Mention the closure property, associative law, commutative law, existence of identity, existence of inverse of each real number for each of the operations

(i) addition (ii) multiplication on real numbers.

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)

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