Transcript Document

Chapter 8
Real Numbers and
Introduction to
Algebra
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
8.1
Symbols and Sets of
Numbers
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Set of Numbers
• Natural Numbers: {1, 2, 3, 4, 5, 6 . . .}
• Whole Numbers: {0, 1, 2, 3, 4 . . .}
• Integers: {. . . –3, –2, –1, 0, 1, 2, 3 . . .}
• Rational Numbers: the set of all numbers that can
be expressed as a quotient of integers, with
denominator  0.
• Irrational Numbers: nonrational numbers that
correspond to points on a number line.
• Real Numbers: all numbers that correspond to
pints on a number line.
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Equality and Inequality Symbols
Symbol
a=b
ab
a<b
a>b
ab
ab
Meaning
a is equal to b.
a is not equal to b.
a is less than b.
a is greater than b.
a is less then or equal to b.
a is greater than or equal to b.
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The Number Line
A number line is a line on which each point is
associated with a number.
–5 –4 –3 –2 –1
0
– 4.8
1
2
3
4
5
1.5
Negative
numbers
Positive
numbers
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Order Property for Real Numbers
For any two real numbers a and b, a is less than b if a is to the left of
b on the number line.
• a < b means a is to the left of b on a number line.
• a > b means a is to the right of b on a number line.
a
b
a < b or also b > a
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Example
Determine whether each mathematical statement is true.
a. 4 < 5
True
b.
27 ≥ 27 True
c.
0>5
d. 16 ≤ 9
False
False
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Example
Translate each sentence into a mathematical statement.
a. Thirteen is less than or equal to nineteen.
13
≤
19
b.
Five is greater than two.
5
>
2
c. Seven is not equal to eight.
7
≠
8
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Example
Graph the numbers on a number line.
1
3
1 ,2.5, , 3.75
4
2
3.75 1 1
4
3 2.5
2
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Example
Insert <, >, or = between the pairs of numbers to
form true statements.
a. 4.7 > 4.697
b.
32.61 = 32.61
c.
– 4 > –7
d.
1
4
<
2
3
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Absolute Value
The absolute value of a real number a, denoted by |a|,
is the distance between a and 0 on the number line.
| –4| = 4
Symbol for
absolute
value
|5| = 5
Distance of 4
–5 –4 –3 –2 –1
Distance of 5
0
1
2
3
4
5
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Example
Find the absolute value of each number.
a. 9  9
b. 6  6
c.  4  4
5 5
d. 0  0
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Example
Insert <, >, or = between the pairs of numbers to
form true statements.
a. 4 > 0
b. 5 < 9
2
1
c. 6 > 6
3
3
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