### Class 8, Mathematics - NCERT Guide

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__RATIONAL NUMBERS__

__RATIONAL NUMBERS__

*Properties of Closure*

__Whole Numbers__
>> Whole Numbers are closed under
addition. In general,

*a + b*is a Whole Numbers for any two Whole Numbers*a*and*b.*
>> Whole Numbers are not closed under
subtraction. 5 - 7 = -2, this is not a Whole Number.

>> Whole Numbers are closed under multiplication. In
general, if

*a*and*b*are any two Whole Numbers, their product*ab*is Whole Number.
>> Whole Numbers are not closed under division..

__Integers__
>> Integers are closed under addition. In general,

*a + b*is an integer for any two integers*a*and*b*.
>> Integers are closed under subtraction. For any two
integers

*a*and*b*,*a - b*is an integer.
>> Integers are closed under multiplication. For any
two integers

*a*and*b*,*a*x*b*is an integer.
>> Integers e not closed under division.

__Rational Numbers__
>> Sum of two rational numbers is again a rational
number. Rational Numbers are closed under addition.

>> The difference of two rational numbers is a rational
number. Rational numbers are closed under subtraction

>> Rational numbers are closed under multiplication.

>> Rational numbers are not closed under division.

*Properties of Commutativity*

__Whole Numbers__
>> Addition is commutative.

>> Subtraction is not commutative.

>> Multiplication is commutative.

>> Division is commutative.

__Integers__
>> Addition is commutative.

>> Subtraction is not commutative.

>> Multiplication is commutative.

>> Division is commutative.

__Rational Numbers__
>> Addition is commutative.

>> Subtraction is not commutative.

>> Multiplication is commutative.

>> Division is not commutative.

*Properties of Associativity*

__Whole Numbers__
>> Addition is
associative.

>> Subtraction
is not associative.

>>
Multiplication is associative.

>> Division is
not associative.

__Integers__
>> Addition is
associative.

>> Subtraction
is not associative.

>>
Multiplication is associative.

>> Division is
not associative.

__Rational Numbers__
>> Addition is
associative.

>> Subtraction
is not associative.

>>
Multiplication is associative.

>> Division is
not associative.

*Points to Remember*
>> Zero is called the identity for
addition of rational numbers. It is additive identity for integers and whole
numbers.

>> 1 is multiplicative identity for
rational numbers.

>> The additive inverse of the rational
number
is

*a / b**(- a / b)*and vice-versa.
>> The
multiplicative or reciprocal inverse of the rational number

*a / b*is*c / d*if,*a / b*x*c / d = 1*
>> Rational
numbers can be represented on a number line.

>> Between any
two rational numbers there are countless rational numbers.

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