Chapter 1 Rational Numbers

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Rational Number represents as fraction p/q where p could be any number and q could not be Zero (0). 

Example 

1] 2 / 3

2] 15 / 18

3]  -6 / 11

4] -78 / 27 

Full video  🔗 

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Closer with whole numbers and integers table 

Can we represent rational number in number line? 

Yes, rational numbers can be represented on a number line.

Table 1: Closure Property of Whole Numbers

Operation Example Result Is it a Whole Number? Closure Property Holds?

Addition

4 + 5= 9 Yes ✅ Yes
0 + 8= 8 Yes ✅ Yes

Subtraction

5 − 3= 2 Yes ✅ Yes
3 − 5= -2 ❌ No ❌ No

Multiplication

6 × 3= 18 Yes ✅ Yes
0 × 7= 0 Yes ✅ Yes

Division

4 ÷ 2= 2 Yes ✅ Yes
5 ÷ 2= 2.5 ❌ No (Not a Whole No.) ❌ No


📌 Conclusion:

Whole numbers are closed under addition and multiplication.

Not closed under subtraction and division.

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Table 2: Closure Property of Integers

Operation Example Result Is it an Integer? Closure Property Holds?

Addition

(−3) + 5  = 2 Yes ✅ Yes
(−4) + (−6) = −10 Yes ✅ Yes

(-55) +70 =  20       Yes.        ✅ Yes

(-888) + 0 =  -888   Yes         ✅ Yes 

We can say that if we add two integers, the result of the addition will be always an integer by following examples.

Integers are closed under addition. For any two integers, A and B,  A + bB is an integer.

Subtraction

(−2)  − 7 = −9 Yes ✅ Yes
4  −  (−3) = 7 Yes ✅ Yes

(-14)  -  (-14) = 0.   Yes ✅ Yes

(-33)  -  0 =  -33       Yes ✅ Yes 

As shown in the example, we can say that the difference between two integers is always an integer. 

Integers are closed under subtraction. If A and B are two integers, then A-B is also an integer.

Multiplication

(−3) × 4= −12 Yes ✅ Yes
(−2) × (−5)= 10 Yes ✅ Yes

79 x (-70) = -5530. Yes ✅ Yes 

55 x -21=1155    Yes ✅ Yes 

By above examples we can say that integers are closed under multiplication. If A and B are two integers and we multiply  AxB , we always get an integer. 

Division

(−6) ÷ 2 = −3 Yes ✅ Yes
7 ÷ 3  = 2.33 ❌ No (Not Integer)❌ No

-11 ÷ 5 =- 2.2        ❌   No Integer ❌ No

Above examples are showing that integers are not showing closure properties in distributive properties. It is not 100% sure that we get an integer if we divide 2 integers a and b.


📌 Conclusion:

Integers are closed under addition, subtraction, and multiplication.

Not closed under division.

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