What Are the 5 Properties of Whole Numbers With Examples?
The concept of properties of whole numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Mastering the properties of whole numbers—closure, commutative, associative, identity, and distributive—helps students perform calculations faster, understand number systems, and answer questions accurately in school and competitive exams.
What Are the Properties of Whole Numbers?
Properties of whole numbers describe how whole numbers behave under mathematical operations such as addition, subtraction, multiplication, and division. Understanding these rules makes problem-solving easier for students from Class 6 onward and is foundational to topics like algebra, mental maths, and logical reasoning. You’ll find these properties applied in areas such as order property, integer operations, and number system analysis.
List of Properties of Whole Numbers
- Closure Property: Whole numbers are closed under addition and multiplication.
- Commutative Property: Changing the order of addition or multiplication doesn’t affect the result.
- Associative Property: The grouping of numbers does not change the sum or product.
- Identity Property: 0 is the identity for addition and 1 is the identity for multiplication.
- Distributive Property: Multiplication distributes over addition.
Summary Table: Properties and Operations
| Operation | Closure | Commutative | Associative | Identity | Distributive |
|---|---|---|---|---|---|
| Addition | Yes | Yes | Yes | 0 | - |
| Multiplication | Yes | Yes | Yes | 1 | a×(b+c)=a×b+a×c |
| Subtraction | No | No | No | - | - |
| Division | No | No | No | - | - |
Closure Property of Whole Numbers
Closure property means when we add or multiply two whole numbers, the answer is always a whole number. For example: 4 + 5 = 9 (whole number); 3 × 7 = 21 (whole number). However, subtraction or division may not be closed: 3 − 5 = -2 (not a whole number), 4 ÷ 3 = 1.33 (not a whole number).
Commutative and Associative Properties
The commutative property says you can add or multiply whole numbers in any order: 2 + 3 = 3 + 2 and 4 × 5 = 5 × 4. The associative property means how you group numbers doesn’t affect the sum or product: (1 + 2) + 3 = 1 + (2 + 3).
Identity Property
Identity property for addition means that 0 can be added to any whole number without changing its value (e.g., 7 + 0 = 7). For multiplication, 1 is the identity: any whole number multiplied by 1 remains unchanged (e.g., 6 × 1 = 6).
Distributive Property of Whole Numbers
Distributive property combines addition and multiplication: a × (b + c) = a × b + a × c. For example, 3 × (4 + 5) = (3 × 4) + (3 × 5) = 12 + 15 = 27.
Step-by-Step Illustration
1. Calculate 12 × (5 + 4).2. Apply distributive property: 12 × 5 + 12 × 4.
3. Solve: 60 + 48 = 108.
4. Final Answer: 108
Speed Trick or Vedic Shortcut
To quickly multiply numbers ending in 5, such as 25 × 35:
- Multiply the tens numbers: 2 × 3 = 6.
- Add one to one of them: 2 + 1 = 3; Multiply 2 × 3 = 6.
- Attach 25 to the result: 6 25.
- Final answer is 875.
Such calculation hacks, as practiced on Vedantu, save valuable seconds during exams!
Try These Yourself
- Is 0 a whole number? Is -3 a whole number?
- Does 6 − 8 follow the closure property?
- Solve: 13 × (4 + 2) using distributive property.
- Which property: 5 + 2 = 2 + 5?
Frequent Errors and Misunderstandings
- Assuming subtraction and division are always closed—remember they are not.
- Confusing the order of operations in associative property.
Relation to Other Concepts
The idea of properties of whole numbers connects closely with the properties of numbers and whole numbers. Mastering these rules forms a base for learning about rational numbers and complex algebra.
Classroom Tip
A quick way to remember the distributive property: “Multiply before you add!” Visual aids and Vedantu’s formula charts simplify these rules for Class 6 and up.
Printable Reference Table
| Property | Addition | Multiplication | Example |
|---|---|---|---|
| Closure | Yes | Yes | 2 + 8 = 10; 5 × 6 = 30 |
| Commutative | Yes | Yes | 3 + 7 = 7 + 3; 4 × 9 = 9 × 4 |
| Associative | Yes | Yes | (2 + 4) + 5 = 2 + (4 + 5) |
| Identity | 0 | 1 | 8 + 0 = 8; 7 × 1 = 7 |
| Distributive | – | Yes | 2 × (3 + 5) = (2 × 3) + (2 × 5) |
We explored properties of whole numbers—definition, classic examples, tricks, mistakes, and their link to bigger maths ideas. Practice more at Vedantu to boost your Maths for board and competitive exams. For advanced study, check out commutative property and distributive property with solved examples.
FAQs on Properties of Whole Numbers: A Complete Guide for Students
1. What are the main properties of whole numbers?
Whole numbers have four main properties:
- Closure
- Associativity
- Commutativity
- Identity
2. What does closure property mean for whole numbers?
The closure property states that when you add or multiply any two whole numbers, the result is also a whole number. For example, $3+4=7$ and $2\times5=10$ are both whole numbers. This property does not hold for subtraction or division.
3. Can you explain the commutative property of whole numbers?
The commutative property means the order of numbers does not change the result for addition or multiplication. For whole numbers, $a + b = b + a$ and $a \times b = b \times a$. But it does not apply to subtraction or division.
4. What is the associative property in whole numbers?
The associative property of whole numbers means the way numbers are grouped does not affect the sum or product. For example, $(2 + 3) + 4 = 2 + (3 + 4)$ and $(2 \times 3) \times 4 = 2 \times (3 \times 4)$.
5. What is the identity property of addition and multiplication for whole numbers?
For addition, the identity element is 0, since $a + 0 = a$. For multiplication, the identity element is 1, because $a \times 1 = a$. These identity properties hold for every whole number.
6. Are whole numbers closed under subtraction and division?
No, whole numbers are not closed under subtraction and division. Subtracting or dividing two whole numbers may not result in a whole number. For example, $3-5=-2$ and $4 \div 3=1.33$, which are not whole numbers.
7. What is the distributive property of multiplication over addition in whole numbers?
The distributive property states that multiplying a number by the sum of two others is the same as multiplying separately and adding results: $a \times (b + c) = (a \times b) + (a \times c)$. This property holds for whole numbers.
8. Can whole numbers be negative?
No, whole numbers cannot be negative. Whole numbers are the set of numbers starting from 0 and include only non-negative integers:
- 0
- 1
- 2
- 3
- …
9. Is zero a whole number and what is its role?
Yes, zero is a whole number. Its role is as the additive identity, which means adding zero to any whole number does not change its value: $a + 0 = a$. Zero is always included in the whole numbers set.
10. Why is the concept of whole numbers important in math?
Understanding whole numbers is important because they form the foundation of basic math operations, such as addition, subtraction, multiplication, and division. They help in counting, ordering, and other mathematical calculations used in daily activities and problem-solving.
11. How are whole numbers different from natural numbers?
The main difference is that whole numbers include zero, while natural numbers start from one. So, the set of natural numbers is $\{1, 2, 3, ...\}$ and whole numbers are $\{0, 1, 2, 3, ...\}$.
