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

(Marvins-Underground-K-12) #1

Teaching Notes 7.10: Simplifying Complex Numbers


Simplifying complex numbers is similar to simplifying binomials. A common error students make
in simplifying complex numbers is substituting fori^2 incorrectly.


  1. Explain to your students that complex numbers can be expressed in the formx+yiwherex
    andyare real numbers. Emphasize thatxis called the ‘‘real’’ part ofx+yiandyis called the
    ‘‘imaginary’’ part. Examples include 3+ 2 i,− 4 − 2 i,and6− 7 i.

  2. Review the information and examples on the worksheet with your students. Explain that
    operations with complex numbers are the same as operations with binomials. Because com-
    plex numbers are used in several operations, fully discuss each step, especially for multiplica-
    tion and division, with your students. Be sure that your students understand the application
    of FOIL.

  3. Emphasize that the conjugates differ only by a sign. In the example for division, 6–7iand
    6 + 7 iare the conjugates of the denominator.


EXTRA HELP:
If your answer containsi^2 ,replacei^2 with−1 and simplify your answer.

ANSWER KEY:


(1) 7 + 2 i (2)− 2 − 3 i (3) 3 − 5 i (4)− 4 + 7 i (5)− 5 − 12 i (6) 2 −6i (7) 1 +i (8)
18 +i
13
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(Challenge)Andrew is wrong. It appears that he multiplied (3−i)(2−i). The correct solution is
5 − 2 i.
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