Download 6.1 Real Numbers, Order, and Absolute Value

248
CHAPTER 6
The Real Numbers and Their Representations
As mathematics developed, it was discovered that the counting, or natural, numbers
did not satisfy all requirements of mathematicians. Consequently, new, expanded number systems were created. The mathematician Leopold Kronecker (1823–1891) once
made the statement, “God made the integers, all the rest is the work of man.” In this
chapter we look at those sets that, according to Kronecker, are the work of mankind.
The Origins of Zero The
Mayan Indians of Mexico and
Central America had one of the
earliest numeration systems that
included a symbol for zero. The
very early Babylonians had a
positional system, but they placed
only a space between “digits” to
indicate a missing power. When
the Greeks absorbed Babylonian
astronomy, they used the letter
omicron, , of their alphabet or to represent “no power,” or “zero.”
The Greek numeration system was
gradually replaced by the Roman
numeration system.
The Roman system was the
one most commonly used in
Europe from the time of Christ
until perhaps 1400 A.D., when the
Hindu-Arabic system began to take
over. The original Hindu word for
zero was sunya, meaning “void.”
The Arabs adopted this word as
sifr, or “vacant.” The word sifr
passed into Latin as zephirum,
which over the years became
zevero, zepiro, and finally, zero.
Real Numbers, Order, and Absolute Value
Sets of Real Numbers The natural numbers are those numbers with which we
count discrete objects. By including 0 in the set, we obtain the set of whole numbers.
Natural Numbers
1, 2, 3, 4, . . . is the set of natural numbers.
Whole Numbers
0, 1, 2, 3, . . . is the set of whole numbers.
These numbers, along with many others, can be represented on number lines
like the one pictured in Figure 1. We draw a number line by locating any point on
the line and calling it 0. Choose any point to the right of 0 and call it 1. The distance
between 0 and 1 gives a unit of measure used to locate other points, as shown in Figure 1. The points labeled in Figure 1 and those continuing in the same way to the
right correspond to the set of whole numbers.
0
1
2
3
4
5
6
FIGURE 1
All the whole numbers starting with 1 are located to the right of 0 on the number line. But numbers may also be placed to the left of 0. These numbers, written
1, 2, 3, and so on, are shown in Figure 2. (The negative sign is used to show
that the numbers are located to the left of 0.)
Zero
6.1
Negative numbers
–5
–4
–3
–2
–1
Positive numbers
0
1
2
3
4
5
FIGURE 2
The numbers to the left of 0 are negative numbers. The numbers to the right of
0 are positive numbers. The number 0 itself is neither positive nor negative. Positive numbers and negative numbers are called signed numbers.
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6.1
The Origins of Negative
Numbers Negative numbers
can be traced back to the Chinese
between 200 B.C. and 220 A.D.
Mathematicians at first found
negative numbers ugly and
unpleasant, even though they kept
cropping up in the solutions of
problems. For example, an Indian
text of about 1150 A.D. gives the
solution of an equation as 5 and
then makes fun of anything so
useless.
Leonardo of Pisa (Fibonacci),
while working on a financial
problem, was forced to conclude
that the solution must be a
negative number (that is, a
financial loss). In 1545 A.D., the
rules governing operations with
negative numbers were published
by Girolamo Cardano in his Ars
Magna (Great Art).
Real Numbers, Order, and Absolute Value
249
There are many practical applications of negative numbers. For example, temperatures sometimes fall below zero. The lowest temperature ever recorded in meteorological records was 128.6F at Vostok, Antarctica, on July 22, 1983. Altitudes
below sea level can be represented by negative numbers. The shore surrounding the
Dead Sea is 1312 feet below sea level; this can be represented as 1312 feet.
The set of numbers marked on the number line in Figure 2, including positive
and negative numbers and zero, is part of the set of integers.
Integers
. . . , 3, 2, 1, 0, 1, 2, 3, . . . is the set of integers.
1
Not all numbers are integers. For example, 2 is not; it is a number halfway
1
between the integers 0 and 1. Also, 3 4 is not an integer. Several numbers that are
not integers are graphed in Figure 3. The graph of a number is a point on the number line. Think of the graph of a set of numbers as a picture of the set. All the
numbers in Figure 3 can be written as quotients of integers. These numbers are
examples of rational numbers.
– 3_2
–2
– 2_3
1_
2
–1
0
11_3
1
23
__ 3 1_
8
4
2
3
4
FIGURE 3
Notice that an integer, such as 2, is also a rational number; for example, 2 21.
Rational Numbers
x x is a quotient of two integers, with denominator not equal to 0 is
the set of rational numbers.
(Read the part in the braces as “the set of all numbers x such that x is a
quotient of two integers, with denominator not equal to 0.”)
The set symbolism used in the definition of rational numbers,
1
1
x x has a certain property ,
2
1
1
FIGURE 4
is called set-builder notation. This notation is convenient to use when it is not possible, or practical, to list all the elements of the set.
Although a great many numbers are rational, not all are. For example, a square
that measures one unit on a side has a diagonal whose length is the square root of 2,
written 2. See Figure 4. It will be shown later that 2 cannot be written as a quotient of integers. Because of this, 2 is not rational; it is irrational.
Irrational Numbers
x x is a number on the number line that is not rational is the set of
irrational numbers.
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Examples of irrational numbers include 3, 7, 10, and , which is
the ratio of the distance around a circle (its circumference) to the distance across it
(its diameter).
All numbers that can be represented by points on the number line are called
real numbers.
Real Numbers
x x is a number that can be represented by a point on the number line
is the set of real numbers.
Real numbers can be written as decimal numbers. Any rational number can be
written as a decimal that will come to an end (terminate), or repeat in a fixed “block”
of digits. For example, 25 .4 and 27100 .27 are rational numbers with terminating decimals; 13 .3333 . . . and 311 .27272727 . . . are repeating decimals. The decimal representation of an irrational number will neither terminate nor
repeat. Decimal representations of rational and irrational numbers will be discussed
further later in this chapter.
Figure 5 illustrates two ways to represent the relationships among the various
sets of real numbers.
Irrational numbers
Rational numbers
4_ – 5_ 11
__
, ,
9
– 8
8 7
15
Integers
–11, –6, –4
23
π
Whole
numbers
0
π
__
4
Natural
numbers
1, 2, 3, 4,
5, 37, 40
Irrational numbers
Real
numbers
Zero
Whole numbers
Integers
Rational
numbers
All numbers shown are real numbers.
Negative integers
Non-integer rational numbers
(a)
(b)
FIGURE 5
EXAMPLE
1
List the numbers in the set
5, 2
13
, 0, 2, , 5, 5.8
3
4
that belong to each of the following sets of numbers.
(a) natural numbers
The only natural number in the set is 5.
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Natural
numbers
6.1
The TI-83 Plus calculator will
convert a decimal to a fraction.
See Example 1(d).
The calculator returns a 1 for these
statements of inequality, signifying
that each is true.
The symbol for equality, ,
was first introduced by the
Englishman Robert Recorde in his
1557 algebra text The Whetstone
of Witte. He used two parallel line
segments because, he claimed, no
two things can be more equal.
The symbols for order
relationships, and , were
first used by Thomas Harriot
(1560 – 1621), another
Englishman. These symbols
were not immediately adopted by
other mathematicians.
Real Numbers, Order, and Absolute Value
251
(b) whole numbers
The whole numbers consist of the natural numbers and 0. So the elements of the
set that are whole numbers are 0 and 5.
(c) integers
The integers in the set are 5, 0, and 5.
(d) rational numbers
The rational numbers are 5, 23, 0, 134, 5, and 5.8, since each of these
numbers can be written as the quotient of two integers. For example, 5.8 5810 295.
(e) irrational numbers
The only irrational number in the set is 2.
(f) real numbers
All the numbers in the set are real numbers.
Order Two real numbers may be compared, or ordered, using the ideas of equality and inequality. Suppose that a and b represent two real numbers. If their graphs on
the number line are the same point, they are equal. If the graph of a lies to the left
of b, a is less than b, and if the graph of a lies to the right of b, a is greater than b.
We use symbols to represent these ideas.
When read from left to right, the symbol means “is less than,” so “7 is less
than 8” is written
7 8.
The symbol means “is greater than.” We write “8 is greater than 2” as
8 2.
Notice that the symbol always points to the smaller number. For example, write “8
is less than 15” by pointing the symbol toward the 8:
8 15 .
The symbol means “is less than or equal to,” so
59
means “5 is less than or equal to 9.” This statement is true, since 5 9 is true. If
either the part or the part is true, then the inequality is true.
The symbol means “is greater than or equal to.” Again,
9
5
is true because 9 5 is true. Also, 8 8 is true since 8 8 is true. But it is not true
that 13 9 because neither 13 9 nor 13 9 is true.
EXAMPLE
The inequalities in Example 2(a)
and (d) are false, as signified
by the 0. The statement in
Example 2(e) is true.
(a)
(b)
(c)
(d)
(e)
2
Determine whether each statement is true or false.
66
The statement is false because 6 is equal to 6.
5 19
Since 5 is indeed less than 19, this statement is true.
15 20
The statement is true, since 15 20.
25 30
Both 25 30 and 25 30 are false, so 25 30 is false.
12 12
Since 12 12 , this statement is true.
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The Real Numbers and Their Representations
Additive Inverses and Absolute Value
For any real number x (except 0),
there is exactly one number on the number line the same distance from 0 as x but
on the opposite side of 0. For example, Figure 6 shows that the numbers 3 and 3
are both the same distance from 0 but are on opposite sides of 0. The numbers 3 and
3 are called additive inverses, or opposites, of each other.
Distance is 3.
Distance is 3.














–3
0
3
FIGURE 6

























The additive inverse of the number 0 is 0 itself. This makes 0 the only real number that is its own additive inverse. Other additive inverses occur in pairs. For example, 4 and 4, and 5 and 5, are additive inverses of each other. Several pairs of
additive inverses are shown in Figure 7.






–6
–4
–1
0
1
4
6
FIGURE 7
The additive inverse of a number can be indicated by writing the symbol in front of the number. With this symbol, the additive inverse of 7 is written 7.
The additive inverse of 4 is written 4, and can also be read “the opposite of
4” or “the negative of 4.” Figure 7 suggests that 4 is an additive inverse of 4.
Since a number can have only one additive inverse, the symbols 4 and 4 must
represent the same number, which means that
4 4 .
This idea can be generalized as follows.
Double Negative Rule
For any real number x,
Number
4
Additive
Inverse
(x) x.
4 or 4
0
0
19
19
2
3
2
3
The chart shows several numbers and their additive inverses. An important
property of additive inverses will be studied later in this chapter: a a a a 0 for all real numbers a.
As mentioned above, additive inverses are numbers that are the same distance
from 0 on the number line. See Figure 7. This idea can also be expressed by saying
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6.1
Real Numbers, Order, and Absolute Value
253
that a number and its additive inverse have the same absolute value. 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.” For example, the distance between 2 and 0 on the number line
is 2 units, so
2 2 .
Because the distance between 2 and 0 on the number line is also 2 units,
2 2 .
Since distance is a physical measurement, which is never negative, the absolute
value of a number is never negative. For example, 12 12 and 12 12, since
both 12 and 12 lie at a distance of 12 units from 0 on the number line. Also, since
0 is a distance of 0 units from 0,
0 0.
In symbols, the absolute value of x is defined as follows.
Formal Definition of Absolute Value
x x
x
if x 0
if x 0
By this definition, if x is a positive number or 0, then its absolute value is x itself.
For example, since 8 is a positive number, 8 8. However, if x is a negative number, then its absolute value is the additive inverse of x. This means that if x 9,
then 9 9 9, since the additive inverse of 9 is 9.
The formal definition of absolute value can be confusing if it is not read carefully. 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. The absolute value of a number is never
negative.
EXAMPLE
3
Simplify by finding the absolute value.
(a) 5 5
(b) 5 5 5
(c) 5 5 5
(d) 14 14 14
(e) 8 2 6 6
(f ) 8 2 6 6
Part (e) of Example 3 shows that absolute value bars are also grouping symbols.
You must perform any operations that appear inside absolute value symbols before
finding the absolute value.
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CHAPTER 6
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Applications
A table of data provides a concise way of relating information.
EXAMPLE 4
The projected annual rates of employment change (in percent)
in some of the fastest growing and most rapidly declining industries from 1994
through 2005 are shown in the table.
Industry (1994 –2005)
Percent Rate
of Change
Health services
5.7
Computer and data processing services
4.9
Child day care services
4.3
Footware, except rubber and plastic
6.7
Household audio and video equipment
4.2
Luggage, handbags, and leather products
3.3
Source: U.S. Bureau of Labor Statistics.
What industry in the list is expected to see the greatest change? the least change?
We want the greatest change, without regard to whether the change is an increase or a decrease. Look for the number in the list with the largest absolute value.
That number is found in footware, since 6.7 6.7. Similarly, the least change is
in the luggage, handbags, and leather products industry: 3.3 3.3.
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In Exercises 1–6, give a number that satisfies the given condition.
1. An integer between 3.5 and 4.5
2. A rational number between 3.8 and 3.9
3. A whole number that is not positive and is less than 1
4. A whole number greater than 4.5
5. An irrational number that is between 11 and 13
6. A real number that is neither negative nor positive
In Exercises 7–10, decide whether each statement is true or false.
7. Every natural number is positive.
8. Every whole number is positive.
9. Every integer is a rational number.
10. Every rational number is a real number.
In Exercises 11 and 12, list all numbers from each set that are (a) natural numbers; (b) whole numbers;
(c) integers; (d) rational numbers; (e) irrational numbers; (f ) real numbers.
11.
9, 7, 1
3
1
, , 0, 5, 3, 5.9, 7
4
5
13. Explain in your own words the different sets of numbers introduced in this section, and give an example
of each kind.
12.
5.3, 5, 3, 1, 1
, 0, 1.2, 1.8, 3, 11
9
14. What two possible situations exist for the decimal
representation of a rational number?