Book: RS Aggarwal & V Aggarwal - Mathematics
Chapter: 1. Real Number
Subject: Mathematics - Class 9th
Q. No. 7 of Exercise 1D
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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)
