CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 11. Complex Numbers

The absolute value of a complex number like| 4 + 3 i|is defined as the distance from the complex number to the origin.
You can use the Pythagorean Theorem to get the absolute value. In this case,| 4 + 3 i|=

√

42 + 32 =

√

25 =5.

Example A
Multiply and simplify the following complex expression.
( 2 + 3 i)( 1 − 5 i)− 3 i+ 8
Solution:( 2 + 3 i)( 1 − 5 i)− 3 i+ 8

= 2 − 10 i+ 3 i− 15 i^2 − 3 i+ 8

= 10 − 10 i+ 15

= 25 − 10 i

Example B
Compute the following power by hand and use your calculator to support your work.
(√
3 + 2 i

) 3

Solution:

(√

3 + 2 i

)

·

(√

3 + 2 i

)

·

(√

3 + 2 i

)

=

(

3 + 4 i

√

3 − 4

)(√

3 + 2 i

)

=

(

− 1 + 4 i√ 3

)(√

3 + 2 i

)

=−

√

3 − 2 i+ 12 i− 8

√

3

=− 9 √ 3 + 10 i

A TI-84 can be switched to imaginary mode and then compute exactly what you just did. Note that the calculator
will give a decimal approximation for− 9

√

3.