Whole Numbers - Definition | Examples - Cuemath

Whole Numbers

Whole numbers are a set of numbers including all natural numbers and 0. They are a part of real numbers that do not include fractions, decimals, or negative numbers. Counting numbers are also considered as whole numbers. Let us learn everything about whole numbers, the whole numbers definition, along with whole number examples in this article.

1.	What are Whole Numbers?
2.	Natural Numbers and Whole Numbers
3.	Whole Numbers on Number Line
4.	Properties of Whole Numbers
5.	FAQs on Whole Numbers

What are Whole Numbers?

Natural numbers along with zero (0) are referred to as whole numbers. We know that natural numbers refer to a set of counting numbers starting from 1, 2, 3, 4 and so on. In simple words, whole numbers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. Or we can say that whole numbers are the set of non-negative integers. The primary difference between natural and whole numbers is the presence of zero in the whole numbers set.

Whole Number Definition

Whole Numbers are those counting positive numbers that start from 0 and go on till infinity. For example, 0, 1, 2, 3, 4, and so on.

Set of Whole Numbers

The set of whole numbers includes the set of natural numbers along with the number 0. The set of whole numbers in mathematics is given as {0, 1, 2, 3, ...} which is denoted by the symbol W.

W = {0, 1, 2, 3, 4, …}

Here are some facts about whole numbers, which will help you understand them better:

• All natural numbers are whole numbers.
• All counting numbers are whole numbers.
• All positive integers including zero are whole numbers.
• All whole numbers are real numbers.
• Every whole number is a rational number.

Whole Number Symbol

The symbol used to represent whole numbers is the alphabet ‘W’ in the uppercase, W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

Smallest Whole Number

0 is the smallest whole number because whole numbers start from 0 (from the definition of whole numbers). Zero is a number that lies between the positive and negative numbers on a number line. Although zero carries no value, it is used as a placeholder. So, zero is neither a positive number nor a negative number.

List of Whole Numbers

The list of whole numbers starts from 0, 1, 2, 3, 4, 5, and goes on till infinity.

Natural Numbers and Whole Numbers

From the above definitions, we can understand that every whole number other than 0 is a natural number. Also, every natural number is a whole number. So, the set of natural numbers is a part of the set of whole numbers or a subset of whole numbers.

Difference Between Whole numbers and Natural Numbers

Let us understand the difference between natural numbers and whole numbers with the help of the table given below:

Whole Number	Natural Number
The set of whole numbers is, W = {0,1,2,3,...}.	The set of natural numbers is, N = {1,2,3,...}.
The smallest whole number is 0.	The smallest natural number is 1.
Every natural number is a whole number.	Every whole number is a natural number, except 0.

Whole Numbers on Number Line

The set of natural numbers and the set of whole numbers can be shown on the number line as given below. All the positive integers or the integers on the right-hand side of 0 represent the natural numbers, whereas all the positive integers along with zero, altogether represent the whole numbers. Both sets of numbers can be represented on the number line as follows:

Properties of Whole Numbers

The basic operations on whole numbers: addition, subtraction, multiplication, and division, lead to four main properties of whole numbers that are listed below:

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

Closure Property

The sum and product of two whole numbers is always a whole number. For example, 7 + 3 = 10 (whole number), 7 × 2 = 14 (whole number).

Associative Property

The sum or product of any three whole numbers remains the same even if the grouping of numbers is changed. For example, when we add the following numbers we get the same sum: 10 + (7 + 12) = (10 + 7) + 12 = 29. Similarly, when we multiply the following numbers we get the same product no matter how the numbers are grouped: 3 × (2 × 4) = (3 × 2) × 4 = 24.

Commutative Property

The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers. This property states that a change in the order of addition does not change the value of the sum. Let 'a' and 'b' be two whole numbers. According to the commutative property a + b = b + a. For example, a = 10 and b = 19 ⇒ 10 + 19 = 19 + 10 = 29 . This property also holds true for multiplication, but not for subtraction and division. For example: 7 × 9 = 63 and 9 × 7 = 63.

Additive Identity

When a whole number is added to 0, its value remains unchanged, i.e., if a is a whole number then a + 0 = 0 + a = a. For example, 3 + 0 = 3 + 0 = 3.

Multiplicative Identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if a is a whole number then a × 1 = a = 1 × a. For example. 4 × 1 = 1 × 4 = 4.

Distributive Property

This property states that the multiplication of a whole number is distributed over the sum or difference of the whole numbers. It means that when two numbers, for example, a and b are multiplied with the same number c, and are then added, then the sum of a and b can be multiplied by c to get the same answer. This property can be represented as: a × (b + c) = (a × b) + (a × c). Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 + 7) = 270 and (10 × 20) + (10 × 7) = 200 + 70 = 270. The same property is true for subtraction as well. For example, we have a × (b − c) = (a × b) − (a × c). Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 − 7) = 130 and (10 × 20) − (10 × 7) = 200 − 70 = 130.

Multiplication by Zero

When a whole number is multiplied by 0, the result is always 0, i.e., a × 0 = 0 × a = 0. For example, 4 × 0 = 0.

Division by Zero

The division of a whole number by 0 is not defined, i.e., if a is a whole number then a/0 is not defined.

For more information about the properties of whole numbers, click on the link provided.

Important Points related to Whole numbers

• 0 is a whole number and not a natural number.
• The first five whole numbers are 0, 1, 2, 3, 4.
• The smallest whole number is 0.
• Whole numbers do not include negative numbers, fractions, and decimals.

☛ Related Topics

• Rational Numbers
• Prime Numbers
• Composite Numbers
• Even Numbers
• Odd Numbers
• Real Numbers
• Natural Numbers
• Irrational Numbers
• Counting Numbers
• Cardinal Numbers

Cuemath is one of the world's leading math learning platforms that offers LIVE 1-to-1 online math classes for grades K-12. Our mission is to transform the way children learn math, to help them excel in school and competitive exams. Our expert tutors conduct 2 or more live classes per week, at a pace that matches the child's learning needs.