Thinking
Mathematically
Number Theory and the Real Number
System
5.3 The Rational Numbers
The Rational Numbers
The set of rational numbers is the set of all numbers
which can be expressed in the form a/b, where a and b
are integers and b is not equal to 0. The integer a is
called the numerator and the integer b is called the
denominator.
Note that every integer is a rational number. For
example 17 can be written 17/1.
Where do rationals fit on the number line?
The Fundamental Principle of
Rational Numbers
a
If b is a rational number and c is any number other
than 0, a  c  a
b c
a
ac
The rational numbers
and
are called
b

c
b
b
equivalent fractions
Fraction Basics
• The fraction denominator (bottom) is the size of
the piece.
• The fraction numerator (top) is the number of
those pieces.
• A mixed number combines a whole number with a
fraction.
• In a proper fraction, the numerator is smaller than
the denominator. Not so in an improper fraction.
Lowest Terms
Lowest Terms – GCD of numerator and denominator is 1
A rational number can be reduced to its lowest
terms by dividing the numerator and denominator
by the greatest common divisor.
Exercise Set 5.3 #7
Reduce to lowest terms
60
?
108
Converting a 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 in
the mixed number.
Exercise Set 5.3 #15
Convert to an improper fraction
3
7
5
Converting a 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:
remainder
quotient
original denominator
Exercise Set 5.3 #21
Convert to a mixed number
76

9
Rational Numbers and Decimals
Any rational number can be expressed as a
decimal.
Meaning of decimals?
The resulting decimal will either terminate, or it
will have a digit that repeats or a block of digits
that repeat.
Exercise Set 5.3 #27, #31
Express as a decimal
7
20
9
11
Expressing a Repeating Decimal as a
Quotient of Integers
Step 1 Let n equal the repeating decimal.
Step 2 Multiply both sides of the equation in step 1
by 10 if one digit repeats, by 100 if two digits
repeat, by 1000 if three digits repeat, and so on.
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
an appropriate number and solve for n.
Example: Decimal to Fraction
Exercise Set 5.3 #39, #53
Express as a quotient of integers
0.4
0.36
Multiplying Rational Numbers
The product of two rational numbers is the
product of their numerators divided by the
product of their denominators.
Cancel/reduce to achieve a result in lowest terms
Exercise Set 5.3 #57, #61
3 7
 ?
8 11
 2  9 
      ?
 3  4 
Dividing Rational Numbers
The quotient of two rational numbers is the product
of the first number and the reciprocal of the
second number.
If a/b and c/d are rational numbers, and c/d is not 0,
then
a c a d a d
   
b d b c
b c
How else can a fraction divided by a fraction be written?
Example: Dividing Rational
Numbers
Exercise Set 5.3 #67
7 15
  ?
8 16
Adding and Subtracting Rational
Numbers with Identical
Denominators
The sum (difference) of two rational numbers with
identical denominators is the sum (difference) of
their numerators over the common denominator.
If a, b, and c are integers (b not 0) numbers, then
a c a c
a c a c
 
and  
b b
b
b b
b
Example: Subtracting Rational
Numbers (same denominator)
Exercise Set 5.3 #73
5 1
 ?
6 6
Adding and Subtracting Rational
Numbers – Differing Denominators
• The least common multiple of the two
denominators is called the least common
denominator (LCD)
• Replace each fraction with an eqivalent
fraction in which the denominator is the
LCD
Example: Adding and Subtracting
Rational Numbers – Differing
Denominators
Exercise Set 5.3 #79, #83
3 3

?
4 20
13 2
 ?
18 9
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.
Between any two rational numbers, there is
another rational number.
Between any two rational numbers, there is an
infinite number of rational numbers.
Thinking
Mathematically
Number Theory and the Real Number
System
5.3 The Rational Numbers