Introduction of Number System

Chapter -01 NUMBER SYSTEM

NATURAL NUMBERS (N) - Counting numbers are called natural numbers.

The collection of natural numbers is denoted by N and is written as N= (1, 2, 3, 4, 5, 6, ...)

REMARKS

(1) The least natural number is 1.

(2) There are infinitely many natural numbers.

WHOLE NUMBERS All natural numbers together with 0 form the collection W of all whole numbers, written as W=10, 1, 2, 3, 4, 5,...).

REMARKS

(i) The least whole number is 0.

(ii) There are infinitely many whole numbers.

(iii) Every natural number is a whole number.

(iv) All whole numbers are not natural numbers, as O is a whole number which is not a natural number.

INTEGERS All natural numbers, O and negatives of natural numbers form the collection of all integers. It is represented by Z after the German word 'zahlen' meaning 'to count'. Thus, we write Z=1,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, ...).

REMARKS

(1) 0 is neither negative nor positive.

(ii) There are infinitely many integers.

(iii) Every natural number is an integer.

(iv) Every whole number is an integer.

RATIONAL NUMBERS The numbers of the form p/q where p and q are integers and q is not equal to 0 are known as rational numbers. The collection of rational numbers is denoted by Q and is written as

Q= P 1≠0}. 19: p. q are integers, q ≠ 0)

‘Rational’ comes from the word ‘ratio’ and Q comes from the word ‘quotient’.

Thus, ¼, 3/2, 11/79, - 2001/2002 etc., are all rational numbers.

REMARKS

(i) There are infinitely many rational numbers.

(ii) There is no least or greatest rational number.

(iii) 0 is a rational number, since we can write, 0 = 0/1

(iv) Every natural number is a rational number since we can write, 1 = 1/1 2 = 2/1 3 = 3/1 etc.

(v) Every integer is a rational number since an integer a can be written a/1 . – 31 = - 31/1 0 = 0/1 and 79 = 79/1

Hence, rational numbers include natural numbers, whole numbers and integers.

SIMPLEST FORM OF A RATIONAL NUMBER A rational number p/q is said to be in its simplest form, if p and q are integers having no common factor other than 1 (that is, p and g are co-primes) and q not equal to 0 .