Properties of Whole Numbers

Properties of Whole Numbers

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Introduction

Whole numbers are the natural numbers along with the number \(0\).

The set of whole numbers in Mathematics is the set \(\textbf{ \{0,1,2,3,...\} } \).

The set of whole numbers is denoted by the symbol, \(W\).

W = \(\{0,1,2,3,4,5,...\}\)

The four properties of whole numbers are as follows:

  1. Closure Property
  2. Associative Property
  3. Commutative Property
  4. Distributive Property

Let's explore them in details.


1. Closure Property

  • Add any two whole numbers and you will see that the sum is again a whole number.

For example:

\[0+2 = 2\]

Here, \(2\) is a whole number.

  • In the same way, multiply any two whole numbers and you will see that the product is again a whole number.

For example:

\[3 \times 5 = 15\]

Here, \(15\) is a whole number.

Thus the set of whole numbers, \(W\) is closed under addition and multiplication.

The closure property of \(W\) is stated as follows:

For all \(a,b \in W\)

\[a + b \in W\\ {\text{  and  }} \\a \times b \in W\]

You can go ahead entering any two whole numbers in this simulation to see their sum and the product.

The sum and product are always whole numbers!

Interesting, isn’t it?


2. Associative Property

The sum and the product of any three whole numbers remain the same though the grouping of numbers is changed.

Example 1:

\[(1+2)+3 = 1+(2+3)\]

because

\[ \begin{align} (1+2)+3&=3+3=6 \\1+(2+3)&=1+5=6\end{align}  \]

Example 2:

\[  (1\times 2) \times 3 = 1 \times (2 \times 3) \]

because

\[\begin{align} (1\times 2) \times 3&= 2 \times 3 =6 \\1 \times (2 \times 3) &=1 \times 6 =6\end{align}  \]

Thus the set of whole numbers, \(W\) is associative under addition and multiplication.

The associative property of \(W\) is stated as follows:

For all \(a,b,c \in W\)

\[\begin{gathered}
  a + \left( {b + c} \right) = \left( {a + b} \right) + c \\
  {\text{and}} \\
  a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c \\
\end{gathered} \]

You can go ahead entering any three whole numbers and see their sum and the product found in two ways.

The sum and product are NOT changed even when the grouping of numbers is changed.

Have you noticed this?


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3. Commutative Property

The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers.

Example 1:

\[2 +3=3+2 \]

because

\[\begin{align}  2+3&=5\\3+2&=5\end{align}  \]

Example 2:

\[ 2\times 3= 3 \times 2  \]

because

\[\begin{align} 2 \times 3 &=6\\3 \times 2 &=6\end{align}  \]

Thus the set of whole numbers, \(W\) is commutative under addition and multiplication.

The commutative property of \(W\) is stated as follows:

For all \(a,b \in W\)

\[\begin{gathered}
  a + b = b + a \\
  {\text{and}} \\
  a \times b = b \times a \\
\end{gathered} \]

You can go ahead entering any two whole numbers in this simulation to see their sum and the product found in two ways.

The sum and product are NOT changed even when the numbers are interchanged.

Have you observed this?

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

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

Example 1:

\[3 \times (2+5) = 3 \times 2 + 3 \times 5\]

because

\[\begin{align}\ 3 \times (2+5) &= 3 \times 7 =21 \\3 \times 2 + 3 \times 5 &=6+15=21\end{align}  \]

The distributive property of multiplication over subtraction is

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

Example 2:

\[3 \times (2-5) = 3 \times 2 - 3 \times 5\]

because

\[\begin{align} 3 \times (2-5) &= 3 \times (-3) =-9 \\3 \times 2 - 3 \times 5 &=6-15=-9\end{align}  \]

 
Thinking out of the box
Think Tank
  1. Is \(W\) closed under subtraction and division?

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

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


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Solved Examples

Example 1



 

The set of whole numbers is closed under which of the operations?

  1. Addition
  2. Subtraction
  3. Multiplication
  4. Division

Solution:

If we assume any two whole numbers, their sum and the product are also the whole numbers.

But their product and quotient don't need to be the whole numbers.

For example, \(1\) and \(2\) are whole numbers. 

\[1-2=-1\\1 \div 2 =0.5\]

Here, the difference and the quotient are NOT whole numbers.

So the set of whole numbers is closed only under addition and multiplication.

So the answers are:

Options 1 and 3
Example 2



 

Find the following product using the distributive property.

\[72 \times 45 \]

Solution:

By using the distributive property, we can write the given product as follows:

\[\begin{align}72 \!\times\! 45 &\!=\! \!\left(70\!+\!2\right)\! \times \!\left(40\!+\!5\right)\! \\[0.3cm]
&\!=\!70 \!\times\! 40 \!+\! 70 \!\times\! 5 \!+\! 2 \!\times\! 40 \!+\! 2 \!\times\! 5 \\[0.3cm]
&\!=\!-2800 \!+\! 350 \!+\! 80 \!+\! 10\\[0.3cm]
&\!=\!3240\end{align}\]

\[72 \times 45 =3240\]
 
Challenge your math skills
Challenging Questions
 
 
 
 
 
 
  1. Find the product using the distributive property:
    \[28 \times 75\]
  2. The set of whole numbers is commutative under which of the operations?

    (a) Addition (b) Subtraction
    (c) Multiplication (d) Division

Practice Questions

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is the closure property of whole numbers?

The closure property of \(W\) is stated as follows:

For all \(a,b \in W\)

\[a + b \in W\\ {\text{  and  }} \\a \times b \in W\]

For more information, you can refer to the "Closure Property" section of this page.

2. Which property states that the sum of any two whole numbers is always a whole number?

The closure property of whole numbers states that the sum of two whole numbers is always a whole number.

For more information, you can refer to the "Closure Property" section of this page.

  
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