The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

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WORKSHEET 7.10: SIMPLIFYING COMPLEX NUMBERS
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A complex number is a number of the formx+yi, wherexandyare real numbers.xis the
‘‘real’’ part ofx+yi.yis the ‘‘imaginary’’ part. Use the guidelines below to simplify complex
numbers:


  • Addition or subtraction:Combine the real parts, then combine the imaginary parts.

  • Multiplication:Use FOIL. Simplify by combining the real parts, then combine the imagi-
    nary parts. Replacei^2 by−1.

  • Division:Multiply the numerator and denominator by the conjugate of the denominator.
    Simplify the numerator. Simplify the denominator. Replacei^2 by−1.


EXAMPLES
Addition:(3+ 2 i)+(6− 7 i)= 9 − 5 i
Subtraction:(3+ 2 i)−(6− 7 i)= 3 + 2 i− 6 + 7 i=− 3 + 9 i
Multiplication:(3+ 2 i)(6− 7 i)= 18 − 9 i− 14 i^2 = 18 − 9 i−14(−1)= 32 − 9 i
Division:
3 + 2 i
6 − 7 i

=

3 + 2 i
6 − 7 i

·

6 + 7 i
6 + 7 i

=

18 + 33 i+ 14 i^2
36 − 49 i^2

=

18 + 33 i+14(−1)
36 −49(−1)

=

4 + 33 i
85

DIRECTIONS: Simplify each expression.




  1. (3− 6 i)+(4+ 8 i) 2. (1− 2 i)−(3+i)




  2. (1−i)(4−i) 4. (2+ 3 i)(1+ 2 i)
    5. (2− 3 i)(2− 3 i) 6. (4− 2 i)(1−i)






(1+ 3 i)
(2+i)
8.

(3− 4 i)
(2− 3 i)

CHALLENGE:Andrew said that (3−i)+(2−i)= 6 − 5 i+i^2 or 5− 5 i.Ishe
correct? If he is incorrect, explain what he might have done wrong and
provide the correct answer.

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2011 by Judith A. Muschla, Gary Robert Muschla, and Erin Muschla. All rights reserved.

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