Advanced Algebra with Trig
Lesson A.6a Notes
Name: ____________________________________
Intro to Complex Numbers
We have discussed how the square of a real number cannot be negative. Therefore, there is no real number x
for which x 2  1. Thus, we introduce the imaginary unit denoted by i and whose square is -1.
i  1

i 2  ( 1) 2  1
i 3  i 2  i  1 i  i
Ex. i 9  i 4  i 4  i1  11 i  i


Practice: What does i19 

i 4  i 2  i 2  1 1  1


Complex numbers are numbers of the form a + bi where a and b are real numbers. The real number a is
called the real part,
and the real number b is called the imaginary part.
Adding, Subtracting, and Multiplying Complex Numbers
To add complex numbers, simply add the real parts together and add the imaginary parts together.
(a  bi)  (c  di)  (a  c)  (b  d)i
Ex. (2  3i)  (1 4i)  (2 1)  (3 4)i  31i

The same idea holds true for subtracting complex numbers.

(a  bi)  (c  di)  (a  c)  (b  d)i
Ex. (3 i)  (4  3i)  (3 4)  (1 3)i  1 4i

To multiply complex numbers, follow the usually rules for multiplying binomials and remember i 2  1.

(a  bi)  (c  di)  a(c  di)  bi(c  di)
double distribution

Ex. (1 3i)  (4  5i)  1(4  5i)  3i(4  5i)  4  5i 12i 15i 2  4  7i 15  19  7i

Practice: Given the following complex numbers, perform in the indicated operations.
w  5  3i
z  1 4i

1) w  z


2) w  z

3) w  z


Advanced Algebra with Trig
Lesson A.6a Notes (continued)
Conjugates of Complex Numbers
If z  a  bi is a complex number, then its conjugate, denoted z , is defined as
z  a  bi  a  bi
Ex. If z  2  3i then its conjugate is z  2  3i


 a complex numbers with its conjugate!
Let’s see what happens if we multiply

 abi  b2i 2  a2  b2
zz  (a  bi)(a  bi)  a2  abi
zz  a2  b2
Or for short
Ex. (2  3i)(2  3i)  4  6i  6i  9i 2  4  9 13 Or using formula (2  3i)(2  3i)  22  32  4  9 13



Practice. Find the conjugate of the following complex numbers. Then multiply by the conjugate.

1) z  3 4i
2) z  1 8i

3) z  2i

Dividing Complex Numbers
 divide complex numbers, we can multiply the numerator and denominator by the conjugate of the
To
denominator (a one in disguise).
2  3i (4  3i) 8  6i 12i  9i 2 8  6i  9 17  6i 17 6





 i
Ex.
4  3i (4  3i)
4 2  (3) 2
16  9
25
25 25
Practice:

3 i
2  4i
Let’s Practice What We’ve Learned!
Given the following complex numbers, perform the indicated operations. w  2  3i

1) w  z
2) w  z

3) w  z



4)
w
z


z  5  2i