What is Number?
A number is a fundamental unit of mathematics. it is used for counting, measuring objects, and different types of arithmetic calculations.
Numbers are expressed by means of figures- 1,2,3,4,5,6,7,8,9 and 0 called digits.
Out of these, 0 is called insignificant digit whereas the others are called significant digits.
A group of figures, representing a number, is called a numerals
Types of Numbers:
Numbers are divided into the following types:
(i) Natural numbers : The counting numbers 1, 2 , 3 ,4, - - - - called Natural Numbers. The set of natural numbers is denoted by N. Thus N = { 1, 2, 3, 4, - - -}
(ii) Whole numbers: Natural numbers including zero are called whole numbers.The set of whole numbers is denoted by W. Thus W = {0, 1, 2, - - -}.
(iii) Integers : The set of numbers which consists of whole numbers and negative numbers is known as a set of integers. The set of integers is denoted by I or Z.
Thus I (or Z) = {-4, -3, -2, -1, 0, 1, 2, 3, 4, - - - -}
Note: (a) Positive integers I⁺ = {1,2,3- - -}
= Natural numbers
(b) Negative integers I⁻ = {- - - -3, -2, -1}
(c) Non-negative integers(whole numbers) = {0, 1, 2, 3,- - - -}
(d) Non-positive integers = { -3, -2, -1, 0}
(iv) Even numbers : The number which is divisible by 2 is called even numbers.
It is also of the form 2n, where n = whole number. example: 2, 4, 6, 8, 10, - - -
(v) Odd numbers: The number which is not divisible by 2. example: 3, 5, 9, 11,.....,,
(vi)Prime numbers: Natural numbers which are divisible by 1 and itself only are called prime numbers. example: 2, 3, 5, 7, 11,13, 17, 19, 23,29 ......,
(Question: Is 881 a prime number?
Sol. The approximate square root of 881 is 30.
Prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
∵ 881 is not divisible by any of the above numbers, so 881 is a prime number.
(vii) Co- prime numbers: Two natural numbers are called co-prime, if there H.C.F ( Highest common factor) is one.
example: (1,2),(1,3),(3,4),(3,8),(3,10) etc.
These numbers are also called relatively prime numbers.
Note: (a) Two prime numbers are always co- prime but converse need not be true.
(b) Consecutive natural numbers are always co-prime numbers.
(viii) Twin prime numbers: If the diffference between two prime numbers is 2, then the numbers are called twin prime numbers. example: (3,5),(5,7),(11,13),(17,19)
Number between twin prime numbers is divisible by 6 ( except(3,5)).
(ix) Composite numbers: A number, other than 1, which is not a prime number is called a composite number.
example: 4, 6, 8, 9, 12, 14,...
Note: (a) 1 is neither a prime number nor a composite number.
(b) 4 is a smallest composite number.
(c) 2 is the only even prime number.
(x) Rational numbers: All the numbers that can be represented in the form p/q, where p and q are integers and q ≠ 0, are called rational numbers. The set of rational numbers is denoted by Q.
Thus Q = {p/q : p and q are integers and q≠0}.
It may be noted that every integer is a rational number since it cn be written as p/1.
All recurring(repeating) decimals are rational numbers.
(xi) Irrational numbers: The numbers which can not be expressed in p/q form where p, q are integers a7nd q ≠0.
A number is an irrational number, if it has a non- terminating and non- repeating decimal representation.
The set of irrational numbers is denoted by Qᶜ. example: √2, √3, 2+√2 etc.
Note: e ≈2.71 is called Napier's constant and π ≈ 3.14 are irrational numbers.
(xii) Real numbers : Numbers which can be expressed on number line are called real numbers.
The complete set of rational and irrational numbers is the set of real numbers. it is denoted by R. Thus R = Q U Qᶜ.
(xiii)Complex numbers: A number of the form a + ib is called a complex number, where a, b are set of real numbers, and
i = √-1. Complex number is usually denoted by Z.
