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The Rational Numbers
Properties of Rational Number Multiplication
Let and be any rational numbers .
• Closure: is a rational number.
• Commutative:
• Associative:
• Identity: where 1 can be represented as m/m, for m ≠ 0
• Multiplicative Inverse :
,
where the multiplicative inverse is called the
reciprocal .
o Over Addition :
o Over Subtraction :
Division Let and be any rational numbers. Then 
There are three methods used to perform rational number
division , as the following
theorem illustrates.
Theorem Let and be any rational numbers where is nonzero. Then the following are equivalent .

Ordering Rational Numbers
There are four equivalent methods to order rational numbers.
• Number line Approach :
if and only if a/b occurs to the left of c/b on the
number line.
• Common PositiveDenominator Approach: Let b > 0 . Then
if and only if
a < c .
• Addition Approach :
if and only if there is a positive rational number p/q
such that
Alternatively,
if and only if
is positive.
• CrossMultiplication Approach: Let b > 0 and d > 0 .
Then
if and only if
ad < bc .
Properties of Ordering Rational Numbers
• Transitive: If
and
,
then
• Addition: If then
• Multiplication:
o By a positive number
then
o By a negative number then
• Density Property: If
then there exists a rational number
such that
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