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, the numbers are classified into different types.
The natural numbers are those numbers that are used for counting and ordering. The set of natural numbers is often denoted by the symbol
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, is closed under addition and multiplication . If a and b are any two natural numbers, then and is also a natural number. e.g. 7+2=9 and 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 is not closed under subtraction.
Similarly is also not closed under division.
(ii) Commutative property : Addition and multiplication of two natural numbers is commutative. If and are any two natural numbers, then, and .
Subtraction and division of two natural numbers is not commutative.
If a and b are any two natural numbers, then and .
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 and .
e.g. (a) and . Hence,
(b) and . Hence
Subtraction and division of natural numbers is not associative .
It means for any natural number a , b and c and
(iv)Identity element : The additive identity of a natural numbers is zero and multiplicative identity of narural numbers is 1. If a is any natural number, then and
(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.
All natural numbers together with ‘0’ are called whole numbers. The set of Whole numbers is denoted by W and written as
It includes all natural numbers , 0 and negative of natural numbers . It is denoted by ,
representation of integers on number line
Negative integers are on the left side of 0
Positive integers are on the right side of the zero
0 is neither +ve nor -ve.
The numbers of the form where and are integers and , are known as rational numbers. The collection of rational numbers is denoted by and is written as
Thus, , etc. are all rational numbers.
Rational numbers include natural numbers , whole numbers and integers since every natural numbers , whole numbers and integers can be written as
where is either any natual no. or whole number or integer.