5.4
Complex Numbers
Goals p Perform operations with complex numbers.
p Apply complex numbers to fractal geometry.
Your Notes
VOCABULARY
Imaginary unit i The imaginary unit i is defined as
1.
i Complex number A number a bi where a and b are
real numbers and i is the imaginary unit
Standard form of a complex number The form a bi
where a and b are real numbers and i is the imaginary
unit. The number a is the real part of the complex
number and bi is the imaginary part of the complex
number.
Imaginary number A complex number a bi where
b0
Pure imaginary number A complex number a bi
where a 0 and b 0
Complex plane A coordinate plane where each point
(a, b) represents a complex number a bi. The
horizontal axis is the real axis and the vertical axis
is the imaginary axis.
Complex conjugates Two complex numbers of the form
a bi and a bi
Absolute value of a complex number If z a bi, then
the absolute value of z, denoted z , is a nonnegative
real number defined as z  a2 b2 .
Geometrically, the absolute value of a complex
number is the number’s distance to the origin.
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Algebra 2 Notetaking Guide • Chapter 5
Your Notes
THE SQUARE ROOT OF A NEGATIVE NUMBER
Property
Example
1. If r is a positive real number,
then r
i r.
5
i 5
2. By Property (1), it follows
that (i r)2 r.
(i 5
)2 i 2 p 5 5
Example 1
Solving a Quadratic Equation
Solve 2x2 3 15.
Solution
2x2 3 15
2x2
18
x2 9
Write original equation.
Subtract 3 from each side.
Divide each side by 2 .
x 9
Take square roots of each side.
x i 9
Write in terms of i.
x 3i
Simplify the radical.
The solutions are 3i and 3i .
Example 2
Plotting Complex Numbers
Plot the complex numbers in the complex plane.
a. 1 i
b. 2 2i
Solution
a. To plot 1 i, start at the origin,
move 1 unit to the right , and
then 1 unit up
.
b. To plot 2 2i, start at the origin,
move 2 units to the left , and
then 2 units down .
c. 3 3i
imaginary
1i
i
3
1
i
1
real
2 2i
3i
3 3i
c. To plot 3 3i, start at the origin, move
3 units to the right , and then 3 units down .
Lesson 5.4 • Algebra 2 Notetaking Guide
105
Your Notes
Example 3
Adding and Subtracting Complex Numbers
Write the expression (5 ⴙ i) ⴙ (1 ⴚ 2i) as a complex
number in standard form.
(5 i) (1 2i)
( 5 1 ) ( 1 2 )i
6i
Complex addition
Standard form
Checkpoint Complete the following exercises.
1. Solve 5x2 2 8.
2. In which quadrant of the
complex plane is 1 3i?
i 2
Quadrant IV
3. Write 3 (7 8i) (5 6i) as a complex number in
standard form.
1 14i
Example 4
Multiplying and Dividing Complex Numbers
Write the expression as a complex number in standard form.
6 4i
a. (1 4i)(3 5i)
b. 1i
Solution
Use FOIL.
a. (1 4i)(3 5i) 3 5i 12i 20i 2
3 7i 20(1)
Simplify and
use i 2 ⴝ 1 .
23 7i
Standard form
6 4i
6 4i
1 i
b. p 1i
1i
1i
106
Algebra 2 Notetaking Guide • Chapter 5
Multiply by 1 i , the
conjugate of 1 ⴙ i.
6 6i 4i 4i2
1 i i i2
Use FOIL.
2 10i
2
Simplify.
1 5i
Write in standard form.
Your Notes
Checkpoint Write the expression in standard form.
3 2i
5. 2i
4. (4 5i)(4 5i)
4
7
i
5
5
41
Example 5
Finding Absolute Values of Complex Numbers
Find the absolute value of each complex number. Which
number is farthest from the origin in the complex plane?
a. 2 3i
b. 2 i
Solution
a. 2 3i
(2)2 (3
)2
13 ≈ 3.6
b. 2 i
(2)2 (1)2
c. 1 3i
1 3i
imaginary
3i
2i
10
z ____
3
1
13
z ____
2 3i
1
real
5
z ___
3i
5 ≈ 2.2
c. 1 3i
(1)2 (3)2
10 ≈ 3.2
Because 2 3i has the greatest absolute value, it is
farthest from the origin in the complex plane.
Lesson 5.4 • Algebra 2 Notetaking Guide
107
Your Notes
COMPLEX NUMBERS IN THE MANDELBROT SET
To determine whether a complex number c is in the
Mandelbrot set, consider the function f(z) z2 c and this
infinite list of complex numbers: z0 0, z1 f(z0),
z2 f(z1), z3 f(z2), …
• If the absolute values z0, z1, z2, z3, … are all
less than some fixed number N, then c is in
the
Mandelbrot set.
• If the absolute values z0, z1, z2, z3, … become
infinitely large , then c is not in the Mandelbrot set.
Example 6
Complex Numbers in the Mandelbrot Set
Tell whether c i belongs to the Mandelbrot set.
Solution
Let f(z) z2 i.
z0 0
z0 0
z1 f( 0 ) 02 i i
z1 1
z2 f( i ) (i)2 i 1 i
z2 ≈ 1.41
z3 f( 1 i ) (1 i)2 i i
z3 ≈ 1
z4 f( i ) z4 ≈ 1.41
(i)2
i 1 i
2 , so all
The absolute values alternate between 1 and the absolute values are less than N 2 . Therefore,
c i belongs
to the Mandelbrot set.
Checkpoint Complete the following exercises.
6. Find the absolute value
of 8 6i.
7. Tell whether c 2 belongs
to the Mandelbrot set.
Homework
10
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Algebra 2 Notetaking Guide • Chapter 5
no