Whole Numbers
Definition of a Natural Number
Natural numbers are the numbers that are used for counting.
The set of natural numbers in Mathematics is the set \(\textbf {1,2,3,...}\)
This set is denoted by the symbol \(N\).
N = \(\{1,2,3,4,5,...\}\) 
Thus, the smallest natural number is \(1\).
Definition of a Whole Number
Whole Numbers are the set of natural numbers along with the number \(0\).
The set of whole numbers in Mathematics is the set \(\textbf {0, 1,2,3,...}\)
This set of whole numbers is denoted by the symbol \(W\).
W = \(\{0,1,2,3,4…\}\) 
In this set, the linear equation \(x + 2 = 2\) is solvable because the solution of this equation is \(0\), which is a whole number.
Thus, going from the set \(N\) to the set \({\text{W}}\) has increased the solvability of equations.
Every equation which is solvable in \(N\) is solvable in \({\text{W}}\),
However, some equations which were not solvable in \(N\) become solvable in \({\text{W}}\).
Whole Numbers Start from?
Whole numbers start from \(0\) (from the definition of whole numbers).
 Which of the following statement is true?
a) Every natural number is a whole number.
b) Every whole number is a natural 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.
Both sets of numbers can be represented on the number line as follows:
Comparison between Whole numbers and Natural numbers
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 \(1\) 
We cannot determine the largest whole number  We cannot determine the largest natural number 
Each whole number is obtained by adding \(1\) to its previous number  Each natural number is obtained by adding \(1\) to its previous number 
\(0\) is a whole number  \(0\) is NOT a natural number 
Properties of Whole Numbers
1. 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 \(a,b \in W\)
\[a + b \in W\\ {\text{ and }} \\a \times b \in W\] 
2. 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 \(a,b,c \in W\)
\[\begin{gathered} a + \left( {b + c} \right) = \left( {a + b} \right) + c \\ {\text{and}} \\ a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c \\ \end{gathered} \] 
3. 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 \(a,b \in W\)
\[\begin{gathered} a + b = b + a \\ {\text{and}} \\ a \times b = b \times a \\ \end{gathered} \] 
Let us summarise these three properties of whole numbers in a table.
Operation  Closure Property  Associative Property  Commutative Property 

Addition  yes  yes  yes 
Subtraction  no  no  no 
Multiplication  yes  yes  yes 
Division  no  no  no 
4. Distributive Property:
The distributive property of multiplication over addition is
\[\begin{align}a \times (b+c) &= a \times b + a \times c\end{align} \] 
The distributive property of multiplication over subtraction is
\[\begin{align}a \times (bc) = a \times b  a \times c\end{align} \] 
For more information about the properties of whole numbers, click here.
 Consider the equation \(x + 2 = 1\)
We see that this equation is not solvable in \({\text{W}}\). In which number system will it be solvable?

Is \(W\) closed under subtraction and division?

Is \(W\) associative under subtraction and division?

Is \(W\) commutative under subtraction and division?
Solved Examples
Example 1 
Identify the whole numbers among the following numbers.
\( 1, 0, 3, \dfrac{1}{2}, 5\).
Solution:
The set of whole numbers in Mathematics is the set \(\textbf {0, 1, 2, 3, ...}\)
So, among the given numbers, the whole numbers are, \(0, 3 \text{ and }5\).
Example 2 
Identify the whole numbers among the following numbers.
\( 7, 2, 3, \dfrac{3}{5}, 0\).
Solution:
The set of whole numbers in Mathematics is the set \(\textbf {0, 1, 2, 3, ...}\)
So, among the given numbers, the whole numbers are, \(7,2, \text{ and }0\).
 \(0\) 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).
Practice Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Important Topics
Given below are the list of topics that are closely connected to whole numbers. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
Frequently Asked Questions (FAQs)
1. Define whole numbers.
Whole numbers in Maths are \(0, 1, 2, ...\infty\).
2. Can whole numbers be negative?
The set of whole numbers in Mathematics is the set \(\textbf {0, 1, 2, 3, ...}\)
Hence, whole numbers cannot be negative.
3. What are the properties of whole numbers?
The set of whole numbers is closed, associative, and commutative under addition and multiplication.
You can refer to the section “Properties of Whole Numbers” on this page for the same.
4. 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 \(\textbf {1, 2, 3, ...}\)
5. With what numbers do the whole numbers start from?
The whole numbers start with \(0\).
6. What is the symbol for the set of whole numbers?
The set of whole numbers is denoted by the symbol \(W\).
7. What are some examples of whole numbers?
The set of whole numbers in Mathematics is the set \(\textbf {0, 1, 2, 3, ...}\)
Some examples are be \(0, 15, 53, \) etc.
8. 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 \(0\).