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Properties of Rational Numbers

 

There are some properties for rational numbers, like closure property, commutative property and associative property. Let us recall the properties one by one.
 


Closure Property


(i) Whole numbers

If a and b are two whole numbers, then a + b = c should also be a whole number.

The property is that the sum, difference, product or the quotient of whole number should always be a whole number.

 

Whole numbers are closed under addition and multiplication.
 

Example : 2 + 3 = 5, a whole number

5 – 2 = 3, a whole number. But, 2 – 5 = -3, which is not a whole number.

2 × 7 = 14, a whole number
6 ÷ 3 = 2, a whole number, but 5 ÷ 2 = , which is not a whole number.
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(ii) Integers :

If a and b are integers then their sum, difference, product and quotient should also be an integer.

Addition: –1 +3 = 2, an integer

Subtraction: –5 – 6 = –11, an integer.

Multiplication: (-2) × (7) = -14, -14 is also an integer.

Division: –9÷ 2= this is not an integer.

So, integers are not closed for division.

Integers are closed for addition, subtraction and multiplication.

(iii) Rational numbers :

Let us see how rational numbers satisfy the closure property.

If is a rational number, then their sum, difference, product and quotient should also be a rational number.
 

⇒ , a rational number.
 

⇒ , a rational number.
 

⇒ , a rational number.
 

⇒ ÷ =1, a rational number
 

But, ÷ 0 is undefined.


So, division does not satisfy the closure property.


Rational numbers are closed under addition, subtraction and multiplication.





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