__Properties of Natural Numbers__

__Properties of Natural Numbers__

**Properties of Natural Numbers**

** 1. Additive Properties (Properties due to Addition):**

** a) ****Closure Property of Addition: **The Set of Natural Numbers is closed with respect to the operation of Addition.

In general, it means that the sum of Natural Numbers is always a Natural Number.

This is known as Closure Property of Addition in the Set of Natural Numbers.

For example **3** and **5 **are two Natural Numbers, then **3** + **5** = **8** is also a natural Number.

** b) Commutative Law of Addition**: If **a **and **b** are two Natural Numbers, then

**a** + **b** = **b** + **a**.

This is known as Commutative Law of Addition in the Set of Natural Numbers.

For example: **3** and **5** are two Natural Numbers, then

**3** + **5** = **8**

&

**5** + **3** = **8**

Hence,

it is established that, **3** + **5** = **5** + **3**

** c) Associative Law of Addition**: If a, b and c are three Natural Numbers, then

**a** + (**b** + **c**) = (**a** + **b**) + **c**

This is known as Associative Law of Addition in the Set of Natural Numbers.

For example: **3**, **5** and **7** are three Natural Numbers then

** 3** + (**5** + **7**) = **3** + **12**

= **15**

&

(**3** + **5**) + **7** = **8** + **7**

= **15**

Hence,

it is established that, **3** + (**5** + **7**) = (**3** + **5**) + **7**

** 2. Multiplicative Properties (Properties due to Multiplications)**:

** a)** **Closure Property of Multiplication**:

The Set of Natural Numbers is closed with respect to the operation of Multiplication.

In general, it means that the Product of Natural Numbers is always a Natural Number

This is known as Closure Property of Multiplication in the Set of Natural Numbers.

For example **3** and **5** are two Natural Numbers,

then **3** × **5** = **15** is also a Natural Number.

** b) Commutative Law of Multiplication**:

If **a **and **b** are two Natural Numbers, then

** a** × **b** = **b** × **a**.

This is known as Commutative Law of Multiplication in the Set of Natural Numbers.

For example: **3** and **5** are two Natural Numbers, then

**3** × **5** = **15**

&

**5** × **3** = **15**

Hence, it is established that, **3** × **5** = **5** × **3**

** c) Associative Law of Multiplication**: If a, b and c are three Natural Numbers, the

**a** × (**b** × **c**) = (**a** × **b**) × **c**

This is known as Associative Law of Multiplication in the Set of Natural Numbers.

For example: **3**, **5** and **7** are three Natural Numbers then

**3** × (**5** × **7**) = **3** × **35**

= **105**

&

(**3** × **5**) × **7** = **15** × **7**

= **105**

Hence,

it is established that, **3** × (**5** × **7**) = (**3** × **5**) × **7**

** d) Existence of Multiplicative Identity**: For every Natural Number **a**, there exist a Natural Number **1 (one) **such that

**a **× **1 = 1 **× **a** = **a**

**1 (one)** is called the Multiplicative Identity in the Set of Natural Numbers

For example: **5** is a Natural Number, then

**5** × **1** = **5**

&

**1** × **5** = **5**

Hence, it is established that, **5** × **1** = **1** × **5** = **5**

**Also Read More**

*Comparative Study of POINT, LINE, LINE SEGMENT and RAY*

__Algebraical Formulae/Algebraical Identities__

Properties of Natural Numbers

**Fast Facts**:

*1. The Set of Natural Numbers is not closed with respect to the operation of Subtraction. The Difference of two Natural numbers is not always a Natural Number.** Illustrate yourself with suitable example.*

* *

*2. The Set of Natural Numbers is not closed with respect to the operation of Division. The Quotient of two Natural numbers is not always a Natural Number.** Illustrate yourself with suitable example.*