Do you remember the properties of addition and subtraction on rational numbers? They are closure, commutative, associative, identity and existence of inverse properties. In this section, you will see if multiplication of rational numbers is valid on these properties or not. In addition you will also see one additional property which is called the distributive property.

1. Closure property

The product of any two rational numbers is again a rational number. That is, for any two rational numbers

is a rational number. So we say, the set of rational numbers is closed under multiplication.

For example,

are rational numbers. Their product

is also a rational number.

2. Commutative property

For any two rational numbers

it is always true that

Multiplication is commutative on the set of rational numbers

For example,

are rational numbers. Their product

Changing the order of the factors does not change the product.

3. Associative property

For any three rational numbers

it is always true that

[you can group the factors in any order and multiply them.]

Multiplication is associative on the set of rational numbers.

For example,

4. Multiplicative Identity property

For any rational number

When we multiply any rational number by one, the product is always the number itself. So we say, 1 is the multiplicative identity element in ℚ.

5. Multiplicative inverse property

For any non-zero rational number

there is always another rational number

so that

is called the additive inverse of