Transcript complex number - Mrs. C. Phippen

Warm-up
1)Write y = 2(x – 1)2 + 5 in
standard form.
2)Write y = -2(x + 1)2 – 2
in standard form.
GP/Assignment/Pairs
 Textbook
 Example problems p. 61 #34-39
 On a piece of paper, to turn in, DO:
 P. 59 #35,40,41
 1) Make an x,y-chart put into L1, L2
in your calculator, do stat  calc  5
Foil Practice/Refresher
Foil WS
Front – Foiling
Binomials
Back – Foiling
Polynomials
Warm-up
Simplify
*Yet another way to multiply…
Think 4th grade & long multiplication
(3x2 – 9x + 5)(2x2 + 4x – 7)
So the answer is:
6x4 – 6x3 – 47x2 + 83x – 35
Questions?
Another Review…Take
notes if you can’t
remember the rules!
What are the properties of exponents?
IE. Zero Product Property
Multiplication Property
Division Property
Power Property
Zero Power0 Property
5 =1
(-2)0 = 1
(3/8)0 = 1
1.020 = 1
General Property
a0 = 1
Notice 00 is undefined just like 2/0
and 0/0
Negative Exponents
2-1=
2-2=
3-1=
4-1=
4-4=
General rule:
a-n = 1/an
½
¼
1/3
¼
1/64
Write each expression as
a simple
fraction.
-4
1)3
2)(-7)0
3)(-4)-3
4)7-3
5)-3-2
1)1/81
2)1
3)1/64
4)1/343
5)-1/9
Reciprocalsn
How are a and a-n related?
an · a-n
an · 1
1 an
an · a-n = 1
Thus, an and a-n are reciprocals
of each other and cancel one
another out.
Re-write the expression so that
all exponents are positive
1)4yx-3
1)4y/x3
2)1/w-4
2)w4
3)1/x-3
3)x3
4)1/v-2
4)v2
5)w-3
5)1/w3
6)w-3/v-2
6)v2/w3
Powers with the same base:
To multiply powers with
the same base, add the
exponents
am · an = am+n
x3 · x4 = x7
Raising a Power to a Power
To raise a power to a
power, multiply the
exponents
(am)n = amn
3
4
(3·4)
12
(X ) = x
=x
Raising a Product to a Power
To raise a product to a
power, raise each factor in
the product to the power
(ab)n = anbn
(xy)4 = x4y4
(2x)3 = 23x3
Dividing Powers w/ Same Base
To divide powers with
the same base,
subtract the
exponents
m
n
m-n
a /a = a
w5/w2 = w5-2 = w3
Raising a Quotient to a Power
To raise a quotient to a
power, raise the dividend
and the divisor to the
power
(a/b)n = an/bn
(x/y)2 = x2/y2
(2x/4)2 = 22x2/42
*Negative Exponent Property
For any non-zero
number, an integer to a
negative power equals
the reciprocal
a-n = 1/an
7k-8 = 7/k8
SIMPLIFY THE PROBLEM
4
3
(4 )
12
4
SIMPLIFY THE PROBLEM
2
-2
(3a )
3-2a-4
*Negative: send to other side
of fraction
1
Or
1
32a4
9a4
SIMPLIFY THE PROBLEM
2
3
(3a )
3
(a)
SIMPLIFY THE PROBLEM
-2
6x
-3
a
Assignment
Ed Helper “Exponents”
Worksheet
Grade reports?
Warm-up
On a half-sheet of paper (share?) Do
now and turn in. You have 10 minutes
from the bell.
1) (-9h3j6)(-5h5)
2) (-8a-6f6)(-10a3)(3a4f-4)
3) (-5d-5g5)5(-d-6g6)5
Collect HW
A little review…
Prime factorization and GCF
Prime number –
A positive integer, greater than 1, with
exactly two factors.
Ex. 2,3, 5, 7, 11, 13, 17, 19
Composite number –
A positive integer, greater than 1, with more
than two factors.
Ex. 4, 6, 8, 10, 9, 15, 21
Prime Factor Trees!
Prime Factorization can be shown using factor
“trees”.
Break down the given number to prime
factors until all of the factors are prime.
Ex.
825
5
165
5
33
3 11
List the prime factors: 3x5x5x11
Simplifying Radicals
LEQ’s: How do you simplify radicals involving
products and quotients?
How do you solve problems involving radicals?
Recall:
A radical is ____.
A radical expression would be ____.
A radical is in simplest
form when:
 The expression under the radical
sign has no perfect square factors
other than 1 (no pairs left)
 The expression under the radical
sign does not contain a fraction
 The denominator does not contain
a radical expression
Does it break a rule?
1) √20
2) 4 √5
3) 1/ √3
4) √(2/5)
Multiplication Property of √
As long as a and b are not
negative,
√a x √b = √ab
√ab = √a x √b
Ex) √54 = √9 x √6
= 3√6
Simplify each expression
1) 5 √300
2) √13 x √52
3) √(x2y5)
2
4) (3 √5)
Division Property
As long as a and b are not
negative,
√(a/b) = √a / √b
√a / √b = √(a/b)
Ex) √(4/9)
√4/ √9
2/3
Simplify each expression
1) √(144/9)
2) √24 / √8
3) √(25c3/b2)
4) Problems from ws
Assignment
Review of Square
Roots WS
Return Tests
 Go over Tests – Make another “cheat
sheet”? Tests will be taken back up!
 Grade Reports
 Missing Assignments/Re-do
Assignments
(Level 1 talking – some people are
doing quizzes/tests!)
Warm-up
9/1/09
Simplify each expression. Use
positive exponents.
1)c3v9c-1c0
2)(w2k0p-5)-7
3)(1.2)5(1.2)-2
4)2y-9h2(2y0h-4)-6
Quiz for Today…
Standard &
Vertex Form
Assignment
More Practice:
Properties of Exponents
Review WS
Turn in!
Do Now
9/2/09
(EC on today’s Quiz?):
Simplify
3
1)
3
(3a b)
4 2
(3ab )
2)
56
8x
I will circulate the room and
check these at the end of
10 minutes. If you do not
have it correct, you will NOT
get bonus points on your
quiz today.
If you have not done so, turn
in Exponent worksheet that
we began in class
yesterday.
Announcements
 5th block tutoring today
 Math II students go to main building Ms.
Dowdy’s room; make sure you sign in and
initial so you get credit for coming
 Everyone’s Invited; WHO’s COMING?
 If you are a car rider and stay after 3:15, you
may come back to my room; I will be at the
school until 4pm for tutoring/test retakes
 Morning Tutoring? Let me know and I will
sign you up (it will be in main building w/
another teacher)
Peer Tutoring
 If you are interested in peer tutoring, let me
know
 You can stop by and see Ms. Kellie Smith (rm
508) and I will also give her your name and
let her know you are interested.
 Please understand, this new math is NOT
easy, and you need to understand as you go
along. The second you don’t understand, find
a way to figure it out!
Did you know?
 The word Himalayas means the "home of
snow."
 A man filed a lawsuit against his doctor
because he survived longer than what the
doctor had predicted
 It took approximately 2.5 million blocks to
build the Pyramid of Giza, which is one of the
Great Pyramids
 The platypus uses its bill to find animals that
it feeds on. Its bill can sense the tiny electric
fields that their preys emit
 Central Park located in New York has 125 drinking fountains
 The largest school in the world is City Montessori School in India
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
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and has over 25,000 students in grade levels ranging from
kindergarten to college
Washing machines use anywhere from 40 to 200 liters of water per
load
Australia has had stamps that actually look like gems. In 1995 and
1996 they used a special technology to make the stamps look like
diamonds and opals.
Hummingbirds are the only animal that can fly backwards
In only eight minutes, the Space Shuttle can accelerate to a speed
of 27,000 kilometers per hour.
In a study conducted regarding toilet paper usage, Americans are
said to use the most toilet paper per trip to the bathroom, which
was seven sheets of toilet paper per trip
German cockroaches can survive for up to one month without food
and two weeks without water
George Washington had teeth made out of hippopotamus ivory
Candy Quiz – Draw Names
y = 2(x + 4)(x + 1)
1. Opening Direction
2. Vertex
3. Axis of Symmetry
4. Point of Extrema
5. y-intercept
6. Zeros
y = - (x – 2) 2 + 3
1. Opening Direction
2. Vertex
3. Axis of Symmetry
4. Point of Extrema
5. y-intercept
6. Zeros
y = x2+ 2x + 1
1. Opening Direction
2. Vertex
3. Axis of Symmetry
4. Point of Extrema
5. y-intercept
6. Zeros
Write in Vertex Form
y  x  4x  4
2
Complex Numbers…
Complex Numbers
LEQ: How do you work with complex
numbers (add, subtract)?
From previously, what if you had the
equation:
x2 + 25 = 0 SOLVE IT!
You end up taking the √ of a negative
number!
(Calculator won’t work)
So, a solution to the problem…
The Imaginary Number
In order to deal with the negative square
root, the imaginary number was
“invented”.
Imaginary Number: i
defined as √-1
For now, you’ll probably only use
imaginary numbers in the context of
solving quadratics for their zeros.
From the web…
(Extra Credit Questions on Friday’s Test)
Imaginary Number
 i is the symbol for the imaginary
i
number.
 It is a complex number whose
square root is negative or zero.
 Rene Descartes was coined the
term in 1637 in his book La
Giometrie.
 The numbers are called
imaginary because they are not
always applied in the real world.
Imaginary Number Applications
 In electrical engineering,
when looking at AC
circuitry, the values of
electrical voltage are
expressed as complex
imaginary numbers known
as phasors.
 Imaginary numbers are
used in areas such as
signal processing, control
theory, electromagnetism,
quantum mechanics and
cartography.
Imaginary Number
 In mathematics Imaginary
Numbers,also called an
Imaginary Unit, can be
found when working with
quadratic functions.
 An equation like x2+1=0 has
an imaginary root, and
requires the use of the
quadratic formula to solve it.
Square Root: A number r is a square root
of a number s if r2 = s.
Radical: The expression √ s is called the radical.
s
radical sign
radicand
Complex Numbers:
written in standard form is
a number a + bi where
a and b are real numbers.
The number a is the real part of the complex
number, and the number bi is the imaginary part.
If b  0, then a + bi is an imaginary number. If a = 0
and b  0, then a + bi is a pure imaginary number.
Complex Numbers
Real Numbers
Imaginary Numbers
(a + bi, b 0)
(a + 0i)
2 + 3i
-1
5 - 5i
9
2
Pure Imaginary Numbers
p
3
(0 + bi, b 0)
-4i
6i
Complex Numbers
 A number of the form a + b(i) ,
where a and b are real numbers,
is called a complex number.
Here are some examples:
 2 – i, 2 – √3i
 The number a is called the real
part of a+bi, the number b is
called the imaginary part of
a+bi.
Operations with Complex
Numbers
Adding & Subtracting them: Just like
combining like terms
Ex. 3i + -1i = 2i
(5 + 7i) + (-2 + 6i) Combine like terms,
simplify
5 – 2 + 7i + 6i
3 + 13i
Adding and Subtracting
Complex Numbers
To add or subtract complex numbers,
add or subtract their real parts and
their imaginary parts separately.
Sum of Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference of Complex Numbers
(a + bi) - (c + di) = (a - c) + (b - d)i
Simplifying complex
numbers:
1) i2 = (√-1)(√-1) = -1
2) i3
3) i4
4) i5
5) i6
6) i7
Examples: Add
a. (7 - i) + (5 + 3i)
b. (3 + 2i) + (7 + 6i)
c. (-5 + 3i) + (-2 - 8i)
You try.Add
1. (5 - 2i) + (2 + 7i)
2. (-1 + i) + (6 - 3i)
3. (3 + 2i) + (-5 - i)
Examples: Subtract
d. (6 - 5i) - (1 + 2i)
e. (4 + 2i) - (5 - i)
f. (-6 + 5i) - (-3 + 9i)
You try. Subtract
4. (9 - 4i) - (-2 + 3i)
5. (6 + 2i) - (5 - 4i)
6. (-3 + 7i) - (-3 - i)
Examples:
Write the expression as a complex number
in standard form.
g. 11 - (7 + 7i) + 3i
h. 9 - (10 + 2i) - 5i
i. 5i + (-4 + 4i) + 17
You try.
Write the expression as a complex number in
standard form.
7. 8 - (2 + 4i) + 3i
8. 2 - (7 + 2i) - 4i
9. -7i + (9 + i) + 6
Ticket Out the Door/Quiz.
Simplify:
1.
-√48
2. (4 + 5i) - (-8 + 12i)
 Complete at the end of your notes; I have to
check off your answer as being correct before
you leave (you don’t need to turn it in)
 Once you have finished with the TOTD, begin
your assignment! (Warm-up adds points)
Assignment:
Page 8: (1-4 all, 6 - 26 even, 28, 29)
Multiplying
Distribute, combine like terms,
simplify
Ex) (5 + 7i)(-2 + 6i)
10 + 30i – 14i + 42i2
10 + 16i + 42(-1)
10 + 16i – 42
-32 + 16i
Absolute Value of
Complex #
 This is the distance from the origin on the
complex number plane
 │a + bi│ = √a2 + b2
 Notice that this is just like the pythagorean
theorem
 Graphing imaginary numbers on the coordinate
plane is very similar to graphing (x,y)coordinates. Now, use the x-axis as the real
plane and the y-axis as the imaginary plane.
Instead of being listed as x,y, the numbers will
Summary
1) How do you perform operations
on complex numbers? What is
different when multiplying
complex numbers?
2) Simplify: √-25
3) Simplify: (2+3i)(3-4i)
4) Solve: 3x2 + 27 = 0
Examples:
e. (8 - 2i)(-6 + 5i)
f. (-3 + i)(8 + 5i)