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).

 
Challenge your math skills
Challenging Questions
  1. 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.

Natural Numbers are part of Whole Numbers

Both sets of numbers can be represented on the number line as follows:

Numberline showing whole and natural numbers

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

\[\begin{align}a \times (b-c) = a \times b - a \times c\end{align}  \]

For more information about the properties of whole numbers, click here.

 
Thinking out of the box
Think Tank
  1. 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?

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

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

  4. 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, \(7,2, \text{ and }0\).

 
important notes to remember
Important Notes
  1. \(0\) is a whole number but it is NOT a natural number.
  2. Negative numbers, fractions and decimals are neither natural numbers nor whole numbers unless they can be simplified as a natural number or whole number.
  3. \(W\) 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 \(0\).

 
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