Rational numbers
Class VIII
Solution:
Page number 4
Properties of rational number – closure
Try these
Question 1: Fill in the blanks in the following table.
Numbers Closed under
addition subtraction multiplication division
Rational yes yes ……. No
numbers
Integers ……… Yes …… No
Whole numbers ……… ………. Yes ……….
Natural ……… No …….. ………..
numbers
Solution:
Numbers Closed under
addition subtraction multiplication division
Rational yes yes Yes No
numbers
Integers Yes Yes No No
Whole numbers Yes No Yes No
Natural Yes No No No
numbers
In the table:
i) Rational numbers are closed under addition and subtraction since adding or
subtracting two rational numbers will always result in another rational number.
However, they are not closed under multiplication or division since multiplying or
dividing two rational numbers may result in an irrational number.
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ii) Integers are closed under addition and subtraction since adding or subtracting two
integers will always result in another integer. However, they are not closed under
multiplication or division since multiplying or dividing two integers may result in
a rational number or an irrational number.
iii) Whole numbers are closed under addition and multiplication since adding or
multiplying two whole numbers will always result in another whole number.
However, they are not closed under subtraction or division since subtracting or
dividing two whole numbers may result in a non-whole number.
iv) Natural numbers are closed under addition since adding two natural numbers will
always result in another natural number. However, they are not closed under
subtraction, multiplication, or division since subtracting, multiplying, or dividing
two natural numbers may result in a non-natural number.
Page number 6
Properties of rational number – commutativity
Try these
Question 1:Fill in the blanks in the following table.
Numbers Commutative
under
addition subtraction multiplication division
Rational yes ……. ……. …….
numbers
Integers ……… No …… …….
Whole ……… ………. Yes ……….
numbers
Natural ……… ……. …….. No
numbers
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Solution:
Numbers Commutative
under
addition subtraction multiplication division
Rational yes No Yes No
numbers
Integers Yes No Yes No
Whole Yes No Yes No
numbers
Natural Yes No Yes No
numbers
In the table:
i) Rational numbers are commutative for addition and multiplication since changing
the order of the rational numbers being added or multiplied does not change the
result. However, they are not commutative for subtraction or division since
changing the order of the rational numbers being subtracted or divided can result
in different values.
ii) Integers are commutative for addition and multiplication since changing the order
of the integers being added or multiplied does not change the result. However, they
are not commutative for subtraction or division since changing the order of the
integers being subtracted or divided can result in different values.
iii) Whole numbers are commutative for addition and multiplication since changing
the order of the whole numbers being added or multiplied does not change the
result.
iv) Natural numbers are commutative for addition and multiplication since changing
the order of the natural numbers being added or multiplied does not change the
result. However, they are not commutative for subtraction or division since
changing the order of the natural numbers being subtracted or divided can result in
different values.
Page number 9
Properties of rational number – associativity
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Try these
Question 1: Fill in the blanks in the following table.
Numbers Associative
under
addition subtraction multiplication division
Rational …….. …….. …….. No
numbers
Integers …….. …….. …….. ……..
Whole Yes …….. Yes ……..
numbers
Natural ……. No …….. ……..
numbers
Solution:
Numbers Associative
under
addition subtraction multiplication division
Rational yes No Yes No
numbers
Integers Yes No Yes No
Whole Yes No Yes No
numbers
Natural Yes No Yes No
numbers
In the table:
i) Rational numbers are associative for addition and multiplication since changing
the grouping of rational numbers being added or multiplied does not change the
result. However, they are not associative for subtraction or division since changing
the grouping of rational numbers being subtracted or divided can result in different
values.
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ii) Integers are associative for addition and multiplication since changing the
grouping of integers being added or multiplied does not change the result.
However, they are not associative for subtraction or division since changing the
grouping of integers being subtracted or divided can result in different values.
iii) Whole numbers are associative for addition and multiplication since changing the
grouping of whole numbers being added or multiplied does not change the result.
iv) Natural numbers are associative for addition and multiplication since changing the
grouping of natural numbers being added or multiplied does not change the result.
However, they are not associative for subtraction or division since changing the
grouping of natural numbers being subtracted or divided can result in different
values.
Page number 11
Properties of rational number
Think, discuss and write
Question 1: If a property holds for rational numbers, will it also hold for integers? For
whole numbers? Which will? Which will not?
Solution: If a property holds for rational numbers, it does not necessarily imply that it
will hold for integers or whole numbers.
Rational numbers are a superset of both integers and whole numbers. They include all
integers (positive, negative, and zero) and all whole numbers (positive integers and
zero). However, rational numbers also include numbers that are not integers or whole
numbers, such as fractions and decimals.
Whether a property holds for integers or whole numbers depends on the specific
property in question. Some properties may hold for both integers and rational
numbers, while others may only hold for rational numbers.
For example, the property of closure under addition holds for both rational numbers
and integers. Adding any two rational numbers or two integers will always result in
another rational number or integer, respectively.
On the other hand, the property of closure under division does not hold for integers.
Dividing two integers may result in a rational number that is not an integer. For
example, dividing 4 by 2 gives the rational number 2, which is an integer. However,
dividing 4 by 3 gives the rational number 4/3, which is not an integer.
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Similarly, the property of closure under subtraction does not hold for whole numbers.
Subtracting two whole numbers may result in a rational number that is not a whole
number. For example, subtracting 5 from 3 gives the rational number -2, which is not
a whole number.
In summary, whether a property holds for integers or whole numbers depends on the
specific property in question. Some properties may hold for both, while others may
only hold for rational numbers.
Page number 12
Distributivity of multiplication over addition for rational number
Try these
Question 1: Find using distributivity:
i) ( 75 ) × ( −312 )+( 75 )× ( 125 )
ii) ( ) × ( )+( ) × ( )
9 4 9 −3
16 12 16 9
Solution:
i) To simplify the expression using distributivity, let's perform the multiplication
first:
( 75 × −312 )+( 75 × 125 )
To multiply fractions, we multiply the numerators and denominators:
( 75×−3
×12 ) ( 5× 12 )
+
7×5
Simplifying the numerators and denominators:
( −21
60 ) +( )
35
60
Now, we have fractions with the same denominator, so we can combine them:
−21+35
60
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Calculating the numerator:
-21 + 35 = 14
The simplified expression becomes:
14
60
To further simplify the fraction, we can divide both the numerator and denominator by
their greatest common divisor, which is 2:
14 7
=
60 30
7
Therefore, the simplified expression is .
30
ii) To find the value of the expression using distributivity, we have:
[ ( 169 × 124 )+( 169 × −39 ) ]
First, let's simplify each multiplication separately:
9 4 9×4 36
[ × = = ]
16 12 16 ×12 192
9 −3 9 ×−3 −27
[ × = = ]
16 9 16 ×9 144
Now, let's add the two simplified fractions:
36 −27
[ + ]
192 144
To add these fractions, we need to find a common denominator, which is the least
common multiple (LCM) of 192 and 144. The LCM of 192 and 144 is 576.
Converting the fractions to have a common denominator of 576, we get:
36 −27 36 ×3 −27 × 4 108 −108
[ + = + = + ]
192 144 192 ×3 144 × 4 576 576
Now, we can add the fractions:
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108 −108 108−108 0
[ + = = =0 ]
576 576 576 576
Therefore, the value of the expression is 0.
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