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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.

Question for Properties of Rational Numbers - 1, Maths, Class 8,
Try yourself:Which property of rational numbers states that the product of two rational numbers is always a rational number?
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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

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FAQs on Properties of Rational Numbers - 1, Maths, Class 8,

1. What are rational numbers?
Ans. Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero. Examples of rational numbers include 1/2, 5/3, -3/4, etc.
2. How can we identify if a number is rational or not?
Ans. To identify if a number is rational or not, we need to check if it can be expressed as a ratio of two integers. If it can be expressed as a ratio of two integers, then it is a rational number, otherwise, it is an irrational number.
3. What are the basic operations that can be performed on rational numbers?
Ans. The basic operations that can be performed on rational numbers include addition, subtraction, multiplication, and division. To perform these operations, we need to find a common denominator and then perform the operations on the numerators.
4. Can a rational number be negative?
Ans. Yes, a rational number can be negative. A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. So, both positive and negative numbers can be rational numbers.
5. What is the difference between a rational number and an irrational number?
Ans. A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. An irrational number, on the other hand, is a number that cannot be expressed as a ratio of two integers. Examples of irrational numbers include pi, e, square root of 2, etc.
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