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Whole Numbers

Properties of Whole Numbers

Properties of Whole Numbers: We all are well aware with the definition of the whole numbers. It starts with 0 and has the set of all natural numbers in it. Let us do one interesting activity with whole numbers. Think of any whole number of your choice. Add it with any other whole number. What will you get? Is it a whole number?

Subtract the chosen whole number with any other whole number. You will again get a whole number. But not always. Multiplying and dividing a whole number with another one will give a whole number sometimes. These are some of the properties of whole numbers. Let us learn about the properties of whole numbers in detail.

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Properties of Whole Numbers

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

Properties of Addition

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).

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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. Suppose a = 10 and b = 18 ⇒ 10 + 18 = 28 = 18 + 10.

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. For example, 10 + (5 + 12) = (10 + 5) + 12 = (10 + 12) + 5 = 27

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. For example, 0 + 7 = 7 = 7 + 0.

Properties of Subtraction

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).

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.

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.

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

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.

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.

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.

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.

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.

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.

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

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).

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.

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

Properties of Whole Numbers

Solved Example for You

Question 1: Multiply 24 × 15 by using a property.
Answer : 24 × 15 = 24 × (10 + 5) = 24 × 10 + 24 × 5 = 240 + 100 = 340.

Question 2: Solve 121 × 18 − 121 × 8 by the distributive property.
Answer : 121 × 18 − 121 × 8 = 121 × (18 − 8) = 121 × 10 = 1210.

Question 3: What is the closure property of whole numbers?

Answer: This property of whole numbers states that when we add to whole numbers we get another whole number in its result. Also, according to it, this property closes the numbers under addition.

Question 4: Why are whole numbers important?

Answer:They are important because they are the easiest numbers to understand and use. Moreover, it provides children’s facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers.

Question 5: What is the formula of commutative property?

Answer: This property of multiplication tells us that it doesn’t matter in what order you multiply the numbers the answer will remain the same. However, the formula of commutative property is a  b = b  a. For example, on multiplying 6  5 or 5  6 we will end up with the same answer that is 30.

Question 6: What are the four properties of addition?

Answer: The four properties of addition that are part of mathematics are commutative, associative, identity, and distributive. Furthermore, the commutative property is when we add two numbers; the sum is the same regardless of the order of the addends.

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