 # Whole Numbers

The whole numbers are the part of the number system in which it includes all the positive integers from 0 to infinity. These numbers exist in the number line, hence they are all real numbers. We can say, all the whole numbers are real numbers but not all the real numbers are whole numbers. Also, natural numbers along with ‘0’ are called whole numbers. The examples are: 0,11,25,36, etc. Let us learn now its definition and properties here in this article.

## Whole Numbers Definition

The whole numbers are the number without fractions and it is a collection of positive integers and zero. It is represented as “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}

These numbers are positive including zero and do not include fractional or decimal parts (3/4, 2.2 and 5.3 are not whole numbers). Addition, Subtraction, Multiplication and Division operations are possible with the whole numbers.

If again you have doubt, What is a whole number in math? A more comprehensive understanding of the whole number can be obtained from the following chart: ## Whole Numbers Properties

The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added, subtracted or multiplied, will give a whole number itself. In the division method, we can get a fraction as a result also. Now let us see some more properties here;

### Closure Property

They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x+y is also a whole number.

### Commutative Property of Addition and Multiplication

The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers x+y=y+x and x.y=y.x

When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x+0=0+x=x

### Multiplicative identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.1=1.x=x

### Associative Property

When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y and z are whole numbers then x+(y+z)=(x+y)+z and x.(y.z)=(x.y).z

### Distributive Property

If x,y and z are three whole numbers, the distributive property of multiplication over addition is x. (y+z)=(x.y)+(x.z), similarly, the distributive property of multiplication over subtraction is x. (y-z)=(x.y)-(x.z)

### Multiplication by zero

When a whole number is multiplied to 0, the result is always 0, i.e., x.0=0.x=0

### Division by zero

Division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.

## Difference Between Whole Numbers and Natural Numbers

Difference Between Whole Numbers & Natural Numbers

Whole Numbers Natural Numbers
Whole Numbers: {0, 1, 2, 3, 4, 5, 6,…..} Natural Numbers: {1, 2, 3, 4, 5, 6,……}
Counting starts from 0 Counting starts from 1
All whole numbers are not natural numbers All Natural numbers are whole numbers

Below figure will help us to understand the difference between the whole number and natural numbers : ### Can Whole Numbers be Negative?

The whole number can’t be negative!

As per definition: { 0,1,2,3,4,5,6,7,……till positive infinity} are whole numbers. There is no place for negative numbers.

### Is 0 a whole number?

The set of numbers contains all Natural Numbers, along with Zero. So yes, 0 (zero) is not only a whole number but the first whole number. ## Whole Numbers Examples

Example 1: Are 100, 227, 198, 4321 whole numbers?

Solution: Yes. 100, 227, 198, 4321 are all whole numbers.

Example 2Solve 10 × (5 + 10) using the distributive property.

Solution:  The whole numbers have following distributive properties: x × (y+z) = (x × y)+(x × z)

10 × (5 + 10) = (10 × 5) + (10 × 10)

= 50 + 100

= 150

This implies 10 × (5 + 10) = 150

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