Number System

Number System is a study in which we study different types of numbers,their relationship and rules govern in them.

We use the symbols 0,1,2,3,4,5,6,7,8 and 9 in the Hindu-Arabic system.These symbols are called digits.

Types of Numbers

1.Natural numbers:

Numbers used for counting are called as Natural Numbers.They are denoted by N.

N={1,2,3,4,5,--------}

» All Natural numbers are positive and 1 is the smallest Natural number.

»  Natural numbers are also called as positive integers.

• Even Numbers:

  Numbers which are divisible by 2 is called as an Even numbers.

They are 2,4,6,8,10,12,----etc.

» The units place of every even number will be 0,2,4,6 or 8.

• Odd numbers:

  These numbers are not divisible by 2.

They are 1,3,5,7,9,11,13,--------etc.

» The units place of every odd number will be 1,3,5,7 or 9.

• Prime numbers:

  Numbers which are divisible by 1 and itself only are called prime numbers.

They are 2,3,5,7,11,13,17,------etc.

• 2 is the only even prime number and the smallest prime number.
• 1 is not a prime number.
• Every prime number greater than 3 can be represented by 6n+1,where n is an integer.

• Composite numbers:

  The numbers which have atleast one factor apart from 1 and itself (or) The numbers which are multiples of prime numbers are called Composite numbers.

They are 4,6,8,9,10,12,------etc.

• 1 is neither a prime number nor a composite number.

• Coprime numbers:

  The numbers having only 1 has a common factor are called Coprime numbers or Relatively prime numbers.

For example (3,8), (4,15), (5,12) and so on.

• Twin Prime numbers:

  If the difference of two prime numbers is 2 then they are called Twin prime numbers.
For example (3,5), (5,7), (11,13),---etc.

2.Whole Numbers:

Natural numbers along with Zero are called as Whole Numbers.They are denoted by W.

W={0,1,2,3,4,5,6,------}

»  0 is the smallest whole number.

»  Whole numbers are also called as Non negative integers.

3.Integers:

Whole numbers along with negative numbers are called as Integers.They are denoted by I or Z.

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

»  {1,2,3,4,----} are called as pasitive integers den style= "color:#ff33cc; border-left:8px solid red; padding:3px;"oted by I⁺.

»  {-1,-2,-3,-4,----} are called as negative integers denoted by I^-.

»  0 is neither nagative nor positive.

4.Rational Numbers (Q):

If p, q are integers and q≠0 then the numbers in the form of `p/q` are called Rational numbers.

Q = {x: x= `p/q` where p,q ε z , q ≠ 0}

• There is Infinite number of rational numbers in between two distinct rational numbers.
• A rational number between ‘a’ and ‘b’ =`(a+b)/2`
• Every rational number is expressible either in terminating or in non-terminating repeating decimal form.

• In a rational number, if the prime factorisation of denominator having prime factors either 2 or 5, then the rational number has terminating decimal expansion and vice versa.

  Example: In rational number the denominator 80 = 2 x 2 x 2 x 2 x 5 is having only 2 and 5 as factors. `17/80`= 0.2125 it is a terminating decimal.

• In a rational number, if the prime factorisation of denominator having prime factors other than 2 or 5, then the rational number has non-terminating recurring decimal expansion.

  Example: In rational number `11/12` the denominator 12 = 2 x 2 x 3 is having a factor other than 2.

  `11/12`= 0.91666666------------- it is non-terminating recurring decimal.

5.Irrational Numbers (Q^l):

The numbers that cannot be written in form, where p, q are integers and q≠0, are called irrational numbers.

Example: √2 , √5 , √12 , π, 0.10110111011110 … … , etc.

• Every irrational number is expressible in non-terminating and non-recurring decimal form.
• An irrational number between ‘a’ and ‘b’ = √ab

6.Real Numbers (R):

The union of rational and irrational numbers is called real numbers.𝑹 = 𝑸𝑼𝑸^l

• There are infinite number of real numbers in between two distinct real numbers.
• There are infinite number of rational and irrational numbers in between two distinct real Numbers.