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 .

• Distributive Property :

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 -Positive- Denominator 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.

• Cross-Multiplication 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