Whole Numbers

Whole numbers are number that includes zero and other natural numbers. They are a subset of the real number and can be easily arranged on the number line. Thus, the whole numbers include all the positive numbers starting from zero till infinity. They are represented by the symbol W and include {0, 1, 2, 3,….}

Whole numbers are a part of real numbers, excluding decimal numbers, fractions, and negative numbers. Let’s learn in detail about whole numbers, along with examples.

What are Whole Numbers?

Whole numbers are natural numbers starting with 0. The numbers 0, 1, 2, 3, 4, 5, 6, and (so on) are whole numbers. It can be said that The whole number is a set of numbers without fractions, decimals, and negative numbers. It is a set of only positive numbers, including 0. The number smallest among the whole numbers is 0. Whole numbers are a part of real numbers just like natural numbers. However, the difference between whole numbers and natural numbers is that whole numbers contain all natural numbers and 0. This counts as the major difference between counting numbers and whole numbers. 0 is excluded from counting numbers.

In the image added below, we can clearly see that the measurements in the scale, the numbers in the lock and the numbers in the pressure gauge all are whole numbers.

Whole Number Definition

In mathematics, whole numbers are defined as a set of natural numbers, including 0. If the set of natural numbers is represented as N, then N = {1, 2, 3, 4, …} and if the set of whole numbers is represented as W, then whole numbers in sets will be, W = {0, 1, 2, 3, 4, …}. Integers are a set of both positive and negative numbers, but whole numbers are only the positive set, including 0. Below are some points that should be kept in mind regarding whole numbers,

Whole Number Symbol

The symbol to represent whole numbers is the alphabet ‘W’ in capital letters.

W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…}

Thus, the whole numbers list includes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ….

• All whole numbers come under real numbers.
• All natural numbers are whole numbers but not vice-versa.
• All positive integers, including 0, are whole numbers.

Smallest Whole Number

0 is the smallest whole number. The definition of a whole number says that the whole number generates from 0 and goes up to ∞. Therefore, 0 becomes the smallest whole number that exists. 0 is neither positive nor negative; it is used as a placeholder.

Whole Number Symbol

The set of whole numbers is represented by the symbol “W”. The W is the set representing the whole number and it includes all-natural numbers including zero.

W = {0, 1, 2, 3, 4, 5,…}

Whole Numbers on Number Line

Whole numbers on a number line can easily be observed as the number line, in the general case, represents integers from -∞ to +∞ , and whole numbers are positive integers, including 0. Looking at a number line, it can be observed that the integers lying on the right side of 0, including 0, are all whole numbers, and the integers lying on the right side of 0, excluding 0, are all natural numbers. This visual representation is the best way to conclude that all natural numbers are whole numbers but not vice versa.

Natural Numbers and Whole Numbers

Natural numbers and whole numbers are now known to us. It can be easily observed from the above number line that all natural numbers are whole numbers, the set of natural numbers is a subset of the whole numbers, and hence, the set of whole numbers W is the proper superset of the set of natural numbers N. Below diagram shows how the set of natural numbers, whole numbers, integers, rational numbers are related to each other.

Difference between Natural Numbers and Whole Numbers

Below is the table that contains the difference between natural numbers and whole numbers,

Natural Numbers	Whole Numbers
The smallest natural number is 1.	The smallest whole number is 0.
The set of natural numbers (N) is {1, 2, 3, …}.	The set of whole numbers (W) is {0, 1, 2, 3, …}
Every natural number is a whole number.	Every whole number is not a natural number.

Properties of Whole Numbers

The basic operation done in mathematics lead to the properties of the whole numbers, that is, properties based on addition, subtraction, multiplication, and division. The main properties of whole numbers are:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property

Closure Property

In this property, if x and y are whole numbers, then x + y is a whole number, and xy is also the whole number. The statement for the closure property is that the sum and the product of two whole numbers will always be a whole number. 

x + y = W

x × y = W

For example: Prove the closure property for 2 and 5. 

2 is a whole number, and 5 is a whole number. To prove the closure property, add and multiply 2 and 5.

2 + 5 = 7 (Whole number).

2 × 5 = 10 (Whole number).

Commutative Property of Addition

In the commutative property of addition, the sum of any two whole numbers is the same. i.e., the order of addition doesn’t matter. i.e., 

x + y = y + x

For Example: Prove the commutative property of addition for 5 and 8.

According to the commutative property of addition:

x + y = y + x

5 + 8 = 13

8 + 5 = 13

Therefore, 5 + 8 = 8 + 5

Commutative Property of Multiplication

In the commutative property of multiplication, the multiplication of any two whole numbers is the same. i.e., any number can be multiplied in any order. i.e.,   

x × y = y × x

For example: Prove the commutative property of multiplication for 9 and 0.

According to the commutative property of multiplication:

x + y = y + x

9 × 0 = 0

0 × 9 = 0

Therefore, 9 × 0 = 0 × 9

Additive Identity

In the additive property, When we add the value with zero, then the value of the integer remains unchanged. i.e., 

x + 0 = x

For example: Prove additive property for 7.

According to additive property

x + 0 = x

7 + 0 = 7

Hence, proved.

Multiplicative Identity

In the multiplicative property, When we multiply the value by 1, then the value of the integer remains unchanged. i.e., 

x × 1 = x

For example: Prove multiplicative property for 13.

According to multiplicative property:

x × 1 = x

13 × 1 = 13

Hence, proved.

Associative Property

In the associative property, when adding and multiplying the number and grouped together in any order, the value of the result remains the same. i.e.,

 x + (y + z) = (x + y) + z 

and 

x × (y × z) = (x × y) ×  z

For example: Prove the associative property of multiplication for the whole numbers 10, 2, and 5.

According to the associative property of multiplication:

x × (y × z) = (x × y) ×  z

10 × (2 × 5) = (10 × 2) × 5

10 × 10 = 20 × 5

100 = 100

Hence, Proved.

Distributive Property

In the distributive property, When multiplying the number and distributing them in any order, the value of the result remains the same. i.e., 

x × (y + z) = (x × y) + (x × z)

For example: Prove the distributive property for 3, 6, and 8.

According to the distributive property:

x × (y + z) = (x × y) + (x × z)

3 × (6 + 8) = (3 × 6) + (3 × 8)

3 × (14) = 18 + 24

42 = 42

Hence, Proved.

Multiplication by Zero

Multiplication of the zero is a special multiplication as multiplying any number by zero yields the result zero. i.e. a × 0 = 0.

Example: Find 238 × 0.

= 238 × 0

we know that multiplying any number yield the result zero.

= 0

Division by Zero

Division is the inverse operation of multiplication. But division by zero is undefined i.e. we can not divide any number by zero. i.e. 

a/0 is undefined

Can Whole Numbers be negative?

As we know that whole numbers include zero with the natural numbers and neither zero is negative nor natural numbers are negative. Thus, the whole number can never be negative.

The set of whole numbers W = { 0, 1, 2, 3, 4, 5,… } which does not hold any negative number.

Is 0 a whole number?

Yes, 0 (zero) is a whole number as a whole number includes zero with natural numbers. Thus zero is the first whole number and the set of the whole number starts from zero.

Whole Number Times Fractions

To multiply a whole number with fraction we follow the following steps.

Step 1: Write the whole number in form of fraction with a denominator of 1.

Step 2: Multiply the numerator of fraction with whole number.

Step 3: Multiply the denominator of fraction with whole number.

Step 4: Simplify if needed to get the required answer.

For example, simplify 3×4/3

= (3/1)×(4/3)

= (3×4)/(1×3)

= 12/3 = 4

Read, More

• Rational Numbers
• Irrational Numbers
• Complex Numbers

Solved Examples on Whole Numbers

Example 1: Are the numbers 100, 399, and 457 the whole numbers?

Solution:

Yes, the numbers 100, 399, 457 are the whole numbers.

Example 2: Solve the equation 15 × (10 + 5) using the distributive property.

Solution:

We know that distributive property are:

 x × (y + z) = x × y + x × z. 

So, 15 × 10 + 15 × 5 = 150 + 75 

= 225.

Example 3: Prove the associative property of multiplication for the whole numbers 1, 0, and 93.

Solution:

According to the associative property of multiplication:

x × (y × z) = (x × y) ×  z

1 × (0 × 93) = (1 × 0) × 93

1 × 0 = 0 × 93

0 = 0

Hence, Proved.

Example 4: Write down the number that does not belong to whole numbers:

4, 0, -99, 11.2, 45, 87.7, 53/4, 32.

Solution:

Out of the numbers mentioned above, it can easily be observed that 4, 0, 45, and 32 belong to whole numbers. Therefore, the numbers that do not belong to whole numbers are -99, 11.2, 87.7, and 53/4.

Example 5: Write 3 whole numbers occurring just before 10001.

Solution:

If the sequence of whole numbers are noticed, it can be observed that the whole numbers have a difference of 1 between any 2 numbers. Therefore, the whole numbers before 10001 will be: 10000, 9999, 9998.

Example 6: What is the mean of the first five whole numbers?

Solution:

The mean of the first five whole numbers is:

First five whole numbers = 0, 1, 2, 3, 4

Mean of first five whole numbers = (0 + 1 + 2 + 3 + 4)/5

= 10/5

= 2

Therefore, the mean of the first five whole numbers is 2.

Whole Numbers-FAQs

1. What are Whole Numbers with Examples?

The group of natural number including the number zero is called whole number. It is represented by the symbol ‘W’.

Example of whole number are, 0, 11, 23, 45, 25, etc.

2. Can a Whole Number ever be Negative?

No, a whole number can never be negative as the set of whole numbers “W” is represented as:

W = {0, 1, 2, 3, …}

Therefore, whole numbers do not contain negative numbers.

3. Are all Whole Numbers Real Numbers?

Yes, all whole numbers are real number. i.e. real number include whole number in themselves. But the opposite is not true i.e. all real number are not whole number.

4. What is the Smallest Whole Number?

As we know that whole number starts from 0 and goes to infinity. Thus, the smallest whole number is 0.

5. How many Whole Numbers are between 32 and 53?

The whole number between 32 and 59 are, 19 which include 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, and 52.

6. Is 11 a Whole Number?

Yes, 11 is a whole number.

6. Is 5 a Whole Number?

Yes, 5 is a whole number.