Question

explain by giving example properties of the group are not followed in substraction of rational number​

Answer 1

Answer:

To know the properties of rational numbers, we will consider here the general properties of integers which include associative, commutative and closure properties. Rational numbers are the numbers which can be represented in the form of p/q, where q is not equal to 0. Basically, the rational numbers are the integers which can be represented in the number line. Let us go through all the properties here.

Rational Number

Rational Numbers Class 8 Notes

Rational Numbers Worksheet

Properties of Integers

Properties of rational numbers

Rational means anything which is completely logical whereas irrational means anything which is unpredictable and illogical in nature. The word rational has evolved from the word ratio. In general, rational numbers are those numbers that can be expressed in the form of p/q, in which both p and q are integers and q≠0. We can denote these numbers by Q.

Closure property

For two rational numbers say x and y the results of addition, subtraction and multiplication operations give a rational number. We can say that rational numbers are closed under addition, subtraction and multiplication. For example:

(7/6)+(2/5) = 47/30

(5/6) – (1/3) = 1/2

(2/5). (3/7) = 6/35

Do you know why division is not under closure property?

The division is not under closure property because division by zero is not defined. We can also say that except ‘0’ all numbers are closed under division.

Commutative law

For rational numbers, addition and multiplication are commutative.

Commutative law of addition: a+b = b+a

Commutative law of multiplication: a×b = b×a

For example:

Commutative law example

Subtraction is not commutative property i. e. a-b ≠ b-a. This can be understood clearly with the following example

Commutative law - subtraction LHS

whereas

Commutative law - subtraction RHS

The division is also not commutative i.e. a/b ≠ b/a as,

Commutative law - Division LHS

whereas,

Commutative law - Division RHS

Associative law

Rational numbers follow the associative property for addition and multiplication.

Suppose x, y and z are rational then for addition: x+(y+z)=(x+y)+z

For multiplication: x(yz)=(xy)z.

Some important properties that should be remembered are:

0 is an additive identity and 1 is a multiplicative identity for rational numbers.

For a rational number x/y, the additive inverse is -x/y and y/x is the multiplicative inverse.

Answer 2

Step-by-step explanation:

Given that:Properties of the group are not followed in subtraction of rational number explain by giving example.

Solution:

Following are the properties which are not followed by subtraction of rational numbers

1) Commutative property:

It states that

$\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$

But this is not followed by subtraction

$\frac{a}{b} - \frac{c}{d} \neq \frac{c}{d} - \frac{a}{b}$

Example: Let

$\frac{a}{b} = \frac{2}{3} \\ \\ \frac{c}{d} = \frac{5}{2} \\ \\ \frac{2}{3} - \frac{5}{2} = \frac{4 - 15}{6} \\ \\ = \frac{ - 11}{6} \\ \\ \frac{5}{2} + \frac{2}{3} = \frac{15 - 4}{6} \\ \\ = \frac{11}{6} \\ \\ \frac{ - 11}{6} \neq \frac{11}{6}$

Thus Commutative property is not followed by subtraction of two rational number.

2) Associative property: Associative property is also not followed by subtraction of rational numbers.

$\frac{a}{b} + ( \frac{c}{d} + \frac{e}{f} ) =( \frac{a}{b} + \frac{c}{d} ) + \frac{e}{f}$

But it is not true for subtraction

$\frac{a}{b} - ( \frac{c}{d} - \frac{e}{f} ) \neq( \frac{a}{b} - \frac{c}{d} ) - \frac{e}{f}$

Example:

Let

$\frac{a}{b} = \frac{2}{3} \\ \\ \frac{c}{d} = \frac{5}{6} \\ \\ \frac{e}{f} = \frac{7}{3} \\ \\ \frac{a}{b} - ( \frac{c}{d} - \frac{e}{f} ) = > \\ \\ \frac{2}{3} - ( \frac{5}{6} - \frac{7}{3}) = \frac{2}{3} - ( \frac{5 - 14}{6} ) \\ \\ = \frac{2}{3} - ( \frac{ - 9}{6}) \\ \\ = \frac{2}{3} + \frac{9}{6} \\ \\ = \frac{6 + 9}{6} = \frac{15}{6} \\ \\ \frac{2}{3} - ( \frac{5}{6} - \frac{7}{3}) = \frac{5}{2} \\ \\ ( \frac{a}{b} - \frac{c}{d} ) - \frac{e}{f} = ( \frac{2}{3} - \frac{5}{6} ) - \frac{7}{3} \\ \\ = ( \frac{4 - 5}{6} ) - \frac{7}{3} \\ \\ = \frac{ - 1}{6} - \frac{7}{3} \\ \\ = \frac{ - 1 - 14}{6} \\ \\ = \frac{ - 15}{6} = \frac{ - 5}{2} \\ \\ ( \frac{2}{3} - \frac{5}{6} ) - \frac{7}{3} = \frac{ - 5}{2} \\ \\ \frac{ - 5}{2} \neq \frac{5}{2}$

Thus,associative property is not followed by subtraction of rational numbers.

By this way, one can say that Associative property and Commutative are not followed by subtraction of rational numbers.

Hope it helps you.