Closure property of addition of rational numbers:

The sum of two rational numbers is always a rational number.

If and are any two rational numbers, then is also a rational number.

Example:

Consider the rational numbers and Then,

, is a rational number

Commutative property of addition of rational numbers:

Two rational numbers can be added in any order.

Thus for any two rational numbers and , we have

=

Example:

=

=

=

Existence of additive identity property of addition of rational numbers:

0 is a rational number such that the sum of any rational number and 0 is the rational number itself.

Thus,

= for every rational number

0 is called the additive identity for rationals.

Example:

=

Existence of additive inverse property of addition of rational numbers:

For every rational number , there exists a rational number

such that = = = 0 and similarly, = 0.

Thus,

= is called the additive inverse of

Example:

= and similarly,

Thus, and are additive inverses of each other.

Associative property of addition of rational numbers:

While adding three rational numbers, they can be grouped in any order.

Thus, for any three rational numbers and , we have

= + =

Example:

Consider three rational numbers, Then,

=

=

=

=

Closure property of multiplication of rational numbers:

The product of two rational numbers is always a rational number.

If and are any two rational numbers then is also a rational number.

Example:

Consider the rational numbers and . Then,

Commutative property of multiplication of rational numbers:

Two rational numbers can be multiplied in any order.

Thus, for any rational numbers and , we have:

= =

Example:

Let us consider the rational numbers and Then,

= and

Therefore,

Associative property of multiplication of rational numbers:

While multiplying three or more rational numbers, they can be grouped in any order.

Thus, for any rationals we have:

=

Example:

Consider the rationals , we have

=

=

=

Existence of multiplicative identity property:

For any rational number , we have

1 is called the multiplicative identity for rationals.

Example:

Consider the rational number . Then, we have

=

Existence of multiplicative inverse property:

Every nonzero rational number has its multiplicative inverse .

Thus,

= is called the reciprocal of .

Clearly, zero has no reciprocal.

Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1)