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Various types of numbers: natural numbers, whole numbers, integers and rational numbers

Various types of numbers :

According to the properties and how they are represented in the number line, there are various types of numbers . The numbers are classified into different types, like natural number, integers, whole numbers etc.

Natural numbers

The natural numbers are those numbers that are used for counting and ordering. The set of natural numbers is often denoted by the symbol $\mathbb{N}$ , $\mathbb{N}=\left \{ 1,2,3,4,5,............ \right \}$

The least natural number is 1 and there are infinitely many natural numbers. They are located at the right side of the number line (after 0)

Properties of natural numbers:

(i) Closure property: The sum and multiplication of any two natural numbers is always a natural number. This is called “Closure property of addition and multiplication” of natural numbers. Thus, $\small \mathbb{N}$ is closed under addition and multiplication . If a and b are any two natural numbers, then $\small (a+b)$

and $\small ab$ is also a natural number. e.g. 7+2=9 and $\small 7\times 2=14$

is also a natural number.

The difference between any two natural numbers need not be a natural number.

Example : 3 – 5 = -2 is a not natural number. Hence $\small \mathbb{N}$ is not closed under subtraction.

Similarly $\small \mathbb{N}$

is also not closed under division.

(ii) Commutative property : Addition and multiplication of two natural numbers is commutative. If $\small a$ and $\small b$

are any two natural numbers, then, $\small a + b = b + a$ and $\small ab=ba$

Subtraction and division of two natural numbers is not commutative.

If a and b are any two natural numbers, then $\small (a - b) \neq (b - a)$ and $a \div b$ $\small \neq$

b \div a

e.g. (i) 5 – 3 = 2 and 3 – 5 = -2 . Hence 5 – 3 ≠ 3 – 5

(ii)2 ÷ 1 = 2 and 1 ÷ 2 = 1.5 . Hence 2 ÷ 1 ≠ 1 ÷ 2

Therefore, Commutative property is not true for subtraction and addition.

(iii) Associative property : Addition and multiplication of natural numbers is associative.

If a, b and c are any three natural numbers, then $\small a + (b + c) = (a + b) + c$ and $\small a \times (b \times c) = (a \times b) \times c$

e.g. (a) $\small 2 + (5 + 1) = 2 + (6) = 8$ and $\small (2 + 5) + 1 = (7) + 1 = 8$

. Hence, $\small 2 + (5 + 1) = (2 + 5) + 1$
(b) $\small 2\times (5\times 3)=30$

and $\small (2\times 5)\times 3=30$ . Hence $\small 2\times (5\times 3)=(2\times 5)\times 3$

Subtraction and division of natural numbers is not associative .

It means for any natural number a , b and c $a-(b-c)\neq (a-b)-c$ and $a \div (b \div c)$ $\small \neq$

(a \div b) \div c

(iv)Identity element : The additive identity of a natural numbers is zero and multiplicative identity of natrural numbers is 1. If a is any natural number, then $\small a+0= 0 + a = a$ and $\small a\times 1 = 1 \times a = a.$

(v)Distributive Property: Multiplication of natural numbers is distributive over addition and subtraction . If a, b and c are any three natural numbers, then a x (b + c) = ab + ac and a x (b – c) = ab – ac.