English | Español

Try our Free Online Math Solver!

Online Math Solver












Please use this form if you would like
to have this math solver on your website,
free of charge.

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

Distributive Property :

o Over Addition :

o Over Subtraction :

Let and be any rational numbers. Then

There are three methods used to perform rational number division , as the following
theorem illustrates.

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

Prev Next