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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.

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• 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?

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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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

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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

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