chapter 1 real numbers and algebra. 1.1 describing data with set of numbers natural numbers are...

Chapter 1

Real Numbers and Algebra

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, ……} 

…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

…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

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)

…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

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

…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.

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

1.3 Bases and Positive Exponents

Squared 4 Cubed

4 44 . 4 = 42 4 . 4. 4 = 43

4

4

4

ExponentBase

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

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

… 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

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

…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

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

…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

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.

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.

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 }

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}

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

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

Using Graphing Calculator

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

Xmin Xmax

Ymax

Ymin

Xsc1

}Ysc1

[ -2, 3, 0.5] by [-100, 200, 50]

Viewing Rectangle ( Page 57 )

[ -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)

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