Introduction
Whole numbers are the fundamental concept of mathematics. It represents the set of all non-negative numbers.
Whole Numbers
It is the set of numbers which include all positive integers and zero. It is represented as- \({0,1,2,3,………}\)
- Every natural number is a whole number.
- Only zero is the whole number which is not a natural number.
- Smallest natural number is 1.
- Smallest whole number is 0.
Operations on Whole numbers
Whole numbers are used in various basic arithmetic operations:
- Addition: Combining two Whole numbers to get a Whole number. Example: \(2+3=5\)
- Subtraction: Removing one Whole number from another. Result is a Whole number if minuend is greater or equal to subtrahend. Example: \(8-5=3\)
- Multiplication: Multiplication is the repeated addition of a Whole number, a specific number of times. Example: \(4\times3=12\ (4+4+4=12)\)
- Division: Division is splitting of a Whole number into equal parts. Example: \(\frac{14}{2}=7\)
Properties of Whole numbers
Properties of Whole numbers are defined to operate arithmetic operations in an easy way. By applying these properties, we can easily solve any problem on whole numbers without using calculator.
There are four properties of Whole Numbers:
- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
Let’s discuss the four properties of Whole Numbers in detail:
1. Closure Property
It is applicable on addition and multiplication of whole numbers. It states that addition and multiplication of any two whole numbers is always a whole number.
Closure property is stated as:
\((a+b) ∈ W\) and \((a\times b) ∈ W\), where \(a,b ∈ W, W\) is set of whole numbers.
Example: a) \(2+4=6\)
b) \(2\times 4=8\)
Both 6 and 8 are whole numbers.
Note: In case of subtraction and division, closure property does not work.
2. Associative Property
It holds for addition and multiplication of three whole numbers. It states that sum and product of any three whole numbers remains same regardless of how numbers are placed.
Associative property is stated as:
\(a+(b+c)=(a+b)+c\) and \(a\times(b\times c)=(a\times b)\times c\) where \(a,b,c ∈ W, W\) is set of whole numbers.
Examples: a) \(\left(1+2\right)+3=1+\left(2+3\right)\)
By solving, \(3+3=1+5\)
\(6=6\)
b) \(\left(1\times2\right)\times3=1\times\left(2\times3\right)\)
By solving, \(2\times3=1\times6\)
\(6=6\)
Note: Associative property does not hold true for Subtraction and Division.
3. Commutative Property
It is applicable on addition and multiplication of whole numbers. This property states that sum and product of two whole numbers remains same regardless the order of numbers.
Commutative Property is stated as:
\(a+b=b+a\), \(a\times b=b\times a\) where \(a,b ∈ W, W\) is set of whole numbers.
Examples: a) \(4+3=3+4\)
By solving, \(7=7\)
b) \(4\times2=2\times4\)
By solving, \(8=8\)
Note: This property does not hold true for Subtraction and Division.
Let us understand with help of table:
| Addition | Subtraction | Multiplication | Division | |
| Closure | ✔ | ✗ | ✔ | ✗ |
| Associative | ✔ | ✗ | ✔ | ✗ |
| Commutative | ✔ | ✗ | ✔ | ✗ |
4. Distributive Property
It is of 2 types.
a) Distributive property of multiplication over addition: It is stated as:
\(a\times\left(b+c\right)=\left(a\times b\right)+\left(a\times c\right)\)
or
\(\left(b+c\right)\times a=\left(b\times a\right)+\left(c\times a\right)\) where \(a,b,c ∈ W, W\) is set of whole numbers.
Examples: \(2\times\left(3+4\right)=\left(2\times3\right)+\left(2\times4\right)\)
\(2\times7=6+8\)
\(14=14\)
b) Distributive property of multiplication over subtraction: It is stated as:
\(a\times\left(b-c\right)=\left(a\times b\right)-\left(a\times c\right)\) where \(a,b,c ∈ W, W\) is set of whole numbers.
Examples: \(2\times\left(4-3\right)=\left(2\times4\right)-\left(2\times3\right)\)
By solving, \(2\times1=8-6\)
\(2=2\)
Magic Square
It is a square in which different numbers are arranged such that sum of numbers in every horizontal line, every vertical line and every diagonal number is same.
| 4 | 9 | 2 |
| 3 | 5 | 7 |
| 8 | 1 | 6 |
