Transcript 8 1 8 2 Imaginary Numbers

8-1 and 8-2: Imaginary Numbers
1.
2.
3.
Objectives:
To simplify square
roots of negative
numbers
To plot complex
numbers in the
complex plane and find
their absolute value
To add, subtract,
multiply, and divide
complex numbers
Assignment:
• P. 135-136: 1-18, 25
• Challenge Problems
Warm-Up
This is a highly
complex shape
know as the
Mandelbrot Set.
A set in math is
just a group of
numbers, like
the set of real
numbers.
Warm-Up
The Mandelbrot Set,
M, is group of
numbers that
when graphed
form a fractal.
To see if a number
is part of the set,
you start with the
recursive formula
zn+1 = zn2 + c.
Warm-Up
The Mandelbrot Set,
M, is group of
numbers that
when graphed
form a fractal.
To see if a number
is part of the set,
you start with the
recursive formula
zn+1 = zn2 + c.
In the formula, c is the
number you’re testing
and z0 = 0.
Let’s say we wanted to
test if c = 1 is a
member of the set:
z0  0
z1  02  1  1
Warm-Up
zn+1 = zn2 + c.
In the formula, c is the
number you’re
testing and z0 = 0.
Let’s say we wanted
to test if c = 1 is a
member of the set:
To get z2, you plug z1 = 1
back into the formula,
where c will remain 1,
the number we’re trying
to test. This is called
recursion:
z2  12  1  2
z0  0
z3  2 2  1  5
z1  02  1  1
z4  52  1  26
Warm-Up
As you can probably
see, these numbers
are increasing in
value, getting further
away from zero.
If we were to continue
iterating with the
formula, the resulting
values would
approach infinity.
To get z2, you plug z1 = 1
back into the formula,
where c will remain 1,
the number we’re trying
to test. This is called
recursion:
z2  12  1  2
z3  2 2  1  5
z4  52  1  26
Warm-Up
To be a member of
the M set, these
values must be
bounded; that is,
they cannot
approach infinity.
Therefore, the
number c = 1 is
not a part of M.
Is c = −1 a member
of the Mandelbrot
Set?
Warm-Up
You might be thinking that
there aren't a whole lot
of members in M since
membership is so
exclusive, but you'd be
wrong as this image
clearly indicates. To find
more members, you just
have to know where to
look: within the set of
Click me for a Mandelbrot Zoom!
complex numbers.
Objective 1
You will be able to simplify
square roots of negative
numbers
Exercise 1
Use a graphing
calculator to graph
y = x2 + 1.
What are the xintercepts?
Exercise 2
Solve the quadratic equation x2 + 1 = 0.
The problem here is that −1 is not a real
number since there is no real number that
you can square to get −1.
– This does not mean there is no solution; it’s
more complex than that.
Imaginary Unit
The imaginary unit i
can be used to find
the square roots of
negative numbers.
Just because
a number is
imaginary
doesn’t mean
that it
doesn’t exist!
𝑖 = −1
𝑖 2 = −1
Girolamo Cardano c. 1545
Exercise 3
A pattern exists as a
result of raising i,
an imaginary
number, to n, an
integer greater than
or equal to 1.
According to the
table, what is the
value of i raised
to the 16th power?
in
Solution
i1
−1
−1
i2
i3
i4
i5
i6
− −1
1
−1
−1
Taking Powers of i
As the previous example shows, there are
only 4 possible values for in.
i1 
i2 

1  i
2
1  1

All other powers of 4
just repeat this pattern.
So, how do you think
i 3  i 2  i  1 i  i
you would evaluate i101?
4
2
2
i  i  i   1 1  1
Taking Powers of i
As the previous example shows, there are
only 4 possible values for in.
i1 
i2 

1  i
2
1  1

Divide the exponent by 4,
then use the remainder to
find the result:
i 3  i 2  i  1 i  i
i 4  i 2  i 2   1 1  1
Remainder
Result
1
i
2
−1
3
−i
0
1
Exercise 4
Evaluate each of the following.
1. i54
2.
i120
3.
i89
4.
i39
Protip: Divisible by 4
How can you tell if a number is divisible by
4?
Any multiple of 100 is divisible by 4.
Note that 100𝑛 is multiple of 4, where 𝑛 ∈ ℤ
100𝑛
100
=
𝑛=
4
4
25𝑛
Since this is an integer with
no remainder, any multiple
of 100 is divisible by 4.
Protip: Divisible by 4
How can you tell if a number is divisible by
4?
If the last two digits of a number are
divisible by 4, the whole number is
divisible by 4.
132
100 + 32
=
=
4
4
100 32
+
=
4
4
25 + 8 =
33
Since this is an integer with no remainder, as
long as the last two digits are divisible by 4,
the whole number is divisible by 4.
Protip: Divisible by 4
How can you tell if a number is divisible by
4?
If the last two digits of a number are
divisible by 4, the whole number is
divisible by 4.
228
200 + 28
=
=
4
4
200 28
+
=
4
4
50 + 7 =
57
Since this is an integer with no remainder, as
long as the last two digits are divisible by 4,
the whole number is divisible by 4.
Protip: Divisible by 4
How can you tell if a number is divisible by
4?
If the last two digits of a number are
divisible by 4, the whole number is
divisible by 4.
316
300 + 16
=
=
4
4
300 16
+
=
4
4
75 + 4 =
79
Since this is an integer with no remainder, as
long as the last two digits are divisible by 4,
the whole number is divisible by 4.
Exercise 5
Simplify each of the following.
1.
−36
2.
−13
3.
𝑖 5
2
Negative Square Roots
The Square Root of a Negative Number
Property
1.
2.
Example
If 𝑟 is a positive number,
then −𝑟 = 𝑖 𝑟
𝑖 𝑟
2
= −𝑟
−5 = 𝑖 5
𝑖 5
= −5
2
= 𝑖2 ∙ 5
Exercise 6a
Find the roots of each quadratic equation.
1.
x2 + 1 = 0
2. 2x2 + 18 = −72
Objective 2
You will be able to plot
complex numbers in the
complex plane and find their
absolute value
Complex Numbers
A complex number in standard form is
written
a  bi
Real Part
𝑎, 𝑏 ∈ ℝ
Imaginary Part
• All real numbers are complex numbers
– This happens when b = 0
• For imaginary numbers, b ≠ 0.
• For a pure imaginary number, a = 0.
Exercise 7
Draw a Venn Diagram
that represents the
set of complex
numbers and
includes real,
imaginary, and pure
imaginary numbers.
ℂ: The set of all complex numbers
Complex Plane
All complex
numbers are
essentially
2-dimensional.
– When you graph
a real number, it
appears on a 1-D
number line
Complex Plane
All complex
numbers are
essentially
2-dimensional.
– However,
complex
numbers have
both a real and
an imaginary part
Imaginary
Axis
Real Axis
Exercise 8
Plot the complex
number 4 + 3i in
the complex plane.
How could we find
its distance from
the origin? In
other words, how
could we find its
absolute value?
Absolute Value
Absolute Value of a Complex Number
The absolute value
of a complex
number z = a + bi,
denoted |z|, is a
nonnegative real
number defined as
z 
a 2  b2
Exercise 9
Find the absolute value of each complex
number.
1. 4 – i
2. −3 – 4i
3. 2 + 5i
4. −4i
Objective 3
You will be able to add,
subtract, multiply, and
divide complex
numbers
Adding and Subtracting
To add or subtract two complex numbers,
simply add or subtract their real and
imaginary parts separately.
Sum
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference (a + bi) – (c + di) = (a – c) + (b – d)i
Exercise 10
Write the expression as a complex number
in standard form.
1. (12 – 11i) + (-8 + 3i)
2. (15 – 9i) – (24 – 9i)
3. 35 – (13 + 4i) + i
Multiplying
To multiply complex numbers, you have to
use a combination of the distributive
property and properties of the imaginary
unit.
2i  3  5i   6i  10i 2  6i  10  1  6i  10  10  6i
Exercise 11
Write the expression as a complex number
in standard from.
1. −5i(8 – 9i)
2. (-8 + 2i)(4 – 7i)
Exercise 12
Multiply and classify the product.
1. (5i)(−5i)
2. (3 + 6i)(3 – 6i)
Dividing
To “divide” complex numbers, you have to
multiply by a complex conjugate
The complex numbers
a + bi and a – bi are
complex conjugates
The product of
complex conjugates is
always a real number
2  3i 1  i  2  3i 1  i  2  2i  3i  3i 2
1  5i




1 i 1 i
2
2
1  i 1  i 
Dividing
To “divide” complex numbers, you have to
multiply by a complex conjugate
The complex numbers
a + bi and a – bi are
complex conjugates
The product of
complex conjugates is
always a real number
2  3i 1  i  2  3i 1  i  2  2i  3i  3i 2
1 5



  i
1 i 1 i
2 2
2
1  i 1  i 
Exercise 13
Write the quotient in standard form.
3  4i
5i
Exercise 14
A number c is a
member of the
Mandelbrot Set if
|z| < 2 after each
iteration.
Determine whether
c = i/2 is a member
of the Mandelbrot
Set.
Exercise 14
8-1 and 8-2: Imaginary Numbers
1.
2.
3.
Objectives:
To simplify square
roots of negative
numbers
To plot complex
numbers in the
complex plane and find
their absolute value
To add, subtract,
multiply, and divide
complex numbers
Assignment:
• P. 135-136: 1-18, 25
• Challenge Problems