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

Name Date

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)

   
   


3. 
   

(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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Copyright

©

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