**Natural numbers:**

In mathematics, the set of integers is the largest set of numbers which includes all the other set of numbers. Natural set of numbers contains all the positive integers from set of integers.

Also, natural numbers except zero all are also the whole numbers. We have to think about that why these numbers are named as natural numbers?

So the answer is the natural numbers are the counting numbers starts from 1, 2, 3, 4, 5, 6,… and these numbers are daily required in counting any objects or things in nature. Without these numbers nothing is possible to count in nature.

The set of natural number is denoted by N.

Hence, N = {1, 2, 3, 4, 5, 6, 7, 8, 9 …}

That means, next natural numbers is found by adding 1 to the previous natural number.

**For example:** 1 + 1= 2, 2+1 = 3, 3+1 = 4, 4+1 = 5 and so on.

In this way, we can find the natural numbers and which are up to infinity.

**For example:**

- If we defined a function as f(x) = N, where N ≤ 9 and N is the set of natural numbers.

Then the function f(x) has the values as given below.

Here, f(x) = N = 1, 2, 3, 4, 5, 6, 7 …

But, N ≤ 9 hence, f(x) = 1, 2, 3, 4, 5, 6, 7, 8, and 9.

- And the set of natural numbers is the proper subset of whole numbers and the set of whole numbers is the proper subset of set of integers.

- The set of natural numbers does not contains the zero element, negative integers, decimals and fractions. It contains only whole numbers except zero.
- All the properties satisfied by whole numbers are also satisfies the set of natural numbers.

**There are some properties of whole numbers which are explained as follows.**

1) Closure property

2) Associative property

3) Commutative property

4) Distributive property

**1) Closure property**

- If we add any two whole numbers then their sum is also the whole number.
- For example: 3&6 are the whole numbers then their addition 3+6=9, 9 is also whole number.
- Hence, closure property over addition is satisfied by set of whole numbers.

- Again, if we take subtraction of any two whole numbers then the answer we got is also the whole number.
- For example: 8&2 are the whole numbers, then 8 – 2=6 and 6 is also the whole number.
- Hence, closure property over subtraction is also satisfied by set of whole numbers.

- Also, if we multiply any two whole numbers then their multiplication is also the whole number.
- For example: 4&3 are the whole numbers, then 4*3= 12 and 12 is also the whole number.
- Hence the closure property over multiplication is also satisfied by the set of whole numbers.

**2) Associative property**:

The addition and multiplication is associative in case of set of whole numbers.

For example:

2,3&4 are the whole numbers then,

2+(3+4)=(2+3)+4

And 2*(3*4)=(2*3)*4

Hence, associative property over addition and multiplication is satisfied by set of whole numbers.

**3) Commutative property:**

If we reversed the order of whole numbers in case of addition and multiplication then the answer will not be changed.

For example:

4 & 5 are the whole numbers, then

4+5= 5+ 4 and 4*5= 5*4

Hence, we can say that commutative property over addition and multiplication is satisfied by the set of whole numbers.

**4) Distributive property:**

According to this property, the whole numbers are distributed over addition.

For example:

If we have taken the whole numbers as 2,3&4 then

2*(3+4)=2*3+2*4

Hence, we can say that whole numbers are distributive over addition.

__Additive identity:__

As set of whole numbers includes zero, hence zero is the additive identity because if in zero we add any whole number then we will get the same number.

For example:

5 &0 are the whole numbers then,

5 + 0 = 0 + 5 = 5

Hence, zero acts as additive identity in case of set of whole numbers.

__Multiplicative identity:__

The set of whole numbers includes one, and if we multiply any whole number by one then we will get the same whole number.

For example:

6 & 1 are the whole numbers then,

6*1 = 1*6 = 6

Hence, in a set of whole numbers one acts as multiplicative identity.