Section 2.3:
Rational Numbers
Definition: A rational number is a
number that may be written in the
a
form
where a and b are integers
b
and b is nonzero.
The integer a is called the numerator, and the integer b is called the
denominator.
1
Examples of rational numbers:
8
121
37
−4
−6
13
−5
−10
2
Equivalent Fractions
c
a
and
are
Two rational numbers
b
d
equivalent if and only if
ad = bc.
3
Example:
Determine whether or
17
−119
and
are
not the fractions
91
−13
equivalent.
Solution: Cross multiplying gives:
(−119)(−13) = (91)(17)
1, 547 = 1, 547
Also, notice that
119
7 × 17
=
91
7 × 13
4
Simplest form of a fraction
To find the simplest form (or reduced form) of a fraction, first find
the greatest common factor of the
numerator and denominator. Then
divide both the numerator and the
denominator by this gcf.
That is, find the prime factorization
of the numerator and denominator
and cancel all common factors.
5
Example: Write the following frac168
tion in simplest form:
96
Solution:
168
2·2·2·3·7
=
96
2·2·2·2·2·3
6
Popper 1: Write the following fraction in
18
simplest form:
132
(A)
3
22
3
(B)
11
6
(C)
33
9
(D)
66
7
Proper/Improper Fractions
a
A fraction of the form
is called
b
• a proper fraction if |a| < |b|
• an improper fraction if |a| > |b|
8
Examples
17
• −
is a proper fraction
26
59
•
is an improper fraction
26
9
Mixed Numbers
An improper fraction can be expressed
as a mixed number which consists
of an integer followed by a proper
fraction.
4, 83
Examples: 5 1
,
−3
3
7
5
10
Converting from Improper Fractions to Mixed Numbers
Examples:
16
(a)
3
−25
(b)
7
−120
(c)
−18
11
Popper 2:
Write the following improper
23
fraction in mixed form:
−5
2
(A) −5
5
3
(B) −4
5
3
(C) −5
4
2
(D) −4
3
12
Converting from Mixed Numbers
to Improper Fractions
Examples:
7
(a) 3
9
4
(b) −5
13
5
(c) −1
8
13
Popper 3: Write the following mixed num3
ber as an improper fraction: −6
7
(A) −
9
7
39
(B) −
7
18
(C) −
7
45
(D) −
7
14
Multiplying and Dividing Fractions
The rules for multiplying and dividing fractions are as follows:
ac
a c
•
· =
b d
bd
a c
ad
•
÷ =
b d
bc
15
Multiplying and Dividing Fractions
Examples:
4 15
1. − ·
3 8
4
2. (−2) ÷ −
9
!
16
Popper 4: Express your
answer
as a frac
10
5
÷
tion in simplest form: −
24
36
1
(A) −
2
2
(B) −
3
3
(C) −
4
4
(D) −
5
17
Adding and Subtracting Fractions
We have the following rules for adding
and subtracting fractions with the
same denominator
a
b
a+b
•
+ =
n
n
n
a
b
a−b
•
− =
n n
n
18
We have the following rules for adding
and subtracting fractions with different denominators
1. Find the lowest common multiple (LCM) of the denominators.
2. Express each fraction as an equivalent fraction with the LCM as
the denominator.
3. Use the addition/subtraction rules
for common denominators
19
Adding and Subtracting Fractions
Examples:
7
5
+
1.
12
30
13 11
2.
−
28 42
20
Popper 5: Express your answer as a frac8
4
tion in simplest form:
−
25 15
(A)
9
150
2
(B)
5
4
(C)
75
3
(D)
25
21
Adding and Subtracting Mixed Numbers
1
3
Example: −3 + 2
3
4
22
Popper 6: Express your answer as a mixed
2
3
number: −4 + 5
5
7
(A) 1
1
35
1
(B) −1
35
34
(C)
35
34
(D) −
35
23
Decimal Notation
In base 10, the place values after
1, 1 , 1 ,
the decimal point are 10
102 103
and so on.
For example,
643.875 = 6 × 100 + 4 × 10 + 3 × 1
1
1
1
+8×
+7×
+5×
10
100
1000
24
Non-repeating decimals
Examples:
6
0.6 =
10
62
0.62 =
100
628
0.628 =
1000
6287
0.6287 =
10000
Note that the number of digits after
the decimal point matches the number of zeros in the denominator.
25
Repeating decimals
Examples:
6
0.6666666666 · · · = 0.6 =
9
62
0.6262626262 · · · = 0.62 =
99
628
0.6286286286 · · · = 0.628 =
999
6287
0.6287628762 · · · = 0.6287 =
9999
Note that the number of repeated
digits matches the number of nines
in the denominator.
26
Repeating decimals
Example:
0.0055555555 · · · = 0.005
1
· (0.5)
=
100
1 5
=
·
100 9
5
=
900
27
Repeating decimals
Example:
0.7222222222 · · · = 0.7 + 0.02
7
1
=
+
· (0.2)
10
10
7
1 2
=
+
·
10
10 9
7
2
=
+
10
90
13
=
18
28
Repeating decimals
Example:
0.03415151515 · · · = 0.034 + 0.00015
34
1
=
+
· (0.15)
1000
1000
34
1
15
=
+
·
1000
1000 99
34
15
=
+
1000
99000
3381
=
99000
29
Popper 7:
Write the following repeating
decimal as a fraction in simplest form: 0.24
(A)
24
90
(B)
24
99
(C)
24
100
(D)
11
45
30
Repeating decimals
12
Example: Express the fraction
37
as a repeating decimal.
31
Popper 8: Write the following fraction as
5
a repeating decimal:
11
(A) 0.4
(B) 0.45
(C) 0.45
(D) 0.454
32