Download The set of real numbers does not include all the numbers needed in

8.1 -- Complex Numbers
Wednesday, August 07, 2013
1:11 PM
The set of real numbers does not include all the numbers needed in algebra.
Imaginary Numbers
cannot be solved with real numbers
Complex Numbers
If a and b are real numbers, then any number in the form a + bi is a complex number
In the complex number a + b ,
a = the real part b = the imaginary party
Note:
Ex1: Simplify using as the imaginary unit.
1)
2)
3)
Classroom Ex1: Simplify
2)
3)
Ex2: Solving Quadratic Equations for Complex Solutions
2)
1)
Classroom Ex2: 1)
Chapter 8 Page 1
2)
Ex3: Solving a Quadratic Equation (Complex Solutions)
Solve:
Classroom Ex3: Solve
Caution: When working with negative radicands, use the definition
BEFORE
you do any other operation! You must take out the
before doing anything else!!!
Ex4: Finding Products and Quotients Involving
Multiply or divide, as indicated. Simplify each answer.
2)
3)
4)
Classroom Ex4: Simplify
2)
1)
3)
Ex5: Simplifying a Quotient Involving
Write
in standard form.
Chapter 8 Page 2
2)
4)
1) Write
in standard form.
2)
Adding and Subtracting Complex Numbers
You combine the real numbers and combine the imaginary numbers and make sure your
final answer is in the form a + b .
Ex6: Adding and Subtracting Complex Numbers
1) (3 4 ) + ( 2 + 6 )
2) ( 4 + 3 ) (6 7 )
Classroom Ex6:
1. (4 5 ) + (-5 + 8 )
2. ( 10 + 7 )
(5
3)
Ex7: Multiplying Complex Numbers
1. (2 3 )(3 + 4 )
2. (4 + 3 )2
Classroom Ex7:
1. (5 + 3 )(2 7 )
2. (4
Chapter 8 Page 3
5 )2
3. (8
8)
( 2
5 ) + ( 10 + 3 )
3. (6 + 5 )(6
3. (9
5)
8 )(9 + 8 )
Conjugates: (6 + 5 ) and (6 5 ) are conjugates, in that they only differ in the sign of their
imaginary parts. The product of conjugates is ALWAYS a real number so we use them to
simplify rational expressions that have imaginary numbers in the denominator.
Ex8: Dividing Complex Numbers
Write each quotient in standard form a + bi.
1)
2)
Classroom Ex8:
1)
2)
Chapter 8 Page 4
Chapter 8 Page 5
8.2 -- Trig (Polar) Form of Complex Numbers
Wednesday, August 07, 2013
1:11 PM
The Complex Plane and Vector Representation
Imaginary axis
Each complex number a + b
determines a unique position
vector with initial point (0, 0)
and terminal point (a, b).
imaginary
5+4
1+3
4+
real
Real axis
P
2-3
Note: This geometric
representation is the reason
that a + b is called the
rectangular form (or standard
form) of a complex number.
Recall that (4 + ) + (1 + 3 ) = 5 + 4 .
Graphically the sum of 2 complex
numbers is represented by the vector that
is the resultant of the vectors
corresponding to the 2 numbers, as
shown in the figure above
Ex1: Expressing the Sum of Complex Numbers Graphically
Find the sum of the following complex #'s. Graph both complex #'s and their resultant
1) 6 2 and 4 3
2) 2 + 3 and 4 + 2
Chapter 8 Page 6
Trigonometric (Polar) Form
The figure at the right shows the complex number x + y
that corresponds to a vector OP with direction angle θ
and magnitude r The following are relationships among
x, y, r, and θ.
r
θ
y
x
Trig form =
Trig form =
Ex2: Converting from Trig Form to Rectangular Form
Express
in rectangular form (a + b ).
Classroom Ex2: Express
in rectangular form.
Converting from rectangular form (a + bi) to trigonometric form (r cis Ө)
1. Sketch the graph of a + b
2. Solve for r
3. Find Ө, make sure it is the correct quadrant
4. Once you have r and Ө, then you can write your answer in trig form.
Ex3: Converting from Rectangular Form to Trig Form
Write each complex numbers in trig form
1)
(use radian measure)
Chapter 8 Page 7
2)
(use degree measure)
Classroom Ex3:
Write each complex numbers in trig form
1)
(use radian measure)
2)
(use degree measure)
Ex4: Converting Between Trig and Rectangular Forms Using Calculator Approximations
Write each complex number in its alternative form, using calculator approximations as necessary.
1)
2)
Classroom Ex4: 1)
Chapter 8 Page 8
2)
8.3 -- The Product and Quotient Theorems
Wednesday, August 07, 2013
1:11 PM
Products of Complex Numbers in Trig Form
Ex1: Using the Product Theorem
Find the product of
rectangular form.
• So to multiply
complex numbers in
trig form, you will
multiply their radii
and add their angles
and
Classroom Ex1: Find the product of
Write the result in rectangular form.
Quotients of Complex Numbers in Trig Form
That is, to divide complex #s in trig form,
divide their radii and subtract their angles
Chapter 8 Page 9
Write the result in
and
Ex2: Using the Quotient Theorem
Find the quotient
. Write the result in rectangular form.
Classroom Ex2:
Ex3: Using the Product and Quotient Theorems with a Calculator
Use a calculator to find the following. Write the results in rectangular form.
1) (9.3 cis 125.2°)(2.7 cis 49.8°)
Classroom Ex3:
1) (2.8 cis 38.5°º)(5.4 cis 74.5°)
Chapter 8 Page 10
2)
2)
8.4 -- De Moivre's Theorem; Powers and Roots
of Complex Numbers
Wednesday, August 07, 2013
1:12 PM
Finding the power of a complex number. You will need to find r, Ө, and the power (n). Once
you find those then you will use the following formula:
If the new angle is greater than 360⁰, then you must find the coterminal angle by subtracting
360⁰ until you get an angle between 0⁰ and 360⁰
Ex1: Finding a Power of a Complex Number
Find
and express the result in rectangular form.
Classroom Ex1:
Find
and express the result in rectangular form.
Chapter 8 Page 11
nth Root Theorem
If n is any positive integer, r is a positive real #, and θ is in degrees, then the nonzero complex
number r(cos θ + sin θ) has exactly n distinct nth roots, given the following.
1. You will need to find r and Ө of the original (a + b )
2. If you are finding square roots then there will be 2 angles, if you are finding cube roots then
there will be 3 angles and so on…
3. The roots that you are finding will have the following:
and the first angle will
equal the following angles will be found by adding
to the first angle you found.
Remember that the number of angles will equal the root you are finding, and all of the angles
you find will be between 0⁰ and 360⁰
** reminder:
Ex2: Finding Complex Roots
Find the two square roots of 9 . Write the roots in rectangular form.
Classroom Ex2: Find the 3 cube roots of -64
Ex3: Finding Complex Roots
Chapter 8 Page 12
Ex3: Finding Complex Roots
Find all 4th roots of
. Write the roots in rectangular form.
Classroom Ex3: Find all 4th roots of
Chapter 8 Page 13
.
8.5 -- Polar Equations and Graphs
Wednesday, August 07, 2013
1:12 PM
Ex1: Plotting Points with Polar Coordinates
Plot each point in the polar coordinate system. Then determine the rectangular coordinates for
each point.
1. P(2, 30˚)
2.
Chapter 8 Page 14
3.
Classroom Ex1: Plot each point in the polar coordinate system. Then determine the rectangular
coordinates for each point.
1. P(4, 135˚)
2.
3.
** While a given point in the plane can have only one pair of rectangular coordinates, this same
point can have an infinite number of pairs of polar coordinates. For ex, (2,30˚) is the same point
as (2,390˚), (2, 330˚) and ( 2,210˚)
Ex2: Giving Alterative Forms for Coordinates of a Point
1. Give 3 other pairs of polar coordinates for the point P(3, 140˚)
2. Determine 2 pairs of polar coordinates for the point with rectangular coordinates ( 1, 1).
Chapter 8 Page 15
Classsroom Ex2:
1. Give 3 other pairs of polar coordinates for the point P(5, 110˚)
2. Determine 2 pairs of polar coordinates for the point with rectangular coordinates
Graphs of Polar Equations: Equations in and are rectangular (or Cartesian) equations. An equation in which
and are the variables instead of and is a polar equation. To graph a polar equation evaluate for various
values of until a pattern appears, then join the points with a smooth curve.
Ex4: Graphing a Polar Equation (Cardioid)
Graph
To graph this find some ordered pairs in the table.
Classroom Ex4: Graph
Ex5: Graphing a Polar Equation (Rose)
Chapter 8 Page 16
.
Ex5: Graphing a Polar Equation (Rose)
Graph
Classroom Ex5:Graph
Ex6: Graphing Polar Equation (Lemniscate)
Graph
The values between 45 and 135 degrees are not included because the
corresponding values of cos 2θ are negative (QII and III) and do not have real square roots.
Chapter 8 Page 17
Classroom Ex6: Graph
Ex7: Graphing a Polar Equation (Spiral of Archimedes)
Graph r = 2θ (with θ measured in radians).
Classroom Ex7: Graph r = -θ (with θ measured in radians)
Chapter 8 Page 18
Chapter 8 Page 19
8.6 -- Parametric Equations, Graphs, and
Applications
Wednesday, August 07, 2013
1:13 PM
Parametric Equations of a Plane Curve
A plane curve is a set of points (x, y) such that
defined on an interval . The equations
parameter
,
and
and and are both
are parametric equations with
Ex1: Graphing a Plane Curve Defined Parametrically
Let
and
for in [ 3, 3]. Graph the set of ordered pairs (x, y).
Classroom Ex1:
, for in [-3, 3]. Graph the set of ordered pairs (x, y).
Ex2: Finding an Equivalent Rectangular Equation
Chapter 8 Page 20
Ex2: Finding an Equivalent Rectangular Equation
Find the rectangular equation for the plane curve of Ex1 defined as follows:
and
for in [ 3, 3].
Classroom Ex2: Find the rectangular equation for the plane curve of Classroom Ex1
defined as follows:
, for in [ 3, 3].
Ex3: Graphing a Plane Curve Defined Parametrically
Graph the plane curve defined by
for
Solution: It is not productive to solve either equation for . We will use the fact that
to apply another approach. Find
and
and set them equal to 1.
Chapter 8 Page 21
Classroom Ex3: Graph the plane curve defined by
for
Ex4: Finding Alternative Parametric Equation Forms
Give 2 parametric representations for the equation of the parabola.
Classroom Ex4: Give 2 parametric representations for the equation of the parabola.
The Cycloid:
animation of a Cycloid
It is a curve traced out by a point at a given distance from the center of a circle as the circle rolls
along a straight line. The parametric equations of a cycloid are as follows.
for
Chapter 8 Page 22
in
Ex5: Graphing a Cycloid
Graph the cycloid
for in
There is no easy way to
find a rectangular equation of a cycloid, so instead we use a table of selected values for in
.
Classroom Ex5: Graph the Cycloid
for
in
Applications of Parametric Equations: Parametric equations are used to simulate motion. If a ball
is thrown with a velocity of feet per sec at an angle θ with the horizontal, its flight can be
modeled by the parametric equations
and
Where is in seconds and is the ball's initial height in feet above the ground. Here gives the
horizontal position information and gives the vertical position information. The term
occurs because gravity is pulling downward.
Ex6: Simulating Motion with Parametric Equations
Chapter 8 Page 23
** may need to use graphing calculator
Ex6: Simulating Motion with Parametric Equations
Three golf balls are hit simultaneously into the air at 132 ft/sec (90mph) at angles of 30˚, 50˚,
and 70˚ with the horizontal.
1. Assuming the ground is level, determine graphically which ball travels the greatest distance.
2. Which ball reaches the greatest height? Estimate its height.
Use
and
Classroom Ex6: Repeat Ex3 for three golf balls are hit simultaneously into the air at 120 ft/sec
(90mph) at angles of 25˚, 45˚, 65˚ with the horizontal.
Chapter 8 Page 24