Whole Numbers
Whole numbers include natural numbers (that begin from 1 onwards), along with 0. Whole numbers are part of real numbers including all the positive integers and zero, but not the fractions, decimals, or negative numbers. Counting numbers are also considered as whole numbers. In this lesson, we will learn whole numbers and related concepts. In mathematics, the number system consists of all types of numbers, including natural numbers and whole numbers, prime numbers and composite numbers, integers, real numbers, and imaginary numbers, etc., which are all used to perform various calculations.
We see numbers everywhere around the world, for counting objects, for representing or exchanging money, for measuring the temperature, telling time, etc. There is almost nothing that doesn't involve numbers, be it match scores, for players not scoring any run, we say 0 runs, be it cooking recipes, counting on objects, etc. Whole Numbers is a set of numbers formed, including all positive integers and 0.
1.  What are Whole Numbers? 
2.  Whole Numbers vs Natural Numbers 
3.  Whole Numbers on Number Line 
4.  Properties of Whole Numbers 
5.  FAQs on Whole Numbers 
What are Whole Numbers?
Natural numbers refer to a set of positive integers and on the other hand, natural numbers along with zero(0) form a set, referred to as whole numbers. However, zero is an undefined identity that represents a null set or no result at all.
The whole numbers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. The primary difference between natural and whole numbers is zero.
Whole Number Definition:
Whole Numbers are the set of natural numbers along with the number 0. The set of whole numbers in Mathematics is the set {0, 1,2,3,...}.This set of whole numbers 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.
Whole Number Symbol
The symbol to represent whole numbers is the alphabet ‘W’ in capital letters, such as W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…
Smallest Whole Number
Whole numbers start from 0 (from the definition of whole numbers). Thus, 0 is the smallest whole number. The concept of zero was first defined by a Hindu astronomer and mathematician Brahmagupta in 628. In simple language, zero is a number that lies between the positive and negative numbers on a number line. As such zero carries no value, though it is used as a placeholder. So zero is neither a positive number nor a negative number, but it is an even number.
Whole Numbers Vs Natural 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's understand the difference between whole numbers and natural numbers through 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 righthand side of 0, represent the natural numbers, whereas all the positive integers and zero, altogether represent the whole numbers. Both sets of numbers can be represented on the number line as follows:
Properties of Whole Numbers
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. The closure property of W is stated as follows: For all a,b∈W: a+b∈W and a×b∈W
Associative Property
The sum or product of any three whole numbers remains the same though the grouping of numbers is changed.The associative property of W is stated as follows: For all a,b,c∈ W: a+{b+c}={a+b}+c and a ×{b×c}={a×b}×c. For example, 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29.
Commutative Property
The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers. The commutative property of W is stated as follows: For all a,b∈ W: a+b=b+a and a×b=b×a. This property states that change in the order of addition does not change the value of the sum. Let a and b be two whole numbers, commutative property states that a + b = b + a. For example, a = 10 and b = 19 ⇒ 10 + 19 = 29 = 19 + 10. It means that the whole numbers are closed under addition. This property also holds true for multiplication, but not for subtraction or division. For example: 7 x 9 = 63 or 9 x 7 = 63
Additive identity
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 = x = 1.x
Distributive Property
This property states that the multiplication of a whole number is distributed over the sum of the whole numbers. It means that when two numbers, take 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 situation 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 is true for subtraction as well. For e.g 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.The distributive property of multiplication over addition is a×(b+c)=a×b+a×c. The distributive property of multiplication over subtraction is a×(bc)=a×ba×c.
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.
For more information about the properties of whole numbers, click here.
Important Points
 0 is a whole number but it is NOT a natural number.
 Negative numbers, fractions, and decimals are neither natural numbers nor whole numbers unless they can be simplified as a natural number or whole number.
 W is closed, associative, and commutative under both addition and multiplication (but not under subtraction and division).
Solved Examples on Whole Numbers

Example 1: Identify the whole numbers among the following numbers ( 1, 0, 3, 1/2, 5).
Solution:
The set of whole numbers in mathematics is the set {0, 1, 2, 3, ...}. Therefore, among the given numbers, the whole numbers are, (0, 3, and 5).

Example 2: Is W closed under subtraction and division?
Solution:
Whole numbers include only the positive integers and zero. We know that on subtracting one positive integer by another, we may not get their difference as a positive integer, similarly, on dividing one positive number by another, we may not get the quotient as a defined number for example in the case of 13/0. Thus, for any two whole numbers, their difference and quotient obtained may not be whole numbers. Therefore, W is not closed under subtraction and division.

Example 3: For the whole number values of a, b, and c, that is, a=3, b=2, c=1, prove a×(b+c)=a×b+a×c
Solution:
a×(b+c) = 3×(2+1) = 3×3 = 9 and a×b+a×c = 3×2+3×1 = 6+3 = 9. Since, LHS=RHS, 9=9, thus, a×(b+c)=a×b+a×c, for the given whole number values. Also, this is distributive property of multiplication of whole numbers. Therefore, holds true for the given or other whole numbers.
Whole Numbers Practice Questions
FAQs on Whole Numbers
Can Whole Numbers be Negative?
The set of whole numbers in Mathematics is the set {0, 1, 2, 3, ...}. Hence, whole numbers cannot be negative.
What are the Properties of Whole Numbers?
Properties of whole numbers determine the operations on whole numbers and the set of whole numbers is closed, associative, and commutative under addition and multiplication.
What is a Natural Number?
A natural number is a number that is used for counting. The set of natural numbers in Mathematics is the set {1, 2, 3, ...}
With What Number do the Whole Numbers start from?
The whole numbers start with 0. And then it goes up to infinity.
What is the Symbol for the Set of Whole Numbers?
The set of whole numbers is denoted by the symbol W.
What are Some Examples of Whole Numbers?
The set of whole numbers in Mathematics is the set {0, 1, 2, 3, ...}. Some examples are 0, 15, 53, etc.
What is the Difference Between the Sets of Whole Numbers and Natural Numbers?
The only difference between whole numbers and natural numbers is that the set of whole numbers contains an extra number which is 0.
What are Whole Numbers in Math?.
Whole numbers in Math is the set of positive integers and 0. In other words, it is a set of natural numbers and 0. Decimals, fractions, negative integers are not part of this set. W=0,1,2,...