Chapter 1
Section 4
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Real Numbers and the Number Line
Classify numbers and graph them on number
lines.
Tell which of two real numbers is less than
the other.
Find additive inverses and absolute values of
real numbers.
Interpret the meanings of real numbers from a
table of data.
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Objective 1
Classify numbers and graph
them on number lines.
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Classify numbers and graph them on a
number line.
The natural numbers and the whole numbers, along with many
others, can be represented on a number line like the one below.
We draw a number line by choosing any point on the line and
labeling it 0. Then we choose any point to the right of 0 and label
it 1. The distance between 0 and 1 gives a unit of measure used
to locate other points.
The “arrowhead” is used to indicate the positive direction on a
number line.
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Classify numbers and graph them on a
number line. (cont’d)
The natural numbers are located to the right of 0 on the
number line. For each natural number, we can place a
corresponding number to the left of 0. Each is the opposite, or
negative, of a natural number.
The natural numbers, their opposites, and 0 form a new set of
numbers called the integers.
{. . . , −3, −2, −1, 0, 1, 2, 3, . . .}
Positive numbers and negative numbers are called signed
numbers.
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EXAMPLE 1
Using Negative Numbers in
Applications
Use an integer to express the number in boldface italics
in each application.
Erin discovers that she has spent $53 more than she has in her
checking account.
Solution: −53
The record-high Fahrenheit temperature in the United States
was 134° in Death Valley, California, on July 10, 1913.
(Source: World Almanac and Book of Facts 2006.)
Solution: 134
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Classify numbers and graph them on a
number line. (cont’d)
{ x x is a quotient of two integers, with denominator not 0} is
the set of rational numbers.
(Read as “the set of all numbers x such that x is a quotient of
two integers, with denominator not 0.”)
This is called set-builder notation. This notation is convenient
to use when it is not possible to list all the elements of a set.
Since any integer can be written as a quotient of itself and 1,
all integers are also rational numbers.
−5
Example: −5 =
1
A decimal number that comes to an end (terminates), such as
23
=
0.23 is a rational number. For example, 0.23
.
100
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Classify numbers and graph them on a
number line. (cont’d)
Decimals numbers that repeat in a fixed block of digits, such
1
as 0.3333… = 0.3 , are also rational numbers. Example: 0.3 =
3
To graph a number, we place a dot on the number line at the
point that corresponds to the number. The number is called the
coordinate of the point.
{ x x is a nonrational number represented by a point on the
number} is the set of irrational numbers.
The decimal form of an irrational number neither terminates
nor repeats.
{ x x is a rational or an irrational number} is the set of real
numbers.
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EXAMPLE 2
Determining whether a Number
Belongs to a Set
Identify each real number in the set
3
⎧5
⎫
⎨ , −7,1 , 0, 11, π ⎬ as rational or irrational.
5
⎩8
⎭
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Objective 2
Tell which of two real numbers
is less than the other.
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Ordering of 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.
This means that any negative number is less than 0, and any
negative number is less than any positive number. Also, 0 is less
than any positive number.
We can also say that, for any two real numbers a and b, a is
greater than b, if a is to the right of b on the number line.
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EXAMPLE 3
Determining the Order of Real
Numbers
Determine whether the statement is true or false.
−4 > −1
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Objective 3
Find additive inverses and
absolute values of real numbers.
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Finding additive inverses and absolute values
of real numbers.
By a property of the real numbers, for any real number x
(except 0), there is exactly one number, called the additive
inverse, on the number line the same distance from 0 as x, but on
the opposite side of 0.
The additive inverse of a number can be indicated by writing
the − symbol in front of the number.
The additive inverse of −7 is written −(−7) and can be read “the
opposite of −7” or “the negative of −7”
The Double Negative Rule, states that for any real number x,
−(−x) = x
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Finding additive inverses and absolute values
of real numbers. (cont’d)
The absolute value of a real number can be defined as the
distance between 0 and the number on the number line. The
symbol for the absolute value of the number x is |x|, read “the
absolute value of x.”
Distance is a physical measurement, which is never negative.
Therefore, the absolute value of a number is never negative.
The rule of Absolute Value says that, for any real number x,
⎧ x if x ≥ 0
x =⎨
⎩− x if x < 0 .
The “−x ” in the second part of the definition does not represent a
negative number. Since x is negative in the second part, −x represents
the opposite of a negative number—that is, a positive number.
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EXAMPLE 4
Finding the Absolute Value
Simplify by finding the absolute value.
− 32 − 2
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Objective 4
Interpret the meanings of real
numbers from a table of data.
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EXAMPLE 5
Interpreting Data
In the table, which commodity in which year represents the
greatest percent increase?
Solution:
Iron and steal, from 2003 to 2004
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