Download The Rational Numbers

CHAPTER 5
Number Theory and the
Real Number System
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5.3
The Rational Numbers
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Objectives
1. Define the rational numbers.
2. Reduce rational numbers.
3. Convert between mixed numbers and improper fractions.
4. Express rational numbers as decimals.
5. Express decimals in the form a / b.
6. Multiply and divide rational numbers.
7. Add and subtract rational numbers.
8. Use the order of operations agreement with rational numbers.
9. Apply the density property of rational numbers.
10. Solve problems involving rational numbers.
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Defining the Rational Numbers
• The set of rational numbers is the set of all numbers
a
which can be expressed in the form , where a and b
b
are integers and b is not equal to 0.
• The integer a is called the numerator.
• The integer b is called the denominator.
The following are examples of rational numbers:
¼, ½, ¾, 5, 0
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Reducing a Rational Number
a
b
• If is a rational number and c is any number other
than 0,
ac a

bc b
• The rational numbers a and a  c are called equivalent
b
bc
fractions.
• To reduce a rational number to its lowest terms,
divide both the numerator and denominator by their
greatest common divisor.
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Example 1: Reducing a Rational Number
Reduce
130
455
to lowest terms.
Solution: Begin by finding the greatest common divisor
of 130 and 455.
Thus, 130 = 2 · 5 · 13, and 455 = 5 · 7 · 13. The greatest
common divisor is 5 · 13 or 65.
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Example 1: Reducing a Rational Number
(continued)
Divide the numerator and the denominator of the given
rational number by 5 · 13 or 65.
130 2  5  13 2


455 5  7  13 7
or
130 130  65 2


455 455  65 7
There are no common divisors of 2 and 7 other than 1.
2
Thus, the rational number is
in its lowest terms.
7
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Mixed Numbers and Improper Fractions
• A mixed number consists of the sum of an integer and
a rational number, expressed without the use of an
addition sign.
Example:
• An improper fraction is a rational number whose
numerator is greater than its denominator.
Example: 19
19 is larger than 5
5
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Converting a Positive Mixed Number to an Improper
Fraction
1. Multiply the denominator of the rational number by
the integer and add the numerator to this product.
2. Place the sum in step 1 over the denominator of the
mixed number.
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Example 2: Converting a Positive Mixed Number to an
Improper Fraction
Example: Convert
to an improper fraction.
Solution:
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Converting a Positive Improper Fraction to a Mixed
Number
1. Divide the denominator into the numerator. Record
the quotient and the remainder.
2. Write the mixed number using the following form:
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Example 3: Converting from an Improper Fraction to a
Mixed Number
42
Convert
to a mixed number.
5
Solution: Step 1 Divide the denominator into the
numerator.
Step 2 Write the mixed number using
Thus,
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Rational Numbers and Decimals
• Any rational number can be expressed as a decimal
by dividing the denominator into the numerator.
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Example 4: Expressing Rational Numbers as
Decimals
Express each rational number as a decimal.
a. 5
b. 7
8
11
Solution: In each case, divide the denominator into the
numerator.
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Example 4: Expressing Rational Numbers as Decimals
(continued)
a.
0.625
8 5.000
48
20
16
40
40
0
Notice the decimal stops.
This is called a
terminating decimal.
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b.
0.6363
11 7.000
66
40
33
70
66
40
Notice the digits 63
repeat over and over
indefinitely. This is
called a repeating
decimal.
33
70
15
Expressing Decimals as a Quotient of Two
Integers
• Terminating decimals can be
expressed with denominators of
10, 100, 1000, 10,000, and so on.
• Using the chart, the digits to the
right of the decimal point are the
numerator of the rational number.
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Example 5: Expressing Decimals as a Quotient of
Two Integers
Express each terminating decimal as a quotient of
integers:
a. 0.7
b. 0.49
c. 0.048
Solution:
7
a.0.7 =
because the 7 is in the tenths position.
10
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Example 5: Expressing Terminating Decimals in
a/b form (continued)
49
100
b. 0.49 =
because the digit on the right, 9, is in the
hundredths position.
48
1000
c. 0.048 =
because the digit on the right, 8, is in the
thousandths position. Reducing to lowest terms,
48
48  8
6


1000 1000  8 125
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Example 6: Expressing a Repeating Decimal in
a/b Form
Express 0.6 as a quotient of integers.
Solution: Step1 Let n equal the repeating decimal such
that n = 0.6 , or 0.6666…
Step 2 If there is one repeating digit, multiply both
sides of the equation in step 1 by 10.
n = 0.66666…
Multiplying by
10 moves the
10n = 10(0.66666…)
decimal point
one place to
10n = 6.66666…
the right.
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Example 6: Expressing a Repeating Decimal in a/b
Form (continued)
Step 3 Subtract the equation in step 1 from the equation
in step 2.
Step 4 Divide both sides of the equation in step 3 by the
number in front of n and solve for n.
9n  6
We solve 9n = 6 for n:
9n 6

9 9
6 2
n 
9 3
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Thus, 0.6 
.
3
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Repeating Decimal to Rational
•
__
Express 0.63 as a rational number.
• Solution
n = 0.6363…
10n = 6.6363…
10n = 6.6363…
- n = 0.6363…
-------------------9n = 6
n = 2/3
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Multiplying Rational Numbers
• The product of two rational numbers is the product of
their numerators divided by the product of their
denominators.
a c a c
a
c
• If and are rational numbers, then  
.
b
d
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b d
bd
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Example 8b: Multiplying Rational Numbers
Multiply. If possible, reduce the product to its lowest
terms:
 2  9 
    
 3  4 
1
 2  9   2 9 18 3  6 3



or 1
     
3 4
12 2  6 2
2
 3  4  Multiply
across.
Simplify to
lowest terms.
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Dividing Rational Numbers
• The quotient of two rational numbers is a product of
the first number and the reciprocal of the second
number.
a
c
If b and d are rational numbers, then
a c a d ad
   
b d b c bc
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Example 9b: Dividing Rational Numbers
Divide. If possible, reduce the quotient to its lowest
terms:
3 7


5 11
3 7
3 11
3  11
33
    

5 11
5 7
57
35
Change to
multiplication
by using the
reciprocal.
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Multiply across.
25
Adding and Subtracting Rational Numbers with
Identical Denominators
The sum or difference of two rational numbers with
identical denominators is the sum or difference of their
numerators over the common denominator.
a
If c and are rational numbers, then
b
b
a c ac
 
b b
b
and
a c ac
 
b b
b
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Example 10: Adding & Subtracting Rational Numbers
with Identical Denominators
Perform the indicated operations:
1 
3
11
5
3
2
a. 
b. 
c.  5    2 
7
4 
12 12
7
4
Solution:
a. 3  2  3  2  5
7
7
7
7
b. 11  5  11  5  6  1  6  1
12
c.
12
12
12
26
2
1  3
21  11 
21 11 21  11 10
5
5   2           


4  4
4  4
4 4
4
4
2
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or  2
1
2
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Adding and Subtracting Rational Numbers
Unlike Denominators
• If the rational numbers to be added or subtracted have
different denominators, we use the least common
multiple of their denominators to rewrite the rational
numbers.
• The least common multiple of their denominators is
called the least common denominator or LCD.
a a c ac
  
b b c bc
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Example 11: Adding Rational Numbers
Unlike Denominators
3 1
Find the sum of  .
4 6
Solution: Find the least common multiple of 4 and 6 so
that the denominators will be identical. LCM of 4 and
6 is 12. Hence, 12 is the LCD.
3 1
3 3 1 2 We multiply the first rational
    
number by 3/3 and the second one
4 6
4 3 6 2 by 2/2 to obtain 12 in the
denominator for each number.
9
2


Notice, we have 12 in the
12 12
denominator for each number.
11

Add numerators and put this sum over the
12
least common denominator.
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Density of Rational Numbers
If r and t represent rational numbers, with r < t, then there
is a rational number s such that s is between r and t:
r < s < t.
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Example 14: Illustrating the Density Property
Find a rational number halfway between ½ and ¾.
Solution: First add ½ and ¾. 1 3 1 2 3 2 3
5
      
2 4 2 2 4 4 4 4
Next, divide this sum by 2.
5
5 2 5 1 5 1 5
2    

4
4 1 4 2 42 8
The number 5 is halfway between ½ and ¾. Thus,
8
1 5 3
2

 .
8 4
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Problem Solving with Rational Numbers
• A common application of rational numbers involves
preparing food for a different number of servings than
what the recipe gives.
• The amount of each ingredient can be found as
follows:
amount of ingredient needed 
desired serving size
 ingredient amount
recipe serving size
in the recipe
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Example 15: Changing the Size of a Recipe
A chocolate-chip recipe for five dozen cookies requires
¾ cup of sugar. If you want to make eight dozen
cookies, how much sugar is needed?
Solution:
desired serving size
Amt. sugar needed 
 ingredient amount in the recipe
recipe serving size
8 dozen 3

 cup
5 dozen 4
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Example 15: Changing the Size of a Recipe (continued)
The amount of sugar, in cups, needed is determined by
multiplying the rational numbers:
8 3 8  3 24 6
1
 

 1
5 4 5  4 20 5
5
Thus, 1 1 cups of sugar is needed.
5
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Proportion
x
60”
240”
960”
Find the height x of the tree, when a 60”-man casts a
shadow 240” long.
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