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Here, we will learn the properties of whole numbers for basic arithmetic operations like addition, subtraction, multiplication, and division. The properties of whole numbers are given below.

**Properties of Addition****1. Closure Property**

Two whole numbers add up to give another whole number. This is the closure property of the whole numbers. It means that the whole numbers are closed under addition. If a and b are two whole numbers and** a + b = c**, then c is also a whole number. 3 + 4 = 7 (whole number).

**2. Commutative Property**

This property of the whole numbers tells that the order of addition does not change the value of the sum. Let a and b are two whole numbers, then **a+b = b+a****Example:** Suppose a = 10 and b = 18 ⇒ 10 + 18 = 28 = 18 + 10.

**3. Associative Property**

When we add three or more whole numbers, the value of the sum remains the same. The order of addition of numbers is not important. Or, in other words, the numbers can be grouped in any manner. The sum remains the same. This is the associative property of addition. If a, b, and c are three whole numbers, then **a + (b + c) = (a + b) + c = (a + c) + b****Example:** 10 + (5 + 12) = (10 + 5) + 12 = (10 + 12) + 5 = 27

**4. Additive Identity**

This is the property of zero by which the value of the whole number remains the same when added to any whole number. Zero is the additive identity of whole numbers. If w is a whole number, then **w + 0 = w = 0 + w**.**Example:** 0 + 7 = 7 = 7 + 0.**PROPERTIES OF SUBTRACTION**

**1. Closure Property**

When one whole number is subtracted from another, the difference is not always a whole number. This means that the whole numbers are not closed under subtraction. If a and b are two whole numbers and **a − b = c**, then c is not always a whole number. Take a = 7 and b = 5, a − b = 7 − 5 = 2 and b − a = 5 − 7 = −2 (not a whole number).

**2. Commutative Property**

Subtraction of two whole numbers is not commutative. This means we cannot subtract two whole numbers in any order and get the same result. Let a and b be two whole numbers, then **a − b ≠ b − a**. Take a = 7 and b = 5, 7 − 5 = 2 ≠ 5 − 7 = −2.

**3. Associative Property**

An associative property does not hold for the subtraction of whole numbers. This means that we cannot group any two whole numbers and subtract them first. Order of subtraction is an important factor. If ‘a’, ‘b’, and ‘c’ are the three whole numbers then, **a − (b − c) ≠ (a − b) − c**. Consider the case when a = 8, b = 5 and c = 2, 8 − (5 − 2) = 5 ≠ (8 − 5) − 2 = 1.

**4. Subtractive Property of Zero**

When we subtract zero from a whole number, the value of the whole number remains the same. Take an example, a = 98, a − 0 = 98 − 0 = 98.**PROPERTIES OF MULTIPLICATION**

**1. Closure Property**

Multiplication of two whole numbers will result in a whole number. Suppose, a and b are the two whole numbers and **a × b = c**, then c is also a whole number. Let a = 10, b = 5, 10 × 5 = 50 (whole number). The whole number is closed under multiplication.

**2. Commutative Property**

The value of the product does not change when the order of multiplication gets changed. This is the commutative property of multiplication. Let the two whole numbers be a and b, then **a × b = b × a** ⇒ 4 × 9 = 36 = 9 × 4.

**3. Associative Property**

When we multiply three or more whole numbers, the value of the product remains the same when they are grouped in any manner. The associative property of multiplication holds for whole numbers. Thus, if ‘a’, ‘b’, and ‘c’ are three whole numbers, then **a × (b × c) = (a × b) × c = (a × c) × b**. For example, 6 × (7 × 2) = (6 × 7) × 2 = (6 × 2) × 7 = 84.

**4. Multiplicative Identity**

When we multiply 1 with any whole number, the product is the number itself. 1 is the multiplicative identity of the whole numbers. If w is a whole number, then **w × 1 = 1 × w**.

**5. Multiplicative Property of Zero**

The product of a whole number and 0 is always 0 i.e., **w × 0 = 0 = 0 × w**. For example, 813 × 0 = 0 = 0 × 813.

**6. Distributive Property of Multiplication over Addition**

This property shows that multiplication of a whole number is distributed over the sum of the whole numbers. If a, b, and c are the three whole numbers. We have, **a × (b + c) = (a × b) + (a × c)**. Let a = 10, b = 20 and c = 5 ⇒ 10 × (20 + 5) = 250 and (10 × 20) + (10 × 5) = 200 + 50 = 250.

**7. Distributive Property of Multiplication over Subtraction**

This property tells that multiplication of a whole number is distributed over the difference of the whole numbers. Suppose ‘a’, ‘b’, and ‘c’ are three whole numbers. From this property we have,**a × (b − c) = (a × b) − (a × c)**. Let a = 10, b = 20 and c = 5 ⇒ 10 × (20 − 5) = 150 and (10 × 20) − (10 × 5) = 200 − 50 = 150.**PROPERTIES OF DIVISION**

**1. Closure Property**

The closure property of the division tells that the result of the division of two whole numbers is not always a whole number. Whole numbers are not closed under division i.e., a ÷ b is not always a whole number. From the property, we have, 14 ÷ 7 = 2 (whole number) but 7 ÷ 14 = ½ (not a whole number).

**2. Commutative Property**

Division of the whole numbers is not commutative. If a and b are the two whole numbers, then **a ÷ b ≠ b ÷ a**. Take an example of a = 14, b = 7, 14 ÷ 7 ≠ 7 ÷ 14.

**3. Associative Property**

The Associative property does not hold for the division of whole numbers. If ‘a’, ‘b’, and ‘c’ are the three whole numbers then, **a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c**

⇒ 100 ÷ (25 ÷ 5) = 20 ≠ (100 ÷ 25) ÷ 5 = 4 ÷ 5.

**Easy Way to Remember the Properties of Whole Numbers**

Whole numbers consists of all Natural numbers and the number zero.

W = {0,1,2,3...}

**Solved Example****Q 1: Multiply 24 × 15 by using a property.****Sol: **24 × 15 = 24 × (10 + 5) = 24 × 10 + 24 × 5 = 240 + 100 = 340.

**Q 2: Solve 121 × 18 − 121 × 8 by the distributive property.****Sol:** 121 × 18 − 121 × 8 = 121 × (18 − 8) = 121 × 10 = 1210.

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