1.3. THE RATIONAL NUMBERS
1.3
31
The Rational Numbers
We begin with the definition of a rational number.
Rational Numbers. Any number that can be expressed in the form p/q,
where p and q are integers, q 6= 0, is called a rational number. The letter Q is
used to represent the set of rational numbers. That is:
p
: p and q are integers, q 6= 0
Q=
q
Because −2/3, 4/5, and 123/(−12) have the form p/q, where p and q are
integers, each is an example of a rational number. If you think you hear
the word “fraction” when we say “rational number,” you are correct in your
thinking. Any number that can be expressed as a fraction, where the numerator
and denominator are integers, is a rational number.
Every integer is also a rational number. Take, for example, the integer
−12. There are a number of ways we can express −12 as a fraction with
integer numerator and denominator, −12/1, 24/(−2), and −36/3 being a few.
Reducing Fractions to Lowest Terms
First, we define what is meant by the greatest common divisor of two integers.
The Greatest Common Divisor. Given two integers a and b, the greatest
common divisor of a and b is the largest integer that divides evenly (with no
remainder) into both a and b. The notation GCD(a, b) is used to represent the
greatest common divisor of a and b.
For example, GCD(12, 18) = 6, GCD(32, 40) = 8, and GCD(18, 27) = 9.
We can now state when a fraction is reduced to lowest terms.
Lowest Terms. A fraction a/b is said to be reduced to lowest terms if and
only if GCD(a, b) = 1.
A common technique used to reduce a fraction to lowest terms is to divide both
numerator and denominator by their greatest common divisor.
You Try It!
EXAMPLE 1. Reduce 8/12 to lowest terms.
Reduce: −48/60
32
CHAPTER 1. THE ARITHMETIC OF NUMBERS
Solution: Note that GCD(8, 12) = 4. Divide both numerator and denominator by 4.
8
8÷4
=
12
12 ÷ 4
Divide numerator and denominator
by GCD(8, 12) = 4.
2
=
3
Answer: −4/5
Simplify numerator and denominator.
Thus, 8/12 = 2/3.
Recall the definition of a prime number.
Prime Number. A natural number greater than one is prime if and only if
its only divisors are one and itself.
For example, 7 is prime (its only divisors are 1 and 7), but 14 is not (its divisors
are 1, 2, 7, and 14). In like fashion, 2, 3, and 5 are prime, but 6, 15, and 21
are not prime.
You Try It!
Reduce 18/24 to lowest
terms.
EXAMPLE 2. Reduce 10/40 to lowest terms.
Solution: Note that GCD(10, 40) = 10. Divide numerator and denominator
by 10.
10
10 ÷ 10
=
40
40 ÷ 10
=
Divide numerator and denominator
by GCD(10, 40) = 10.
1
4
Simplify numerator and denominator.
Alternate solution: Use factor trees to express both numerator and denominator as a product of prime factors.
10
2
40
5
10
4
2
2
2
5
Hence, 10 = 2 · 5 and 40 = 2 · 2 · 2 · 5. Now, to reduce 10/40 to lowest terms,
replace the numerator and denominator with their prime factorizations, then
cancel factors that are in common to both numerator and denominator.
1.3. THE RATIONAL NUMBERS
10
2·5
=
40
2·2·2·5
2·
5
=
2
·
2
·
2·5
1
=
4
33
Prime factor numerator and denominator.
Cancel common factors.
Simplify numerator and denominator.
When we cancel a 2 from both the numerator and denominator, we’re actually
dividing both numerator and denominator by 2. A similar statement can be
made about canceling the 5. Canceling both 2 and a 5 is equivalent to dividing
both numerator and denominator by 10. This explains the 1 in the numerator
when all factors cancel.
Answer: 3/4
Example 2 demonstrates an important point.
When all factors cancel. When all of the factors cancel in either numerator
or denominator, the resulting numerator or denominator is equal to one.
Multiplying Fractions
First, the definition.
Multiplication of Fractions. If a/b and c/d are two fractions, then their
product is defined as follows:
a c
ac
· =
b d
bd
Thus, to find the product of a/b and c/d, simply multiply numerators and
multiply denominators. For example:
1 3
3
2 7
14
5
1
5
· =
and
− · =−
and
− · −
=
2 4
8
5 3
15
8
6
48
Like integer multiplication, like signs yield a positive answer, unlike signs yield
a negative answer.
Of course, when necessary, remember to reduce your answer to lowest terms.
EXAMPLE 3. Simplify: −
14 10
·
20 21
You Try It!
27
8
Simplify: − · −
9
20
34
CHAPTER 1. THE ARITHMETIC OF NUMBERS
Solution: Multiply numerators and denominators, then reduce to lowest terms.
−
14 10
140
·
=−
20 21
420
Multiply numerators
and denominators.
2·2·5·7
2·2·3·5·7
2·
2·
5·
7
=−
2
·
2
·
3
·
5
·7
1
=−
3
=−
Answer: 6/5
Prime factor.
Cancel common factors.
Simplify.
Note that when all the factors cancel from the numerator, you are left with a
1. Thus, (−14/20) · (10/21) = −1/3.
Cancellation Rule. When multiplying fractions, cancel common factors according to the following rule: “Cancel a factor in a numerator for an identical
factor in a denominator.”
The rule is “cancel something on the top for something on the bottom.”
Thus, an alternate approach to multiplying fractions is to factor numerators
and denominators in place, then cancel a factor in a numerator for an identical
factor in a denominator.
You Try It!
Simplify:
35
6
− · −
45
14
EXAMPLE 4. Simplify:
15
14
· −
8
9
Solution: Factor numerators and denominators in place, then cancel common
factors in the numerators for common factors in the denominators.
14
2·7
3·5
15
· −
· −
=
Factor numerators
8
9
2·2·2
3·3
and denominators.
3·5
2·7
· −
=
Cancel a factor in a
2·2·2
3·3
numerator for a common.
factor in a denominator.
35
=−
Multiply numerators and.
12
denominators.
Answer: 1/3
Note that unlike signs yield a negative product. Thus, (15/8) · (−14/9) =
−35/12.
35
1.3. THE RATIONAL NUMBERS
Dividing Fractions
Every nonzero rational number has was it called a multiplicative inverse or
reciprocal.
The Reciprocal. If a is any nonzero rational number, then 1/a is called the
multiplicative inverse or reciprocal of a, and:
a·
1
=1
a
Note that:
2·
1
=1
2
and
3 5
· =1
5 3
and
−
4
7
· −
= 1.
7
4
Thus, the reciprocal of 2 is 1/2, the reciprocal of 3/5 is 5/3, and the reciprocal
of −4/7 is −7/4. Note that to find the reciprocal of a number, simply invert
the number (flip it upside down).
Now we can define the quotient of two fractions.
Division of Fractions. If a/b and c/d are two fractions, then their quotient
is defined as follows:
c
a d
a
÷ = ·
b
d
b c
That is, dividing by c/d is the same as multiplying by the reciprocal d/c.
The above definition of division is summarized by the phrase “invert and multiply.”
You Try It!
35
10
4 27
EXAMPLE 5. Simplify: − ÷ −
Simplify: − ÷
21
12
9 81
Solution: Invert and multiply, then factor in place and cancel common factors
in a numerator for common factors in a denominator.
35
10
12
35
− ÷ −
=− · −
Invert and multiply.
21
12
21
10
5·7
2·2·3
=−
· −
Prime factor.
3·7
2·5
2
5
·2·3
·7
· −
Cancel common factors.
=−
3·
7
2·
5
2
=
Multiply numerators and denominators.
1
=2
Simplify.
36
Answer: −4/3
CHAPTER 1. THE ARITHMETIC OF NUMBERS
Note that when all the factors in a denominator cancel, a 1 remains. Thus,
(−35/21) ÷ (−10/12) = 2. Note also that like signs yield a positive result.
Adding Fractions
First the definition.
Addition of Fractions. If two fractions have a denominator in common,
add the numerators and place the result over the common denominator. In
symbols:
a b
a+b
+ =
c
c
c
For example:
3 7
4
− + =
5 5
5
and
4
7
11
− + −
=−
3
3
3
and
4
5
1
+ −
=−
7
7
7
If the fractions do not posses a common denominator, first create equivalent
fractions with a least common denominator, then add according to the rule
above.
Least Common Denominator. If the fractions a/b and c/d do not share
a common denominator, the least common denominator for b and d, written
LCD(b, d), is defined as the smallest number divisible by both b and d.
You Try It!
Simplify:
5 1
− +
6 9
Answer: −13/18
3
5
EXAMPLE 6. Simplify: − +
8 12
Solution: The least common denominator in this case is the smallest number
divisible by both 8 and 12. In this case, LCD(8, 12) = 24. We first need to
make equivalent fractions with a common denominator of 24.
3
5
3 3
5 2
− +
=− · +
·
Make equivalent fraction with
8 12
8 3 12 2
a common denominator of 24.
9
10
=− +
Multiply numerators and denominators.
24 24
1
=
Add: −9 + 10 = 1.
24
Note how we add the numerators in the last step, placing the result over the
common denominator. Thus, −3/8 + 5/12 = 1/24.
37
1.3. THE RATIONAL NUMBERS
Order of Operations
Rational numbers obey the same Rules Guiding Order of Operations as do the
integers.
Rules Guiding Order of Operations. When evaluating expressions, proceed in the following order.
1. Evaluate expressions contained in grouping symbols first. If grouping
symbols are nested, evaluate the expression in the innermost pair of
grouping symbols first.
2. Evaluate all exponents that appear in the expression.
3. Perform all multiplications and divisions in the order that they appear
in the expression, moving left to right.
4. Perform all additions and subtractions in the order that they appear in
the expression, moving left to right.
You Try It!
EXAMPLE 7. Given x = 2/3, y = −3/5, and z = 10/9, evaluate xy + yz.
Solution: Following Tips for Evaluating Algebraic Expressions, first replace
all occurrences of variables in the expression xy + yz with open parentheses.
Next, substitute the given values of variables (2/3 for x, −3/5 for y, and 10/9
for z) in the open parentheses.
xy + yz =
=
+
Replace variables with parentheses.
2
10
3
3
−
+ −
Substitute: 2/3 for x, −3/5
3
5
5
9
for y, and 10/9 for z.
Given a = −1/2, b = 2/3,
and c = −3/4, evaluate the
expression a + bc and
simplify the result.
38
CHAPTER 1. THE ARITHMETIC OF NUMBERS
Use the Rules Guiding Order of Operations to simplify.
6
30
+ −
15
45
2
2
=− + −
5
3
2 5
2 3
=− · + − ·
5 3
3 5
6
10
=− + −
15
15
16
=−
15
=−
Answer: −1
Multiply.
Reduce.
Make equivalent fractions with a
least common denominator.
Add.
Thus, if x = 2/3, y = −3/5, and z = 10/9, then xy + yz = −16/15
You Try It!
Simplify:
(−1/3)4
EXAMPLE 8. Given x = −3/5, evaluate −x3 .
Solution: First, replace each occurrence of the variable x with open parentheses, then substitute −3/5 for x.
3
−x3 = − ( )
3
3
=− −
5
3
3
3
=− −
−
−
5
5
5
27
=− −
125
=
Answer: 1/81
27
125
Replace x with open parentheses.
Substitute −3/5 for x.
Write −3/5 as a factor
three times.
The product of three negative
fractions is negative. Multiply
numerators and denominators.
The opposite of −27/125 is 27/125.
Hence, −x3 = 27/125, given x = −3/5.
You Try It!
Given x = −3/4 and
y = −4/5, evaluate x2 − y 2 .
EXAMPLE 9. Given a = −4/3 and b = −3/2, evaluate a2 + 2ab − 3b2 .
Solution: Following Tips for Evaluating Algebraic Expressions, first replace all
occurrences of variables in the expression a2 + 2ab − 3b2 with open parentheses.
39
1.3. THE RATIONAL NUMBERS
Next, substitute the given values of variables (−4/3 for a and −3/2 for b) in
the open parentheses.
2
2
a + 2ab − 3b =
=
2
+2
−3
2
2
2
3
3
4
4
−
+2 −
−
−3 −
3
3
2
2
Next, evaluate the exponents: (−4/3)2 = 16/9 and (−3/2)2 = 9/4.
16 2
+
=
9
1
4
3
3 9
−
−
−
3
2
1 4
Next, perform the multiplications and reduce.
16 24 27
+
−
9
6
4
16
27
=
+4−
9
4
=
Make equivalent fractions with a common denominator, then add.
16 4
36 27 9
−
· +4·
·
9 4
36
4 9
64 144 243
=
+
−
36
36
36
35
=−
36
=
Thus, if a = −4/3 and b = −3/2, then a2 + 2ab − 3b2 = −35/36
Answer: −31/400
Fractions on the Graphing Calculator
We must always remember that the graphing calculator is an “approximating
machine.” In a small number of situations, it is capable of giving an exact
answer, but for most calculations, the best we can hope for is an approximate
answer.
However, the calculator gives accurate results for operations involving fractions, as long as we don’t use fractions with denominators that are too large
for the calculator to respond with an exact answer.
40
CHAPTER 1. THE ARITHMETIC OF NUMBERS
You Try It!
Simplify using the graphing
calculator:
4 8
− +
5 3
EXAMPLE 10. Use the graphing calculator to simplify each of the following
expressions:
(a)
2 1
+
3 2
(b)
2 5
·
3 7
(c)
3 1
÷
5 3
Solution: We enter each expression in turn.
a) The Rules Guiding Order of Operations tell us that we must perform divisions before additions. Thus, the expression 2/3 + 1/2 is equivalent to:
2 1
+
3 2
4 3
= +
6 6
7
=
6
2/3 + 1/2 =
Divide first.
Equivalent fractions with LCD.
Add.
Enter the expression 2/3+1/2 on your calculator, then press the ENTER
key. The result is shown in the first image in Figure 1.12. Next, press the
MATH button, then select 1:IFrac (see the second image in Figure 1.12)
and press the ENTER key again. Note that the result shown in the third
image in Figure 1.12 matches the correct answer of 7/6 found above.
Figure 1.12: Calculating 2/3 + 1/2.
b) The Rules Guiding Order of Operations tell us that there is no preference
for division over multiplication, or vice-versa. We must perform divisions
and multiplications as they occur, moving from left to right. Hence:
2
× 5/7
3
10
=
/7
3
10 1
=
×
3
7
10
=
21
2/3 × 5/7 =
Divide: 2/3 =
Multiply:
2
3
2
10
×5=
3
3
Invert and multiply.
Multiply:
10 1
10
× =
3
7
21
41
1.3. THE RATIONAL NUMBERS
This is precisely the same result we get when we perform the following
calculation.
2 5
10
× =
Multiply numerators and denominators.
3 7
21
Hence:
2 5
2/3 × 5/7
is equivalent to
×
3 7
Enter the expression 2/3×5/7 on your calculator, then press the ENTER
key. The result is shown in the first image in Figure 1.13. Next, press the
MATH button, then select 1:IFrac (see the second image in Figure 1.13)
and press the ENTER key again. Note that the result shown in the third
image in Figure 1.13 matches the correct answer of 10/21 found above.
Figure 1.13: Calculating 2/3 × 1/2.
c) This example demonstrates that we need a constant reminder of the Rules
Guiding Order of Operations. We know we need to invert and multiply in
this situation.
3 1
3 3
÷ = ×
Invert and multiply.
5 3
5 1
9
=
Multiply numerators and denominators.
5
So, the correct answer is 9/5.
Enter the expression 3/5/1/3 on your calculator, then press the ENTER
key. Select 1:IFrac from the MATH menu and press the ENTER key
again. Note that the result in the first image in Figure 1.14 does not
match the correct answer of 9/5 found above. What have we done wrong?
If we follow the Rules Guiding Order of Operations exactly, then:
3
/1/3
5
3
= /3
5
3 1
= ×
5 3
1
=
5
3/5/1/3 =
3
5
3
3
Divide: /1 =
5
5
Divide: 3/5 =
Invert and multiply.
Multiply:
3 1
1
× =
5 3
5
42
CHAPTER 1. THE ARITHMETIC OF NUMBERS
This explains the answer found in the first image in Figure 1.14. However,
it also show that:
3/5/1/3
is not equivalent to
3 1
÷
5 3
We can cure the problem by using grouping symbols.
3 1
/
5 3
3 1
= ÷
5 3
(3/5)/(1/3) =
Parentheses first.
/ is equivalent to ÷.
Hence:
3 1
÷
5 3
Enter the expression (3/5)/(1/3) on your calculator, then press the ENTER key. Select 1:IFrac from the MATH menu and press the ENTER
key again. Note that the result in the second image in Figure 1.14 matches
the correct answer of 9/5.
(3/5)/(1/3)
is equivalent to
Figure 1.14: Calculating (3/5)/(1/3).
Answer: 28/15
43
1.3. THE RATIONAL NUMBERS
❧ ❧ ❧
Exercises
❧ ❧ ❧
In Exercises 1-6, reduce the given fraction to lowest terms by dividing numerator and denominator by
the their greatest common divisor.
20
50
36
2.
38
10
3.
48
1.
36
14
24
5.
45
21
6.
36
4.
In Exercises 7-12, reduce the given fraction to lowest terms by prime factoring both numerator and
denominator and cenceling common factors.
153
170
198
8.
144
188
9.
141
7.
171
144
159
11.
106
140
12.
133
10.
In Exercises 13-18, for each of the following problems, multiply numerators and denominators, then
prime factor and cancel to reduce your answer to lowest terms.
18
14
20
3
13.
· −
16. − · −
8
13
2
6
2
18
16 19
14.
· −
17. − ·
16
5
8 6
19
18
14 7
15. − · −
18. − ·
4
13
4 17
In Exercises 19-24, for each of the following problems, first prime factor all numerators and denominators, then cancel. After canceling, multiply numerators and denominators.
5
12
21
36
19. − · −
20. − · −
6
49
17
46
44
21. −
21 12
·
10 55
22. −
49 52
·
13 51
CHAPTER 1. THE ARITHMETIC OF NUMBERS
55
54
· −
29
11
55
7
· −
24.
13
49
23.
In Exercises 25-30, divide. Be sure your answer is reduced to lowest terms.
27 45
50
5
28. − ÷
÷ −
25.
28 23
39
58
13
7
31
4
÷
−
29.
−
26.
÷ −
10
28
25
5
4
48
60 34
30. − ÷ −
27. − ÷
13
35
17 31
In Exercises 31-38, add or subtract the fractions, as indicated, and simplify your result.
5 1
2
1
31. − +
35. − − −
6 4
4
9
1 5
1
1
32. − +
36.
−
−
−
7 8
2
8
8
1
8 4
33. − + −
37. − −
9
3
9 5
1
1
4 1
34. − + −
38. − −
3
2
7 3
In Exercises 39-52, simplify the expression.
8 5 2 39. − − 9
2 5
8 7 1 40. − − 5
6 2
2 5
7
1
+ −
−
41. −
6
2
3
2 3
1
5
42.
+ −
2
2
8
9
8
9
1
43. −
−
+
−
5
7
5
2
44.
45.
46.
47.
48.
2
5
6
1
−
+
−
−
3
7
3
7
9
5 7
− +
−
8 2
2
1
3 9
+
−
2 2
4
2
7
9
2
−
− −
5
2
5
2
3
2
1
−
4
3
4
45
1.3. THE RATIONAL NUMBERS
4
−
9
3 5
1
50. −
−
2 6
3
49.
8
9
2
4
−
−
−
3
7
7
8
3
1
5
1
52. −
−
−
2
3
8
8
6 2
−
5 5
51.
In Exercises 53-70, evaluate the expression at the given values.
53. xy − z 2 at x = −1/2, y = −1/3, and
z = 5/2
62. ab+bc at a = −8/5, b = 7/2, and c = −9/7
63. x3 at x = −1/2
2
54. xy −z at x = −1/3, y = 5/6, and z = 1/3
2
64. x2 at x = −3/2
2
55. −5x + 2y at x = 3/4 and y = −1/2.
56. −2x2 + 4y 2 at x = 4/3 and y = −3/2.
57. 2x2 − 2xy − 3y 2 at x = 3/2 and y = −3/4.
2
65. x−yz at x = −8/5, y = 1/3, and z = −8/5
66. x − yz at x = 2/3, y = 2/9, and z = −3/5
67. −x2 at x = −8/3
2
58. 5x − 4xy − 3y at x = 1/5 and y = −4/3.
59. x + yz at x = −1/3, y = 1/6, and z = 2/5.
68. −x4 at x = −9/7
60. x + yz at x = 1/2, y = 7/4, and z = 2/3.
69. x2 + yz at x = 7/2, y = −5/4, and
z = −5/3
61. ab+bc at a = −4/7, b = 7/5, and c = −5/2
70. x2 +yz at x = 1/2, y = 7/8, and z = −5/9
71. a + b/c + d is equivalent to which of the
following mathematical expressions?
73. a + b/(c + d) is equivalent to which of the
following mathematical expressions?
b
+d
c
a+b
(c)
+d
c
(a) a +
(b)
a+b
c+d
(d) a +
b
c+d
72. (a + b)/c + d is equivalent to which of the
following mathematical expressions?
b
+d
c
a+b
(c)
+d
c
(a) a +
(b)
a+b
c+d
(d) a +
b
c+d
75. Use the graphing calculator to reduce
4125/1155 to lowest terms.
b
+d
c
a+b
(c)
+d
c
(a) a +
(b)
a+b
c+d
(d) a +
b
c+d
74. (a + b)/(c + d) is equivalent to which of
the following mathematical expressions?
b
+d
c
a+b
(c)
+d
c
(a) a +
(b)
a+b
c+d
(d) a +
b
c+d
76. Use the graphing calculator to reduce
2100/945 to lowest terms.
46
CHAPTER 1. THE ARITHMETIC OF NUMBERS
77. Use the graphing calculator to simplify
45 70
· .
84 33
78. Use the graphing calculator to simplify
34 13
+ .
55 77
❧ ❧ ❧
79. Use the graphing calculator to simplify
35
28
− ÷ −
.
33
44
80. Use the graphing calculator to simplify
11
11
.
− − −
84
36
Answers
❧ ❧ ❧
1.
2
5
25. −
580
39
3.
5
24
27. −
930
289
5.
8
15
29.
7.
9
10
31. −
7
12
9.
4
3
33. −
11
9
3
11.
2
35. −
1
36
45
13. −
13
37. −
76
45
39. −
109
90
171
15.
26
19
17. −
3
19.
10
49
21. −
23. −
126
275
270
29
98
65
41.
79
36
43.
53
35
45. −
131
8
47. −
323
50
49.
62
45