i )
10+6i
8
3. (2  4i)(5  8i)
4. (2  3i)(4  5i)
-42 - 4i
-7- 22i
Name _____________________________________________________Date____________Per. _______
Quiz: Multiply Complex Numbers
Multiply. Write as a +bi.
1.
(3  5i )(2i )
3. (2  4i)(5  8i)
2.
(8i )(i )
4. (2  3i)(4  5i)
Name __________________________KEY__________________________ Date__________ Per._____
Test Complex Numbers A
Simplify.
1.
5
4.   81
3.  25
2.  18
5.
 48
6.
 12
Find the additive inverse.
7.  18i
8.  37  2i
Complete the operation.
9. (2  3i )  ( 8-i)
12. (1  5i )  (19  i )
14. (1  5i )  (19  i )
10. (-1  5i )  ( 3  9i )
11. (-1  i )  (  3  8i )
13. (-8  3i )  (5  10i )
14. (-9  i )  (3  12i )
15. (-8  3i )  (5  10i )
16. (-9  i )  (3  12i )
Complete the operation. Show all work.
18. (8i )(4i )
19. (5)(3  4i )
20. (4i )(2  3i )
21. (8i )(3  5i )
22. (5  2i )(3  6i )
23. (6  9i )(4  2i )
24. (4  9i )(7  3i )
17. (12i )(2i )
25. (5  6i )(3  8i )
26. Explain in complete sentences how simplifying the two types of expressions below is
different.
a. (3  2i )(5  4i )
b. (3  2 x)(5  4 x)
Name ________________________________________________________ Date__________ Per._____
Test Complex Numbers B
Find the additive inverse.
2. 8  3i
1. 7i
Simplify and write as a+bi. Show all work.
3.
6 2
6.   64  10
4.  27  3
7.
 72  12
5.   75  9
8.
8 3
Complete the operation. Show all work.
9. (2i )  ( 7-4i )
10. (-3)  ( 8  9i )
11. (-2  2i )  ( 6  7i )
11. (3  5i )  ( 6-i)
12. (-1  4i )  ( 5  8i )
13. (-1  i )  (  2  7i )
14. (1  4i )  (18  i )
15. (-7  4i )  (4  11i )
16. (-7  i )  (2  13i )
Complete the operation. Show all work.
18. (7i )(3i )
19. (6)(4  3i )
20. (5i )(3  4i )
21. (7i )(4  3i )
22. (6  2i )(3  8i )
23. (6  8i )(5  2i )
24. (4  8i )(6  3i )
25. (5  4i )(3  7i )
17. (11i )(3i )
26. Explain in complete sentences how simplifying the two types of expressions below is
different.
a. (3  6i )(2  4i )
b. (3  6 x)(2  4 x)
Name __________________________KEY__________________________ Date__________ Per._____
Test Complex Numbers A
Simplify.
1.
5
3.  25
2.  18
𝑖 5
4.   81
5𝑖
3𝑖 2
5.
 48
-9i
6.
 12
4𝑖 3
2𝑖 3
Find the additive inverse.
7.  18i
8.  37  2i
18i
37-2i
Complete the operation.
9. (3i )  ( 9-2i )
10. (-4)  ( 5  9i )
-9+5i
1+9i
11. (2  3i )  ( 8-i)
12. (-1  5i )  ( 3  9i )
-6 +4i
2+14i
14. (1  5i )  (19  i )
18+4i
15. (-8  3i )  (5  10i )
-13+7i
11. (-5  2i )  ( 7  3i )
-12+5i
13. (-1  i )  (  3  8i )
2 +9i
16. (-9  i )  (3  12i )
-12+11i
Complete the operation. Show all work.
17. (12i )(2i )
18. (8i )(4i )
19. (5)(3  4i )
-24
32
-15 - 20i
20. (4i )(2  3i )
21. (8i )(3  5i )
22. (5  2i )(3  6i )
-12+8i
-40-24i
27+36i
23. (6  9i )(4  2i )
24. (4  9i )(7  3i )
25. (5  6i )(3  8i )
42+24i
1-75i
63-22i
26. Explain in complete sentences how simplifying the two types of expressions below is
different.
a. (3  2i )(5  4i )
b. (3  2 x)(5  4 x)
a. requires an extra step to
simplify i2 and then combine like
terms.
Name ____________KEY________________________________________ Date__________ Per._____
Test Complex Numbers B
Find the additive inverse.
2. 8  3i
1. 7i
-7i
-8+3i
Simplify and write as a+bi. Show all work.
3.
6 2
4.  27  3
−2 + 𝑖 6
6.   64  10
−10 − 8𝑖
5.   75  9
3 + 3𝑖 3
7.
 72  12
−12 + 6𝑖 2
9 − 5𝑖 3
8.
8 3
−3 − 2𝑖 2
Complete the operation. Show all work.
9. (2i )  ( 7-4i )
10. (-3)  ( 8  9i )
11. (-2  2i )  ( 6  7i )
-7+6i
5+9i
-8+9i
11. (3  5i )  ( 6-i)
12. (-1  4i )  ( 5  8i )
13. (-1  i )  (  2  7i )
-3+6i
4+12i
3+8i
14. (1  4i )  (18  i )
15. (-7  4i )  (4  11i )
16. (-7  i )  (2  13i )
17-5i
-11+7i
-9+12i
Complete the operation. Show all work.
17. (11i )(3i )
18. (7i )(3i )
19. (6)(4  3i )
-33
21
-24+18i
20. (5i )(3  4i )
21. (7i )(4  3i )
22. (6  2i )(3  8i )
-20+15i
-21-28i
2+24i
23. (6  8i )(5  2i )
24. (4  8i )(6  3i )
25. (5  4i )(3  7i )
46+28i
60i
-13-23i
26. Explain in complete sentences how simplifying the two types of expressions below is
different.
a. (3  6i )(2  4i )
b. (3  6 x)(2  4 x)
a. requires an extra step to
simplify i2 and then combine like
terms.
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