chapter 1 real numbers and algebra. 1.1 describing data with set of numbers natural numbers are...
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Chapter 1
Real Numbers and Algebra
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1.1 Describing Data with Set of numbers
Natural Numbers are counting numbers
and can be expressed as N = { 1, 2, 3, 4, 5, 6, …. }
Set braces { }, are used to enclose the elements of a set. A whole numbers is a set of numbers, is
given by
W = { 0, 1, 2, 3, 4, 5, ……}
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…Continued
The set of integers include both natural and the whole numbers and is given by I = { …, -3, -2, -1, 0, 1, 2, 3, ….} A rational number is any number can be written as the ratio of two integers where q = 0. Rational numbers can be written as fractions and include all integers. Some examples of rational numbers are , 1.2, and 0.
q
p
7
22,
2
7,
5
3,
3
2,
1
8
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…continued
Rational numbers may be expressed in decimal form that either repeats or terminates.
The fraction may be expressed as 0.3, a repeating decimal, and the fraction may be expressed as 0.25, a terminating decimal. The overbar indicates that 0.3 = 0.3333333….
Some real numbers cannot be expressed by fractions. They are called irrational numbers.
2, 15, and are examples of irrational numbers.
3
13
1
4
1
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Identity Properties For any real number a, • a + 0 = 0 + a = a, 0 is called the additive identity and• a . 1 = 1 . a = a, The number 1 is called the multiplicative identity.
Commutative Properties For any real numbers a and b,
a + b = b + a (Commutative Properties of addition)
a.b = b.a (Commutative Properties of multiplication)
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…Continued
Associative Properties For any real numbers a, b, c,
(a + b) + c = a + (b + c) (Associative Properties of addition)
(a.b) . c = a . (b . c) (Associative Properties for multiplication)
Distributive PropertiesFor any real numbers a, b, c,
a(b + c) = ab + ac anda(b- c) = ab - ac
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1.2 Operation on Real Numbers
The Real Number Line
-3 -2 -1 0 1 2 3
Origin
-3 -2 -1 0 1 2 3
Origin -2 2 -2 = 2 Absolute value
cannot be negative 2 = 2
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…ContinuedIf a real number a is located to the left of areal number b on the number line, we saythat a is less than b and write a<b.
Similarly, if a real number a is located to theright of a real number b, we say that a isgreater than b and write a>b.
Absolute value of a real number a, written a , is equal to its distance from the origin onthe number line. Distance may be eitherpositive number or zero, but it cannot be anegative number.
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Arithmetic OperationsAddition of Real NumbersTo add two numbers that are either both positive or bothnegative, add their absolute values. Their sum has the samesign as the two numbers.Subtraction of real numbersFor any real numbers a and b, a-b = a + (-b).Multiplication of Real NumbersThe product of two numbers with like signs is positive.The product of two numbers with unlike signs is negative.Division of Real NumbersFor real numbers a and b, with b = 0, = a . That is, to divide a by b, multiply a by the reciprocal of b.
b
a
b
1
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1.3 Bases and Positive Exponents
Squared 4 Cubed
4 44 . 4 = 42 4 . 4. 4 = 43
4
4
4
ExponentBase
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Powers of TenPower of 10 Value
103 1000
102 100
101 10
100 1
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
10
1
100
1
1000
1
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1.3 Integer Exponents Let a be a nonzero real number and n be a
positive integer. Then an = a. a. a. a……a (n factors of a ) a0 = 1, and a –n = a -n b m
b -m = a n
a -n b n
b = a
na
1
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… cont
The Product Rule
For any non zero number a and integers m and n,
am . an = a m+n
The Quotient Rule For any nonzero number a and integers m and
n, am
= a m – n
a n
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Raising Products To Powers For any real numbers a and b and integer n, (ab) n = a n b n Raising Powers to Powers For any real number a and integers m and n, (am)n = a mn
Raising Quotients to Powers For nonzero numbers a and b and any integer a n = an
bbn
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…Continued
A positive number a is in scientific notation when a is written as b x 10n, where 1 < b < 10 and n is an integer.
Scientific Notation
Example : 52,600 = 5.26 x 104 and 0.0068 = 6.8 x 10 -3
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1.4 Variables, Equations , and Formulas
A variable is a symbol, such as x, y, t, used to represent any unknown number or quantity.
An algebraic expression consists of numbers, variables, arithmetic symbols, parenthesis, brackets, square roots.
Example 6, x + 2, 4(t – 1)+ 1, X + 1
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…cont
An equation is a statement that says two mathematical expressions are equal.
Examples of equation 3 + 6 = 9, x + 1 = 4, d = 30t, and x + y = 20 A formula is an equation that can be used to
calculate one quantity by using a known value of another quantity.
The formula y = computes the no. of yards in x feet. If x= 15, then y= = 5. 3
x
3
15
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Square roots
The number b is a square root of a number a if b2 = a. Example - One square root of 9 is 3 because 32 = 9.The other square root of 9 is –3 because (-3)2 = 9. We use thesymbol to 9 denote the positive or principal square root of 9.That is, 9 = +3. The following are examples of how to evaluate the square root symbol. A calculator is sometimes needed to
approximate square roots, 4 = + 2 -The symbol ‘ + ‘ is read ‘plus or minus’. Note that 2 representsthe numbers 2 or –2.
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Cube roots
The number b is a cube root of a number a if b3 = a
The cube root of 8 is 2 because 23 = 8, which may
be written as 3 8 = 2. Similarly 3 –27 = -3 because
(- 3)3 = - 27.
Each real number has exactly one cube root.
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1.5 Introduction to graphing
Relations is a set of Ordered pairs.
If we denote the ordered pairs in a relation(x,y), then the set of all x-values is called theDomain (D) of the relation and the set of all yvalues is called the Range (R)
S = {(2, -2), (3, 4), (8, 9), (11, 13 )}D= {2, 3, 8, 11}R= { -2, 4, 9, 13 }
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Example 1. Find the domain and range for the relationgiven by S = {( -3, -1), (0,3), (2, 4), (4,5), (6,5)} Solution The domain D is determined by the firstelement in each ordered pair, or D ={-3, 0,2, 4,6} The range R is determined by the secondelement in each ordered pair, or R = {-1,3,4,5}
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The Cartesian Coordinate System
Quadrant II y Quadrant I y
Quadrant III Quadrant IV
The xy – plane Plotting a point
x x
Origin
-2 -1 1 2 -2 -1 1 2
3
2
1
-1
-2
(1, 3)
21
0-1-2
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Scatterplots and Line Graphs
If distinct points are plotted in the xy- plane, the resulting graph is called a scatterplot.
1 2 3 4 5 6 7
7
6
5
4
3
2
1
X
Y
(1, 1)
(3, 4)
(4, 6)
(5, 0)
(6, 3)
0
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Using Graphing Calculator
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Using Graphing Calculator
Go to Y= and enter Go to 2nd then table set and enter Go to 2nd then table
Make a table for y = , starting at x = 10 and incrementing by 10 and compareThe table for example 4 ( pg 41)
Graph
9
2x
9
2x
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Xmin Xmax
Ymax
Ymin
Xsc1
}Ysc1
[ -2, 3, 0.5] by [-100, 200, 50]
Viewing Rectangle ( Page 57 )
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[ -4, 4, 1] by [-4, 4, 1]
Go to Stat Edit then enter points Go to 2nd then stat plot
Scatter plot
Making a scatterplot with a graphing calculator
Plot the points (-2, -2), (-1, 3), (1, 2) and (2, -3) in [ -4, 4, 1] by [-4, 4, 1] (Example 10, page 58)
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Example 11Cordless Phone Sales
[1985, 2002, 5] by [0, 40, 10]
Year 1987 1990 1993 1996 2000
Phones (millions)
6.2 9.9 18.7 22.8 33.3
Go to Stat edit and enter data Enter line graph
Enter datas in window Hit graph