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.- 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. - 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. - 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
------------------------------------------------------------------------------------------
(Challenge)Andrew is wrong. It appears that he multiplied (3−i)(2−i). The correct solution is
5 − 2 i.
------------------------------------------------------------------------------------------270 THE ALGEBRA TEACHER’S GUIDE