Number System – An Introduction

Number: Figures representing the quantity of any countable object or item is known as a number.

Types Of Numbers

Natural Number : Numbers used in counting  of any countable item are termed as natural numbers.
Ex. N={ 1, 2, 3, 4, 5, ....}

The smallest natural number is 1, while the greatest natural number is not possible because every natural number has its successor.

Whole Number: Natural numbers including  0 are known  as whole number

Ex. W={0, 1, 2, 3, 4, ......}

The smallest whole number is 0, while the greatest whole number is not possible because every natural number has its successor.

Prime number : A number which is not divisible by other numbers except the number itself and 1 is known as prime number

Ex. 2,3,5,7,11,13,17,19,23,........etc. are prime numbers.

The smallest prime number is 2 as 1 is not a prime number.

Determine whether 419 is a prime number or not?

Solution: The square root of 419 is 20.5 approx. The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19   419 is not divisible by any one of them , so 419 is a prime number.

Composite Numbers: Numbers other than one which have more than two factors and are not prime numbers are known as Composite numbers. 

Ex. 4, 6, 8, 9, 12

Even Numbers: Numbers which are divisible by 2 are known as even numbers.

Ex. 2, 4, 6, 8, 10.............. etc.

In assumptions even numbers are taken as (2x), (2x + 2), (2x + 4), (2x + 6), ........so on

Odd Numbers: The number which is not divisible by 2 are known as odd numbers.

Ex. 1, 3, 7, 9, 11.................. etc. 

In assumptions even numbers are taken as (2x + 1), (2x +  3), (2x + 5), (2x + 7), ........so on

Consecutive numbers: Numbers increasing by one are known as consecutive numbers.

 Ex. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

In assumptions even numbers are taken as (x), (x + 1), (x + 2), (x + 3), ........so on

Integers: The set of numbers which consists of Negative numbers and whole numbers is known as Integers.

I= { ....-5, -4, -3, -2, -1, 0, 1, 2, 3, 4..... } 

Rational Numbers: The numbers which can be written in the form P / Q , where P and Q are integers and Q not equal to zero is known as Rational Numbers.

Ex. (1/2), (3/7), (7/11).

Irrational Numbers: The numbers which cannot be written in ( P / Q) form are known as Irrational Numbers.

Real Numbers: Real number consists of both rational as well as irrational numbers with positive sign are known as real number.

“Series” another an important chapter which I want to discuss with my readers/ students.  There are mainly six types of series which are mainly and frequently asked in the competition examinations, which are as follows :-

Ø  Natural number series. Eg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ……………………….

Ø  Even natural numbers series. Eg. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ………….

Ø  Odd natural numbers series. Eg. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ……………

Ø  Square of natural number series. Eg. 1², 2², 3², 4², 5², ……………………….

Ø  Cube of natural numbers series. Eg. 1³, 2³, 3³, 4³, 5³, ………………………..

Ø  Miscellaneous series.

SERIES 1^st :- Natural Number Series :- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …………….

               Sum of n- natural numbers → 1 + 2 + 3 + 4 + 5 + 6 + 7 + ………. n terms

                                        Sum of n- natural numbers = [{n (n + 1)}/2]

                                                              Where n= number of terms in the given series

Condition :- The above formula is applicable only and only if the series starts from 1.

SERIES 2^nd   :- Even Natural Number Series :- 2, 4, 6, 8, 10, 12, 14,  …………….

               Sum of n- even natural numbers → 2 + 4 + 6 + 8 + 10  ………. n terms

                                        Sum of n- even natural numbers = n (n+1)

                                                              Where n= number of terms in the given series

                                                     n = [last term/2]

Condition :- The above formula is applicable only and only if the series starts from 2.

SERIES 3^rd  :- Odd Natural Number Series :- 1, 3, 5, 7, 9, 11, 13, 15,  …………….

               Sum of n- odd natural numbers → 1 + 3 + 5 + 7 + 9 +  ………. n terms

                                        Sum of n- odd natural numbers = n²

                                                              Where n= number of terms in the given series

                                                  n = [{Last term + 1}/2]

Condition :- The above formula is applicable only and only if the series starts from 1.

SERIES 4^th  :- Square of Natural Number Series :- 1², 2², 3², 4², 5², 6², 7², 8², …………….

        Sum of squares n- natural numbers → 1² + 2² + 3² + 4² + 5² + 6² + 7² + ………. n terms

                                        Sum of squares n- natural numbers =  [{n (n+1) (2n+1)}/6]

                                                              Where n= number of terms in the given series

Condition :- The above formula is applicable only and only if the series starts from 1.

SERIES 5^th  :-Cube of  Natural Number Series :- 1³, 2³, 3³, 4³, 5³, 6³, 7³, 8³,  …………….

     Sum of cubes of n- natural numbers → 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + ………. n terms

                                        Sum of n- natural numbers =[{n (n+1)}/2]²

                                                              Where n= number of terms in the given series

Condition :- The above formula is applicable only and only if the series starts from 1.

Hope you have enjoyed the base concept of this chapter till now. More Conceptual enjoyment with the Various Questions related to this chapter which are frequently asked in the examinations will be presented to your for self study on this blog soon, just keep visiting this blog regularly and writing me your related queries through comment or forum. Best of Luck for Exams. To be continued soon........................