Properties of Rational Numbers - 1, Maths, Class 8, PDF Download

PROPERTIES OF RATIONAL NUMBER

Closure Property

1. Addition : When two rational numbers are added, their sum is always a rational number.

For example, 5/6 + 4/5 = 49/30,

Which is also a rational number. Therefore, rational numbers are closed under addition. It means for any two rational numbers a and b, a b is also a rational number.

2. Subtraction: When two rational numbers are subtracted, the result is always a rational number.

 For example,

 3/4 – 2/3 = 1/12,

Which is also a rational number.

Therefore, rational numbers are closed under subtraction. It means for any two rational numbers, a and b, a-b is also a rational number.

3 . Multiplication: When two rational numbers are multiplied, their product is always a rational number. 

For example 4/11 × 3/7 = 12/77

Which is also a rational number.

Therefore, rational number are closed under multiplication. It means for any two rational number a and b a × b is also a rational number.

4.Division: As for any rational number a, a ÷ 0 is not defined, therefore not all rational numbers are closed under division. We can say that except zero, all rational numbers are closed under division.

Take a look at some examples of division of rational numbers. 

5/7 ÷ 3/8 = 5/7 × 8/3 =40/21,

Which is a rational number. 

-4/5 ÷ -6/7 = -4/5 × 7/-6 = 14/15,

which is a rational number. 

Note: For any rational number a/b, b/a is called its reciprocal.

Commutative

1. Addition: Addition is commutative for a rational numbers. In general, for any two rational numbers a and b,

a b = b a

The following examples prove the commutativity of addition for rational numbers.

3/7 + 5/7 = 8/7  and    5/7 +  3/7 = 8/7

-4/9 -7/9 = -11/9 and    -7/9 -4/9 = -11/9 

2. Multiplication: Multiplication is also commutative for rational numbers. In general, for any two rational numbers a and b,

a × b = b × a 

The following examples prove the commutativity of multiplication for rational numbers.

 2/7 × 5/9 = 10/63    and 5/9 × 2/7 = 10/63

-3/5 × -8/11 =24/55  and -8/11 × -3/5 = 24/55

3. Subtraction: Subtraction is not commutative for rational numbers. In general , for any tow rational numbers a and b, 

a-b ≠ b-a

Look at the following example showing that subtraction of rational numbers is not commutative.

5/6 – 2/3 = 1/6 but 2/3 -5/6 = -1/6 

4.Division: Division is not commutative for rational numbers. In general, for any rational numbers a and b,

 a ÷ b ≠ b ÷a 

Look at the following example showing that division of rational numbers is not commutative.

8/11 ÷ 4/5 = 10/11 but 4/5 ÷ 8/11 = 11/10