Chapter 2
WHOLE NUMBERS
• Counting numbers are called natural numbers. Natural numbers are denoted by N and given by
N = {1, 2, 3, 4, …}
• Natural numbers included with ‘0’ are called whole numbers. Whole numbers are represented by W and given by W = {0, 1, 2, 3, 4, …}
• A number which comes after a given number is called its successor.
Successor = given number + 1
• A number which comes before a given number is called its predecessor.
Predecessor = given number – 1
• Every whole number has a successor. Every whole number except 0, has a
predecessor.
• The following number line represents a whole number line on which whole numbers are represented.
![WHOLE NUMBERS](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_9.png)
The distance between two points is called unit distance.
Number operations of addition, subtraction and multiplication can easily be performed on number line.
• Closure Property for addition and multiplication: If we add or multiply two whole numbers the result is again a whole number. Closure property does not hold good for subtraction and division of whole numbers.
• Commutativity of addition and multiplication: Whole number can be added or multiplied in any order. That is, for any two whole numbers ‘a’ and ‘b’, a + b = b + a and a × b = b × a Commutative property does not hold good for subtraction and division of whole numbers.
• Associativity of addition and multiplication: Whole numbers can be grouped for the convenience of adding or multiplying. That is, for any three whole numbers ‘a’ , ‘b’ and ‘c’,a + (b + c) = (a+ b) + c and a × (b × c) = (a × b) × c
Associative property does not hold good for subtraction and division of whole numbers.
• Distributivity of multiplication over addition: For any three whole numbers ‘a’ , ‘b’ and ‘c’, a × (b + c) = (a × b) + (a × c)
• Identity for addition: 0 is the identity for addition; this means when 0 is added to any whole the result is the whole number itself.
• Identity for multiplication: 1 is the identity for multiplication; this means when 1 is
multiplied with any whole number the result is the whole number itself.
• Division of a whole number by zero is not defined
The numbers 1,2, 3, are called natural numbers or counting numbers.
Let us add one more number i.e., zero (0), to the collection of natural numbers. Now the numbers are 0,1,2, … These numbers are called whole numbers
We can say that whole nos. consist of zero and the natural numbers. Therefore, except zero all the whole nos. are natural numbers.
Facts of Whole numbers
• The smallest natural number is 1.
• The number 0 is the first and the smallest whole nos.
• There are infinitely many or uncountable number of whole-numbers.
• All natural numbers are whole-numbers.
• All whole-numbers are not natural numbers. For example, 0 is a whole-number but it is not a natural number.
The first 50 whole nos. are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50
Number line
A number line is a picture of a graduated straight and horizontal line in which numbers are written. A number written on the left-hand side of the number line is lesser and number written on the right-hand side of the number line is greater. Lets us look into some solved example problems.
![](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_10.png)
In the above figure, we can see a number line where integers are placed. Here, the positive and negative integers are placed on either side of the zero.
Find 12 × 35 using distributivity.
12 × 35 = 12 × (30 + 5)
= 12 × 30 +12 × 5
= 360 + 60 = 420.
Calculate – (2 + 3) + 4 =? = 5+ 4 = 9.
A number line is a pictorial representation of numbers on a straight line. It’s a reference for comparing and ordering numbers. It can be used to represent any real number that includes every whole number and natural number. Just to recollect, the whole number is a set of numbers that include all counting numbers (1, 2, 3,4,5,6 …….) and zero (0), whereas the natural number is the set of all counting numbers i.e. 1, 2, 3, 4, 5, 6……..
Writing numbers on a number line make it easier to compare the numbers. From the above figure, we can see that the integers on the left side are smaller than the integers on the right side. For example, 0 is less than 1, -1 is less than 0, -2 is less than -1, and so on.
Numbers on a Number Line
Arithmetic operations of numbers can be better explained on a number line. To begin with, one must know to locate numbers on a number line. Zero is the middle point of a number line. All (natural numbers) positive numbers occupy the right side of the zero whereas negative numbers occupy the left side of zero on the number line. As we move on to the left side value of a number decreases. For example, 1 is greater than -2. In a number line,
integers, fractions, and decimals can also be represented easily. Check out the links given below to learn more.
Number Line 1 to 100
Here are the number lines from 1 to 100 numbers. As we already we can extend the number line indefinitely, thus students can also try to draw the line beyond 100, to practice.
![WHOLE NUMBERS](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_11.png)
Addition on Number Lines
Adding Positive Numbers When we add two positive numbers, the result will always be a positive number. Hence, on adding positive numbers direction of movement will always be to the right side.
For example, addition of 1 and 5 (1 + 5 = 6)
Here the first number is 1 and the second number is 5; both are positive. First, locate 1 on the number line. Then move 5 places to the right will give 6.
![](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_12.png)
Adding Negative Numbers
When we add two negative numbers, the result will always be a negative number. Hence, adding negative numbers direction of movement will always be to the left side.
For example, the addition of -2 and -3
Here, the first number is -2 and the second number is -3; both are negative. Locate -2 on the number line. Then move 3 places to the left will give -5.
![](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_13.png)
Some other Important terms to remember
![WHOLE NUMBERS](https://gyanchakra.co.in/wp-content/uploads/2023/03/Screenshot_14.png)
Properties of Whole Numbers
Closure Property
Closure property on Addition for Whole Number
0 + 2 = 2
1 + 3 = 4
5 + 6 = 11
So Whole number are closed on Addition
Closure property on Multiplication for Whole Number
0 × 2 = 0
1 × 4 = 4
5 × 1 = 5
So Whole number are closed on Multiplication
Closure property on subtraction of Whole number
5 – 0 = 5
0 – 5 =?
1 – 3 =?
3 – 1 = 2
So Whole number are not closed on Subtraction
Closure property on Division of Whole number
2/1= 2
1/2=?
0/2= 0
2/0=? (Division by Zero is undefined)
So Whole Number are not closed on Division
In short Closure Property
If a and b are any two whole numbers, then a + b, a × b are also whole numbers.
Commutative property
Commutativity property on Addition for Whole Number
So Whole number are Commutative on Addition
0 + 2 = 2 + 0 = 2
Commutativity property on Multiplication for Whole Number
0 × 2 = 0 or 2 × 0 = 0
So Whole number are Commutative on Multiplication
Commutativity property on subtraction of Whole number
5 – 0 = 5 but 0 – 5 =?
So Whole number are not Commutative on Subtraction
Commutativity property on Division of Whole number
2/1= 2 but 1/2=?
So Whole Number are not Commutative on Division
In short
You can add two whole numbers in any order. You can multiply two whole numbers in any order.
Commutative property: If a and b are any two whole numbers, then a + b = b + a and a × b = b × a
Associative property
Associativity property on Addition for Whole Number
0 + (2 + 3) = (0 + 2) + 3 = 50 + (2 + 3) = (0 + 2) + 3 = 5
1 + (2 + 3) = 6 = (1 + 2) + 31 + (2 + 3) = 6 = (1 + 2) + 3
So Whole number are Associative on Addition
Associativity property on Multiplication for Whole Number
0 × (2 × 3) = 0 or (0 × 2) × 3 = 0
So Whole number are Associative on Multiplication
Associativity property on subtraction of Whole number
10 – (2 – 1) = 9 but (10 – 2) – 1 = 7
So Whole number are not Associative on Subtraction
Associativity property on Division of Whole number
16 ÷ (4 ÷ 2) = 8 but (16 ÷ 4) ÷ 2 = 2
So Whole Number are not Associative on Division
So in Short
If a, b and c are any two whole numbers, then (a + b) + c = a + (b + c) and (a × b) × c = a × (b
× c).
Distributive property
If a, b and c are any two whole numbers, then a (b + c) = a × b + a × c
Additive Identity
If a is any whole number, then a + 0 = a = 0 + a
Example
2 + 0 = 2
0 + 3 = 3
5 + 0 = 5
Multiplicative Identity
If a is any whole number, then a × 1 = a = 1 × a
Example
1 × 1 = 1
5 × 1 = 5
6 × 1 = 6
Multiplication by zero
If a is any whole number, then a × 0 = 0 = 0 × a
Example
1 × 0 = 0
5 × 0 = 0
0 × 0 = 0
Division by zero
If a is any whole number, then a ÷ 0 is not defined
![WHOLE NUMBERS](https://gyanchakra.co.in/wp-content/uploads/2023/03/unnamed-3-5-1024x739.jpg)
Also Read :- Mathematics chapter 1 Knowing Our Numbers
NCERT Notes
Nursery Class | Ncert Notes for Class 1 | Ncert Notes For Class 2 | Ncert Notes For Class 3 | Ncert Notes For Class 4 | Ncert Notes for class 5 | Ncert Notes for Class 6 | Ncert Notes For Class 7 | Ncert Notes for Class 8 | Ncert Notes for Class 9 | Ncert Notes for class 10 | Ncert Notes for class 11 Science | Ncert Notes for class 11 Commerce | Ncert Notes for Class 11 Arts | Ncert Notes for class 12 Science | Ncert Notes for class 12 Commerce | Ncert Notes for class 12 Arts
Exams
UPSC Exam | SSC CGL Exam | SBI PO Exam | IBPS PO Exam | RRB NTPC Exam | RBI Exam | JEE Main Exam | NEET Exam | Gate Exam | CLAT Exam | RPSC RAS Exam
Shop at Gyanchakra
E- books | Nursery Books | Class 1 to 12 | Entrance Exams | Govt Exams | Skills | Courses | Games & Toys | Motivational Items
Follows on Social Media
Facebook | Youtube | Instagram | Pinterest | Medium | Linkedin |
Important Govt Sites
CBSE | NCERT | JEE | UPSC | NEET | SSC | PMO | Raj Board | RAS Exam |
Blogs
Current Affairs | Geography | Grammar | UPSC Exam | Higher Education | CBSE | Science Facts For Kids | JEE Exams | JEE Advance Exams | JEE mains Exams | NEET Exams | Science Facts For Students
Free Test Series for Entrance exams
Free Test Series for WBJEE | Free Test series for UPSEE | Free Test Series for AP EAMCET | JEE Mains Advanced | Free Test Series for BITSAT | Free Test Series for TS EAMCET | Free Test Series for Manipal Entrance Test MET | Free Test Series for CIPET JEE | Free Test Series for AIEEA UG Exam ICAR | Free Test Series for IISER SCB | Free test series for NEST National Entrance Screening Test | Maths Question Bank | Free Test Series for MHT CET | Free Test Series for GOA CET | Free Test For SRMJEEE
Healthy Nutrition for students
Healthy Snacks | Low Glycemic index Food | High Protien Food | Heart healthy | Low Calorie Foods | Dairy Free Foods | Gluten Free Foods | Keto Friendly Foods | Gut Healthy
Free Test Series for Govt Exams
Free Test Series for UPSC Prelims | Free test series for UPSC Drug Inspector | Free Test Series for TNPSC | Free Test Series for MPSC | Free Test Series for MPPSC | Free test Series for CGPSC | Free test Series for UPPSC | Free test Series for RPSC | Free Test series for Meghalaya PSC | Free Test Series for JPSC | Free Test Series for KPSC | Free Test Series for UKPSC | Free Test Series for OPSC |
Free Notes for NCERT Class 1
Free Video Notes for NCERT Class 1 EVS | Free Notes for NCERT Class 1 Mathematics | Free Video Notes for NCERT Class 1 Mathematics | Free Notes for NCERT Class 1 English | Free Video Notes for NCERT Class 1 Hindi | Free Video Notes for NCERT विज्ञान | Free Video Notes For NCERT गणित
Free Notes For NCERT Class 2 English medium
Free Video Notes for NCERT Class 2 EVS | Free video Notes for NCERT Class 2 Mathematics | Free Notes for NCERT Class 2 Mathematics | Free Notes for NCERT Class 2 English
Free Notes for NCERT Class 2 Hindi Medium
Free Notes for NCERT Class 2 Hindi | Free video Notes for Ncert Class 2 विज्ञान | Free Video Notes for NCERT Class 2 गणित |
Free Notes for NCERT class 3 English Medium
Free Video Notes for Ncert Class 3 EVS | Free Notes for Ncert class 3 EVS | Free Video Notes For Ncert Class 3 Mathematics | Free Notes for Ncert Class 3 Mathematics | Free Notes for Ncert Class 3 english | Free video Notes for Ncert Class 3 hindi | Free Video Notes for Ncert Class 3 विज्ञान
Free Notes for NCERT class 3 Hindi Medium
Free Video Notes for NCERT class 3 EVS Hindi Medium
Free Notes for NCERT class 4 English Medium
Free Video Notes for Ncert Class 4 EVS | Free Notes for Ncert class 4 EVS | Free Video Notes For Ncert Class 4 Mathematics | Free Notes for Ncert Class 4 Mathematics | Free Notes for Ncert Class 4 english | Free video Notes for Ncert Class 4 hindi
Free Notes for NCERT class 4 Hindi Medium
Free Video Notes for NCERT class 4 EVS Hindi Medium | Free Video Notes for NCERT class 4 Maths Hindi Medium
Free Notes for NCERT class 5 English Medium
Free Video Notes for Ncert Class 5 EVS | Free Notes for Ncert class 5 EVS | Free Video Notes For Ncert Class 5 Mathematics | Free Notes for Ncert Class 5 Mathematics | Free Notes for Ncert Class 5 english | Free video Notes for Ncert Class 5 hindi
Free Notes for NCERT class 5 Hindi Medium
Free Video Notes for NCERT class 5 EVS Hindi Medium | Free Video Notes for NCERT class 5 Maths Hindi Medium
Free Notes for NCERT class 6 English Medium
Free Video Notes for Ncert Class 6 Science | Free MCQ for NCERT Class 6 Science | Free Notes for Ncert class 6 Science | Free Video Notes For Ncert Class 6 Mathematics | Free MCQ for NCERT Class 6 Mathematics | Free Notes for Ncert Class 6 Mathematics | Free Notes for Ncert Class 6 english | Free Video Notes for Ncert Class 6 Social Science | Free MCQ for NCERT Class 6 Social Science | Free Notes for Ncert class 6 Social Science
Free Notes for NCERT class 6 Hindi Medium
Free Video Notes for NCERT class 6 Science Hindi Medium | Free Notes for NCERT class 6 Science Hindi Medium | Free Video Notes for NCERT class 6 Mathematics Hindi Medium | Free Notes for NCERT class 6 Mathematics Hindi Medium | Free Notes for NCERT class 6 English Hindi Medium | Free Notes for NCERT class 6 Social Science Hindi Medium | Free Notes for NCERT class 6 Hindi in Hindi Medium
Free Notes for NCERT class 7 English Medium
Free Video Notes for Ncert Class 7 Science | Free MCQ for NCERT Class 7 Science | Free Notes for Ncert class 7 Science | Free Video Notes For Ncert Class 7 Mathematics | Free MCQ for NCERT Class 7 Mathematics | Free Notes for Ncert Class 7 Mathematics | Free Notes for Ncert Class 7 english | Free Video Notes for Ncert Class 7 Social Science | Free MCQ for NCERT Class 7 Social Science | Free Notes for Ncert class 7 Social Science | Free Notes for Ncert class 7 English
Free Notes for NCERT class 7 Hindi Medium
Free Video Notes for NCERT class 7 Science Hindi Medium | Free Notes for NCERT class 7 Science Hindi Medium | Free Video Notes for NCERT class 7 Mathematics Hindi Medium | Free Notes for NCERT class 7 Mathematics Hindi Medium | Free Notes for NCERT class 7 English Hindi Medium | Free Notes for NCERT class 7 Social Science Hindi Medium | Free Notes for NCERT class 7 Hindi in Hindi Medium
Free Notes for NCERT class 8 English Medium
Free Video Notes for Ncert Class 8 Science | Free MCQ for NCERT Class 8 Science | Free Notes for Ncert class 8 Science | Free Video Notes For Ncert Class 8 Mathematics | Free MCQ for NCERT Class 8 Mathematics | Free Notes for Ncert Class 8 Mathematics | Free Notes for Ncert Class 8 english grammar | Free Video Notes for Ncert Class 8 Social Science | Free MCQ for NCERT Class 8 Social Science | Free Notes for Ncert class 8 Social Science | Free Notes for Ncert class 8 English
Free Notes for NCERT class 8 Hindi Medium
Free Video Notes for NCERT class 8 Science Hindi Medium | Free Notes for NCERT class 8 Science Hindi Medium | Free Video Notes for NCERT class 8 Mathematics Hindi Medium | Free Notes for NCERT class 8 Mathematics Hindi Medium | Free Notes for NCERT class 8 English Hindi Medium | Free Notes for NCERT class 8 Social Science Hindi Medium | Free Notes for NCERT class 8 Hindi in Hindi Medium
Free Notes for NCERT class 9 English Medium
Free Video Notes for Ncert Class 9 Science | Free MCQ for NCERT Class 9 Science | Free Notes for Ncert class 9 Science | Free Video Notes For Ncert Class 9 Mathematics | Free MCQ for NCERT Class 9 Mathematics | Free Notes for Ncert Class 9 Mathematics | Free Notes for Ncert Class 9 english | Free Video Notes for Ncert Class 9 Social Science | Free MCQ for NCERT Class 9 Social Science | Free Notes for Ncert class 9 Social Science
Free Notes for NCERT class 9 Hindi Medium
Free Video Notes for NCERT class 9 Science Hindi Medium | Free Notes for NCERT class 9 Science Hindi Medium | Free Video Notes for NCERT class 9 Mathematics Hindi Medium | Free Notes for NCERT class 9 Mathematics Hindi Medium | Free Notes for NCERT class 9 English Hindi Medium | Free Notes for NCERT class 9 Social Science Hindi Medium | Free Notes for NCERT class 9 Hindi in Hindi Medium
Free Notes for NCERT class 10 English Medium
Free MCQ for NCERT Class 10 Science | Free Notes for Ncert class 10 Science | Free MCQ for NCERT Class 10 Mathematics | Free Notes for Ncert Class 10 Mathematics | Free Notes for Ncert Class 10 english | Free MCQ for NCERT Class 10 Social Science
Free Notes for NCERT class 10 Hindi Medium
Free Notes for NCERT class 10 Science Hindi Medium | Free Notes for NCERT class 10 Mathematics Hindi Medium | Free Notes for NCERT class 10 English Hindi Medium | Free Notes for NCERT class 10 Social Science Hindi Medium | Free Notes for NCERT class 10 Hindi in Hindi Medium
Free Notes for NCERT class 11 PCMB English Medium
Free Notes for Ncert class 11 Science | Free MCQ for NCERT Class 11 Science | Free Video Notes for Ncert class 11 Science | Free Notes for Ncert Class 11 Mathematics | Free MCQ for NCERT Class 11 Mathematics | Free Video Notes for Ncert Class 11 Mathematics | Free Notes for Ncert Class 11 english | Free Notes for NCERT Class 11 Physical Education
Free Notes for NCERT class 11 PCMB Hindi Medium
Free Notes for Ncert class 11 Science Hindi Medium | Free Video Notes for Ncert class 11 Science Hindi Medium | Free Notes for Ncert Class 11 Mathematics Hindi Medium | Free Video Notes for Ncert Class 11 Mathematics Hindi Medium | Free Notes for Ncert Class 11 english | Free Notes for NCERT Class 11 Physical Education Hindi Medium | Free Notes for Ncert Class 11 Hindi
Free Notes for NCERT class 11 Art Hindi Medium
Free Notes for Ncert class 11 Art अर्थशास्त्र सांख्यिकी Hindi Medium | Free Notes for Ncert class 11 Art इतिहास Hindi Medium | Free Notes for Ncert Class 11 Art समाजशास्त्र Hindi Medium | Free Notes for Ncert Class 11 Art भूगोल Hindi Medium | Free Notes for NCERT Class 11 Art राजनीति विज्ञान Hindi Medium | Free Notes for NCERT Class 11 Art हिन्दी Hindi Medium | Free Notes for NCERT Class 11 Art अंग्रेज़ी Hindi Medium | Free Notes for NCERT Class 11 Art शारीरिक शिक्षा Hindi Medium
Free Notes for NCERT class 11 Commerce English Medium
Free Notes for Ncert class 11 Commerce Physical Education | Free Notes for NCERT Class 11 Commerce English | Free Notes for Ncert class 11 Commerce Accountancy | Free MCQ for Ncert Class 11 Commerce Accountancy | Free Notes for NCERT Class 11 Commerce Entrepreneurship | Free Notes for Ncert Class 11 Commerce Economics Micro-Economics | Free MCQ for Ncert Class 11 Commerce Economics Micro-Economics | Free Notes for NCERT Class 11 Commerce Economics Statistics | Free MCQ for NCERT Class 11 Commerce Economics Statistics | Free Notes for NCERT Class 11 Commerce Business Studies | Free MCQ for NCERT Class 11 Commerce Business Studies
Free Notes for NCERT class 11 Commerce Hindi Medium
Free Notes for Ncert class 11 Commerce शारीरिक शिक्षा | Free Notes for NCERT Class 11 Commerce व्यवसाय अध्ययन | Free Notes for Ncert class 11 Commerce अंग्रेज़ी | Free Notes for NCERT Class 11 Commerce हिंदी | Free Notes for Ncert Class 11 Commerce अर्थशास्त्र सांख्यिकी | Free Notes for NCERT Class 11 Commerce अर्थशास्त्र व्यष्टि अर्थशास्त्र | Free Notes for NCERT Class 11 Commerce लेखाशास्त्र | Free Notes for NCERT Class 11 Commerce गणित
Free Notes for NCERT class 12 PCMB English Medium
Free Notes for Ncert class 12 Science | Free MCQ for NCERT Class 12 Science | Free Video Notes for Ncert class 12 Science | Free Notes for Ncert Class 12 Mathematics | Free MCQ for NCERT Class 12 Mathematics | Free Video Notes for Ncert Class 12 Mathematics | Free Notes for Ncert Class 12 english | Free Notes for NCERT Class 12 Physical Education
Free Notes for NCERT class 12 PCMB Hindi Medium
Free Notes for Ncert class 12 Science Hindi Medium | Free Video Notes for Ncert class 12 Science Hindi Medium | Free Notes for Ncert Class 12 Mathematics Hindi Medium | Free Video Notes for Ncert Class 12 Mathematics Hindi Medium | Free Notes for Ncert Class 12 english | Free Notes for Ncert Class 12 Hindi
Free Notes for NCERT class 12 Art Hindi Medium
Free Notes for Ncert class 12 Art अर्थशास्त्र सांख्यिकी Hindi Medium | Free Notes for Ncert class 12 Art इतिहास Hindi Medium | Free Notes for Ncert Class 12 Art समाजशास्त्र Hindi Medium | Free Notes for Ncert Class 12 Art भूगोल Hindi Medium | Free Notes for NCERT Class 12 Art राजनीति विज्ञान Hindi Medium | Free Notes for NCERT Class 12 Art हिन्दी Hindi Medium | Free Notes for NCERT Class 12 Art अंग्रेज़ी Hindi Medium | Free Notes for NCERT Class 12 Art शारीरिक शिक्षा Hindi Medium | Free Notes for NCERT Class 12 Art गृह विज्ञान Hindi Medium | Free Notes for NCERT Class 12 Art समष्टि अर्थशास्त्र Hindi Medium | Free Notes for NCERT Class 12 Art भारतीय अर्थव्यवस्था का विकास Hindi Medium
Free Notes for NCERT class 12 Commerce English Medium
Free Notes for Ncert class 12 Commerce Physical Education | Free Notes for NCERT Class 12 Commerce English | Free Notes for Ncert class 12 Commerce Accountancy | Free MCQ for Ncert Class 12 Commerce Accountancy | Free Notes for NCERT Class 12 Commerce Entrepreneurship | Free Notes for Ncert Class 12 Commerce Economics Micro-Economics | Free MCQ for Ncert Class 12 Commerce Economics Micro-Economics | Free Notes for NCERT Class 12 Commerce Indian Economic Development | Free MCQ for NCERT Class 12 Commerce Indian Economic Development | Free Notes for NCERT Class 12 Commerce Business Studies | Free MCQ for NCERT Class 12 Commerce Business Studies
Free Notes for NCERT class 12 Commerce Hindi Medium
Free Notes for NCERT Class 12 Commerce व्यवसाय अध्ययन | Free Notes for Ncert class 12 Commerce अंग्रेज़ी | Free Notes for NCERT Class 12 Commerce हिंदी | Free Notes for Ncert Class 12 Commerce अर्थशास्त्र सांख्यिकी | Free Notes for NCERT Class 12 Commerce लेखाशास्त्र | Free Notes for NCERT Class 12 Commerce गणित