Whole Numbers: Definition, Properties and Examples

# Whole Numbers: Definition, Properties and Examples

## Whole Numbers

Whole numbers are a part of real numbers which includes positive integers with 0. Whole numbers do not include negative numbers, fractions, decimals or rational numbers. We see many numbers around us such as to represent money, to tell time, to measure temperature, to measure length, weight and capacity. In all these numbers, the counting numbers along with 0 are whole numbers.

## What are Whole Numbers?

When 0 is taken together with natural numbers, it forms a group of whole numbers.

We know that the natural numbers are the set of counting numbers starting from 1, 2, 3, … . On the other hand, the whole numbers are the numbers which includes 0 along with the counting numbers. We can say that the positive integers along with 0 are called whole numbers. Or the set of non-negative integers are known as the whole numbers. The main difference between the whole numbers and the natural numbers is the presence of 0 in whole numbers.

## Whole Numbers Definition

Whole numbers are defined as the group of all natural numbers along with 0 is called a group of whole numbers. It is written as: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ……….

## Examples of Whole Numbers

Some examples of whole numbers are as follows:

0, 67, 251, 8632, 26931, 792102, 4967303, ……….

## Whole Numbers Symbol

Whole numbers are denoted by the symbol, W.

## Set of Whole Numbers

The set of whole numbers is the collection of all the counting numbers along with 0. The set of whole numbers is denoted by the capital letter W. It is written as:

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ………. }

The set of whole numbers can be written in three forms.

1. Description form: In description form, the set of whole numbers is written as:

W = {all counting numbers along with 0}

2. Roaster form: In roaster form, the set of whole numbers is written as:

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …….}

3. Set builder form: In set builder form, the set of whole numbers is written as:

W = {x : x is a non-negative integer}

## Smallest and Largest Whole Numbers

The whole numbers start from 0. So, the smallest whole number is 0. What is the largest whole number? Can you find it?

Suppose I write a very large whole number, for example, 4663094810434086542145698869732190867284684762549577120374618459624

Can you say this is the largest whole number? Of course, no. Because there are many whole numbers greater than this number. Hence, if you write a very large whole number, there are many numbers larger than that number.

Thus, no largest whole number exists.

## Difference Between Whole Numbers and Natural Numbers

The whole numbers are the set of natural numbers including 0. They are written as: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …..

The natural numbers are the set of counting numbers starting from 1. They are written as: 1, 2, 3, 4, 5, 6, 7, 8, 9, …..

 Whole Numbers Natural Numbers The set of whole numbers is denoted by W and written as: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …….} The set of natural numbers is denoted by N and written as: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, ……} The smallest whole number is 0. The smallest natural number is 1. All the natural numbers are whole numbers. All the whole numbers are natural numbers, except 0.

## Whole Numbers on Number Line

Let us draw a ray and mark a point at its starting point. Label this point as 0. Make some more points on it at equal distances, on the right side of 0. Label these points as 1, 2, 3, 4, ... .

This gives you a number line for whole numbers. The number line extends endlessly in one direction.

On the number line for whole numbers, no number is marked to the left of 0. Every number on the number line is greater than any number on its left side.

## Properties of Whole Numbers

### Closure Property

The sum and the product of two or more whole numbers are always whole numbers. But the difference and the quotient of two whole numbers are not always whole numbers. In general, a + b = c and a × b = c. Where c is also a whole number.

Closure property of addition: 5 + 9 = 14; 3 + 8 = 11; 50 + 35 = 85; 80 + 25 = 105, etc.

Thus, the whole numbers are closed under addition.

Closure property of multiplication: 5 × 7 = 35; 6 × 8 = 48; 8 × 9 = 72; 5 × 9 = 45, etc.

Thus, the whole numbers are closed under addition and multiplication.

Closure property of subtraction: Let us take a few numbers from the group of numbers: 6, 9, 25, 7, 12, 18, 14 and subtract.

25 – 9 = 16

18 – 12 = 6

7 – 14 = ?

6 – 9 = ?

In the last two cases, the difference is not a whole number.  Thus, the whole numbers are not closed under subtraction.

Closure property of division: Take a few numbers from the group of numbers: 5, 8, 20, 9, 16, 18, 12 and divide.

20 ÷ 5 = 4

16 ÷ 8 = 2

12 ÷ 9 = ?

8 ÷ 5 = ?

In the last two cases, the quotients are not whole numbers. Therefore, the whole numbers are not closed under division.

### Commutative Property

The sum and the product of two whole numbers remains the same even if we change the order of the whole numbers. In general, a + b = b + a and a × b = b × a.

Commutative property of addition: 7 + 9 = 16 and 9 + 7 = 16. Thus, 7 + 9 = 9 + 7.

Commutative property of multiplication: 6 × 4 = 24 and 4 × 6 = 24. Thus, 6 × 4 = 4 × 6.

Thus, the whole numbers are commutative under addition and multiplication.

Commutative Property of Subtraction: Consider the numbers 64 and 35.

We know that 64 − 35 = 29.

But 35 − 64 ≠ 29

So, 64 − 35 ≠ 35 − 64

Hence, we can say that commutative property does not hold true for subtraction of whole numbers.

Commutative Property of Division: We know 16 ÷ 4 = 4, but 4 ÷16 is a fraction.

So, 16 ÷ 4 ≠ 4 ÷16

Hence, we can say that commutative property does not hold true for division.

### Associative Property

The sum and the product of three whole numbers remains the same even if we change the grouping of the numbers. In general, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.

Associative property of addition: 4 + (3 + 5) = 4 + 8 = 12 and (4 + 3) + 5 = 7 + 5 = 12. Thus, 4 + (3 + 5) = (4 + 3) + 5.

Associative property of multiplication: 5 × (2 × 6) = 5 × 12 = 60 and (5 × 2) × 6 = 10 × 6 = 60. Thus, 5 × (2 × 6) = (5 × 2) × 6.

Thus, the whole number are associative under addition and multiplication.

Associative Property of Subtraction: Let us consider an example.

(15 − 6) − 4 = 9 − 4 = 5

But 15 − (6 − 4) = 15 − 2 = 13

So, (15 − 6) − 4 ≠ 15 − (6 − 4)

Verify this result using some other numbers also. We can say that associative property is not satisfied for the subtraction of whole numbers.

Associative Property of Division: We know that (30 ÷ 6) ÷ 5 ≠ 30 ÷ (6 ÷ 5)

Verify this result with some other numbers also.

Thus, we can say that the associative property is not satisfied for the division of whole numbers.

12 + 0 = 12

45 + 0 = 45

134 + 0 = 134

This is true for all the whole numbers. If you add zero to any whole number, the value of the number will not change. So, zero is called the additive identity for whole numbers.

### Multiplicative Identity

8 × 1 = 8

67 × 1 = 67

156 × 1 = 156

This result is true for all whole numbers.

When a number is multiplied by 1, we get the same number as the product. 1 is called the multiplicative identity for whole numbers.

### Distributive Property

According to the distributive property of multiplication over addition and subtraction, a × (b + c) = a × b + a × c and a × (b – c) = a × b – a × c.

Distributive property of multiplication over addition:

6 × (3 + 5) = 6 × 3 + 6 × 5

6 × 8 = 18 + 30

48 = 48

Distributive property of multiplication over subtraction:

4 × (7 – 3) = 4 × 7 – 4 × 3

4 × 4 = 28 – 12

16 = 16

### Multiplication by Zero

23 × 0 = 0 and 0 × 23 = 0

We can see that 23 × 0 = 0 = 0 × 23. It is true for all whole numbers.

When we multiply a whole number by 0, the product is always 0.

### Dividing 0 by a Whole Number

If 0 is divided by a whole number, we get 0 as the quotient.

For example, 0 ÷ 5 = 0

Similarly, 0 ÷ 12 = 0 and 0 ÷ 100 = 0 and so on.

### Dividing a Whole Number by 1

If a whole number is divided by 1, the quotient is the whole number itself.

For example, 47 ÷ 1 = 47

Similarly, 146 ÷ 1 = 146 and 485 ÷ 1 = 485 and so on.

### Division by Zero

Division by 0 is not possible. So, division of a whole number by 0 is not defined.

## Solved Examples on Whole Numbers

Example 1: Fill in the blanks.

a. The smallest whole number is _____.

b. ______ is a whole number but not a natural number.

c. The first 5 whole numbers are _____________.

Solution:

a. The smallest whole number is 0.

b. 0 is a whole number but not a natural number.

c. The first 5 whole numbers are 0, 1, 2, 3, 4.

Example 2: Identify the whole numbers in the following numbers.

15, -4, 0, 7.2, 9.8, 1/3, 106, -5, 123, 545, ½, -3/8, 65, 4/7, 2156, 25007

Solution: Whole numbers are:

15, 0, 106, 123, 545, 65, 2156, 25007

Example 3: Which property holds true in the following statement:

5 + (8 + 6) = (5 + 8) + 6

Solution: In the given statement, the associative property of addition holds true.

Example 4: Which property holds true in the following statement:

25 × 14 = 14 × 25

Solution: In the given statement, the commutative property of multiplication holds true.

Related Topics:

What is an Addend in Maths

Minuend and subtrahend

Multiplicand and multiplier

Dividend, divisor, quotient and remainder

Natural numbers

Whole numbers

Properties of rational numbers

Are all integers rational numbers?

Find five rational numbers between 3/5 and 4/5