Complex Numbers

1
Complex Numbers
2
Complex Numbers
EXAMPLE
Write as a multiple of i
SOLUTION
3
Complex Numbers
4
Complex Numbers
5
Complex Numbers
EXAMPLE
Perform the indicated operations, writing the
result in the form a bi (a) (-9 2i) (-17
6i) (b) (-2 6i) (4 - i).
SOLUTION
(a) (-9 2i) (-17 6i)
-9 2i 17 6i
Remove the parentheses. Change signs of the real
and imaginary parts being subtracted.
-9 17 2i 6i
Group real and imaginary terms.
(-9 17) (2 6)i
Add real parts and imaginary parts.
8 8i
Simplify.
6
Complex Numbers
CONTINUED
(b) (-2 6i) (4 - i)
-2 6i 4 - i
Remove the parentheses.
-2 4 6i - i
Group real and imaginary terms.
(-2 4) (6 - 1)i
Add real parts and imaginary parts.
2 5i
Simplify.
7
Complex Numbers
EXAMPLE
Find the products (a) -6i(3 5i) (b) (-4
2i)(-4 - 2i).
SOLUTION
(a) -6i(3 5i)
Distribute -6i through the parentheses.
Multiply.
Replace with -1.
Simplify and write in a bi form.
(b) (-4 2i)(-4 2i)
Use the FOIL method.
8
Complex Numbers
CONTINUED
Group real and imaginary terms.
Combine real and imaginary terms.
9
Complex Numbers
10
Complex Numbers
EXAMPLE
Multiply
SOLUTION
Express square roots in terms of i.
The square root of 64 is 8.
11
Complex Numbers
EXAMPLE
Divide and simplify to the form a bi
SOLUTION
The conjugate of the denominator is 4 2i.
Multiplication of both the numerator and the
denominator by 4 2i will eliminate i from the
denominator.
Multiply by 1.
Use FOIL in the numerator and
in the
denominator.
12
Complex Numbers
CONTINUED
Simplify.
Perform the multiplications involving -1.
Combine like terms in the numerator and
denominator.
Express answer in the form a bi.
Simplify.
13
Complex Numbers
EXAMPLE
Divide and simplify to the form a bi
SOLUTION
The conjugate of the denominator, 0 - 4i, is 0
4i. Multiplication of both the numerator and the
denominator by 4i will eliminate i from the
denominator.
Multiply by 1.
Multiply. Use the distributive property in the
numerator.
14
Complex Numbers
CONTINUED
Perform the multiplications involving -1.
Express the division in the form a bi.
Simplify real and imaginary parts.
15
Complex Numbers
16
Complex Numbers
EXAMPLE
Simplify
SOLUTION
17
Complex Numbers
10.7
The notes for this section were made possible
using the Blitzer Introductory and Intermediate
Algebra Text.