Math, asked by roshanmishra9428, 10 months ago

Explain the property of rational number with example

Answers

Answered by bhumikaagrawal1906
10

Answer:

Closure Property

1) Addition of Rational Numbers

The closure property states that for any two rational numbers a and b, a + b is also a rational number.

12 + 34

= 4+68

= 108

Or, =  54

The result is a rational number. So we say that rational numbers are closed under addition.

2) Subtraction of Rational Numbers

The closure property states that for any two rational numbers a and b, a – b is also a rational number.

12 – 34

= 4–68

= −28

Or, =  −14

The result is a rational number. So the rational numbers are closed under subtraction.

3) Multiplication of Rational Numbers

The closure property states that for any two rational numbers a and b, a × b is also a rational number.

12 × 34

=  68

The result is a rational number. So rational numbers are closed under multiplication.

4) Division of Rational Numbers

The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.

12 ÷ 34

= 1×42×3

=  23

The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.

Commutative Property

1. Addition

For any two rational numbers a and b, a + b = b+ a

−23+  57 and 57+  −23 = 121

so, −23+  57 = 57+  −23

We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.

2. Subtraction

For any two rational numbers a and b, a – b ≠ b –  a. Given are the two rational numbers 53 and  14,

53 – 14 = 20−312

= 1712

But, 14 – 53 = 3−2012

= −1712

So subtraction is not commutative for ratioanl numbers.

3. Multiplication

For any two rational numbers a and b, a × b = b × a

−73+  65 = 65+  −73

= −4215 =  −4215

We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.

4. Division

For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers 53 and  14

53 ÷ 14 = 5×43×1

= 203

But, 14 ÷ 53 = 1×34×5

= 320

We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.

Associative Property

Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.

For example, given numbers are 5, -6 and 23

( 5 – 6 ) + 23

= -1 + 23

= – 13

Now, 5 + ( -6 + 23 )

=  – 13

In both the groups the sum is the same.

Addition and multiplication are associative for rational numbers.

Subtraction and division are not associative for rational numbers.

Distributive Property

Distributive property states that for any three numbers x, y and z we have

 x × ( y + z ) = (x × y) +( x × z)

Answered by Divyanshverma1
1

Answer:

CLOSURE PROPERTY WITH REFERENCE TO RATIONAL NUMBERS - DEFINITION

Closure property states that if for any two numbers a and b, a∗b is also a rational number, then the set of rational numbers is closed under addition.

∗ represents +,−,× or ÷

For eg:-

2

1

and

4

3

2

1

+

4

3

=

2×4

1×4+3×2

=

8

4+6

=

8

10

=

4

5

is a rational number

2

1

4

3

=

2×4

1×4−3×2

=

8

4−6

=

8

−2

=

4

−1

is a rational number

2

1

×

4

3

=

2×4

1×3

=

8

3

is a rational number

4

3

2

1

=

2×3

1×4

=

1×3

1×2

=

3

2

is a rational number

Hence, set of rational number is closed under +,−,× and ÷.

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