Download A.1 The Make Up of the Real Numbers System

E - BOOK FOR COLLEGE ALGEBRA
A1
King Fahd University of Petroleum & Minerals
The Real Numbers System




Set of Numbers
Ordering of Real Numbers
Absolute Value
Interval of Real
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E - BOOK FOR COLLEGE ALGEBRA
King Fahd University of Petroleum & Minerals
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 . . .}
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E - BOOK FOR COLLEGE ALGEBRA
King Fahd University of Petroleum & Minerals
Set of Numbers

Rational Numbers
Rational numbers are set of all numbers of the form
a
, b0
b
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
3
non-repeating decimals. For example  , e, 2, 7
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
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
Rational numbers
1−π
−e
e
e−π
Integers
-1, -2, -3, - 4, .…
Whole number
0
Natural
numbers
1, 2, 3, ….
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
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
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0
1
2
3
4
5
6
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
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
1.
 x if x  0
x  
 x if x  0
2.
x  0 if and of if x  0
3.
x y  x  y
4.
xy  x y
Definition: We define the distance between
two numbers a and b by │a − b│
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King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Intervals of Real Numbers
Verbal Description
Geometric Description
All real numbers between
a and b
Notation
( a, b )
All real numbers between
a and b inclusive
a
b
a
b
( a, b ]
All real numbers between
a inclusive and b
[ a, b )
All real numbers between
a inclusive and b inclusive
a
b
a
b
[ a, b ]
All real numbers that are
greater than a
( a, ∞ )
a
All real numbers that are
greater than a inclusive
[ a, ∞ )
a
All real numbers that are
less than a
( –∞, a )
a
All real numbers that are
less than a inclusive
All real numbers
( –∞, a ]
a
-∞
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∞
( –∞, ∞ )
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Use the definition of the absolute value
to evaluate
Example 1
because
a)
5 5
b)
 5    5 

.

c)
.2 5   2 5 
d)
e)
5 2
5 0
because 5  0
because
2
.   5     5   5  
because
.  1   1
2
2
because
 5

1
2
2
2


3
3
3

1



1

1

 

. 
 

 


 


2
f)
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because
 3
5
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
For 0<x<1, simplify each absolute
value expression.
Example 2
a)
e  2x
 e  2x
because 2 x  2  e  2.718
b)
3x  
   3x
because 3x  3  
c)
.x3  x 2  x 2  x3
d)
.
.
x  x. 2
1 x
because x3  x 2 for 0  x  1
x  2  x

1 x

2 1  x 
x2 x

1 x
1 x
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
.
 2
because
0  x 1
King Fahd University of Petroleum & Minerals
E - BOOK FOR COLLEGE ALGEBRA
Example 3
a)
Use the absolute value to describe
each statement.
the distance between k and 4 is 2
k4
 2
b) the distance between x and –3 is less than 1
x   3
c)
 1 
x3
 1
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
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 100 km