e - book for college algebra king fahd university of petroleum & minerals a1 e - book for...

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

A1


The Real Numbers System

Set of Numbers Ordering of Real Numbers Absolute Value Interval of Real



Set of Numbers

Natural Numbers The set of natural numbers (counting numbers) is {1, 2, 3,

4, 5 . . .}

Whole Numbers The set of whole numbers (including 0 with the set of

natural numbers) is {0, 1, 2, 3, 4, 5 . . .}

Integers Integers are the expanded set of numbers obtained by

including the negative of the counting numbers. {. . . –4, –3, –2, –1, 0, 1, 2, 3, 4 . . .}



Set of Numbers

Rational Numbers Rational numbers are set of all numbers of the form

where a (called numerator) and b (called denominator) are integers.

Irrational Numbers Irrational numbers are real numbers that are not

rational. Equivalently, they are non-terminating and non-repeating decimals. For example

a, b 0

b

3, , 2, 7e



Set of Numbers

The Real Numbers The set of all the irrational numbers and the rational

numbers constitutes the set of the real numbers.

Irrational numbers

1 − π

− e

e

e − π

Rational numbers

Integers-1, -2, -3, - 4, .…

Whole number0

Natural numbers

1, 2, 3, ….



Ordering of Real Numbers

The real number system has the property of being ordered. This means that for any two different real numbers a and b, one and only one of the following three statements must be true: a < b, a = b or a > b. This property allows the arrangement of the real numbers on any line, L, as shown in the figure below.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6



Absolute Value

The absolute value of a real number is the distance from the number to the origin O on the number line. The absolute value of x is denoted by │x│.

Absolute Value Properties

Definition: We define the distance between two numbers a and b by │a − b│

if 0

if 0

x xx

x x

if and of if0 0x x

xy x yx y x y

1. 2.

3. 4.



Intervals of Real NumbersVerbal Description Geometric Description Notation

All real numbers between a and b

( a, b )

All real numbers between a and b inclusive

( a, b ]

All real numbers between a inclusive and b

[ a, b )

All real numbers between a inclusive and b inclusive

[ a, b ]

All real numbers that are greater than a

( a, ∞ )

All real numbers that are greater than a inclusive

[ a, ∞ )

All real numbers that are less than a

( –∞, a )

All real numbers that are less than a inclusive

( –∞, a ]

All real numbers

( –∞, ∞ )

a b

a b

a b

a b

a

a

a

a

- ∞ ∞



a) because

b) . because

c) . because

d) . because

e) . because

f) . because

Example 1

5 5

55

2 25 2 5 5

5 55

12 2

1

22 23 3

13

11

Use the definition of the absolute value to evaluate

5 0

5 0

2 5

5

12

3



a) because

b) because

c) . because

d) .

. .

.

Example 2For 0<x<1, simplify each absolute value expression.

2 2 2.718x e

3 3x

3 2 fo 0r 1x x x

0 1x

2 2xe x e

33x x

3 2 2 3x xx x

2

1

2 12

1

1

1

2

2

x x

x

xx x

x

x x

x

x

because



Example 3Use the absolute value to describe each statement.

a) the distance between k and 4 is 2k 4 2

b) the distance between x and –3 is less than 1

3 1 3 1x x

c) the city of Jeddah is closer to Makkah than it is to Madinah

Jeddah Makkah Jeddah Madinah

d) the city of Hail is more than 100 kilometers away from Buraidah

Hail Buraida 100 km

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