Number System will be introduced to the mathematics Fraternity gradually, and each new type comes with its own uses or due to complete the shortfall of the previous one and its own challenges.
The main types of numbers /Classification of Numbers  used in modern mathematics are such as:


Classification of Numbers
Classification of Numbers





Classification of Numbers

On the basis of Properties,  Classification of Numbers can be done in two ways according to my opinion - 

(A ) BASIC TYPES OF NUMBERS
  • Natural Numbers (N)
  • Whole Numbers (W)
  • Integers Numbers (I or Z)
  • Rational Numbers (Q)
  • Irrational Numbers 
  • Real Numbers (R)
  • Imaginary Numbers, Complex Number, etc
(B) OTHER STANDARD TYPES OF NUMBERS

  • Even Numbers 
  • Odd Numbers 
  • Prime Numbers 
  • Composite Numbers 
  • Perfect square Numbers 
  • Perfect cubic Numbers
  • Perfect  Numbers
  • Co-Prime Numbers
  • Discrete or continuous etc
(A ) Classification of Numbers - BASIC TYPES OF NUMBERS
  • Natural Numbers (N)
                 OR
       Counting Numbers
                 OR
        Positive integers 
A  number that generally occurs commonly and obviously in our surroundings or nature. From Ancient Time, first of all, Number is used for counting Fruits, Pebbles, Pet's, animals, or Balls, etc...              
The set of natural numbers Generally denoted by boldface “N” (The first letter of Natural Number, It is a general convention adopted in Mathematics).
it's set is written as
N= Set of Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8, …}
More preciously it can be defined as "Those numbers which start from 1 ( One ) and Gradually increased By STEP 1 are known as Natural Numbers or Counting Numbers."
Examples
One Orange
Classification of Numbers

Image source- Google|image by- 
food-healthy-orange-white-42059
 
Two Oranges
Classification of Numbers+  Classification of Numbers  =Two Oranges...
Three Oranges
Classification of Numbers Classification of NumbersClassification of Numbers

=  Three Oranges ...

    • Whole Numbers (W)
                       OR
           Complete Numbers              

    Those numbers that generally occur commonly and obviously in our surroundings or nature are known as Naturals, From Ancient Time of they are used for counting Fruits, Pebbles, pets, animals, or Ball's, etc... 
    If these are sufficient then what is the need for more extension in Naturals, Think ...???   
    Classification of Numbers
    Image source- Google|image by-ExplorerBob (pixabay.com)

     Classification of Numbers

      In the above picture according to the Natural Number concept, we have 2 ( Two ) water bottles (Glass).
               Let us assume if all the above Two water bottles (Glass) BROKEN then how many Number of Water bottles, We have left ...........???
    In Natural numbers we are not able to give an answer to the above problems, So the extension of Naturals is required.
    Mathematically it can be expressed such as:-
if we solve the following equation:- 
    x + 2 = 2                      (for every x belongs to N )
    x =  ????
    In naturals no solution Exists (possible) since ZERO (Null) does not belong to Natural Numbers, Therefore an Extension in Natural Number is required.
    Natural numbers extension along with ‘0’  ( Zero)are called Whole Numbers. 
    The set of Whole Numbers Generally  denoted by usual convention boldface 'W'The first letter of the Whole Number
     (It is a general convention adopted in Mathematics from ancient times).
    W = Set of Whole Numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, …}
    More preciously it can be defined as "Those numbers which start from 0 ( Zero ) and Gradually increased By STEP 1 are known as Whole Numbers."
    OR
    The whole numbers are the set or collection of positive integers and zero. 

  • Integers (I or Z)                     

    If Whole Numbers are sufficient then what is the need for more extension in Wholes, Think ...???   

         Classification of NumbersClassification of Numbers

    In the above picture according to the Number concept  (till Whole Numbers), we have 2 ( Two ) water bottles (Glass).
                 Let us assume if any  Nears & Dears ask to give three water bottles (Glass) to us then how many Number of Water bottles, We have left ...........???
        Up to concepts of Whole Numbers we are not able to give the answer to the above problems, So the extension of Whole Numbers is required and essentials.
          Mathematically it can be expressed such as:-
          if we solve the following equation:- 
              x + 5 = 2                      (for every x belongs to W )
              x =  ????
          In naturals, no solution Exists (possible), since Negative of Naturals (-3) is not belongs to Whole Numbers, Therefore Extension in Whole Numbers is required.
            Therefore Whole Numbers along with the negative of Natural numbers are called Integers. 
            The set of Integers Generally denoted by boldface ' I' The first letter of the Integer,  (It is a general convention adopted in Mathematics) but more popular Notation is boldface  " Z" ( It is the first Letter of Zahlen. It is a German word which means Numbers).
                Z or I = {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}
            More preciously it can be defined as "Those numbers which start from 0 ( Zero ) and Gradually increased By STEP 1 (one only)   on both sides (Positive and Negative) are called Integers."
                OR
            The integers are the set or collection of all whole numbers plus all negative of the Natural Numbers. 

            • Rational Numbers (Q).

            If Integers Numbers are sufficient then what is the need for more extensions in integers, Think ...???   
                    Classification of Numbers
                      Image source- Google|image by-
                      Petr Kratochvil


            In the given picture of an apple according to the Number concept  (till Integers Numbers), we have 1 Apple. Let us assume if we have to divide this apple into our Three friends also.  Then how many  Parts of Apple each friend got, ...........???Up to concepts of Integer Numbers we are not able to give the answer to the above problems, So the extension of Integers is also required and essentials.
              Mathematically it can be expressed in a similar manner such as:-
              if we solve the following equation:- 
                      2x+3=12          (for every x belongs to Z)
                      x =?
              In Integers no solution Exists (possible), since 9/2 fractions do not belong to Integers Numbers, Therefore Extension in IntegersNumbers is required.
                As per the famous Proverb "Necessity is the mother of Invention", so Rationals Discovered by Mathematicians to Solve Above Problems. Therefore Integers along with non zero fractions. In Mathematics, Rational numbers play an important role in Number System.
                The general conventions in the first learner about Rational Numbers are such as:-
                • Rational Numbers are simply quotients/fractions having Denominators not zero.
                • Rational Numbers can be obtained by dividing two integers.
                • The fraction with denominators  ( none zero)is a Rational number.
                • Rational Numbers contains all Natural Numbers, Whole Numbers, Integers.
                • Rational Numbers are expressed in the "Numerator /  Denominator"  form. 
                Although all the above conventions are considered as definition But  the definition on the basis of all above assumptions can be given in a more appropriate way as:

                  Rational Numbers are generally expressed in the form  'Numerator (p) /  Denominator (q) from ancient times, where p and q both are integers, and q is not equal to zero. 

                OR
                If we express in Decimal Fractions then Either Terminating Or Non-Terminating But RECURRING / Periodic / Repeating are known as Rational Numbers.
                Like all other Numbers, Rational numbers are usually denoted by a  Q (boldfaced) Where "Q" is taken from quotients.
                now a few examples are such as:
                2/3,  3/5, 4/7, 11/2, 303/12654, 0/1, 100/1, 51/ 9, -1/2, -2/3, -4/5, -11/2, - 2013/11111,1/2, 3/4, 17/2, 14/13, 41/1  ...
                In decimal representation can be written as 
                3.0, 1.25, 4.33..., 1.232323232323...., 4.1234,14.1234123412341234..., etc
                Few important properties of Rational Number are listed below:-
                1. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, -0/1, 0/2, 0/3, etc. 
                2. But it is important to keep in mind that -1/0, -15/0, 1/0, 2/0, 3/0, etc. are not rational. 
                3. Thus all the fractions where the top and bottom numbers are integers(Bottom number can not be zero); e.g., 2/3,  3/5, 4/7, 11/2, 303/12654, 0/1, 100/1, 51/ 9, -1/2, -2/3, -4/5, -11/2, - 2013/11111,1/2, 3/4, 17/2, ⁻14/13, 41/1  ...  are Rationals.
                4. The denominator of Rationals cannot be 0, but the numerator can be ( Important).
                5. Since the denominator may be equal to 1,  so every Natural Number is a rational number. ( 1/1, 2/1,3/1, etc )
                6. Since the denominator may be equal to 1,  so every Whole Number is a rational number. ( 0/1,1/1, 2/1,3/1, etc )
                7. Since the denominator may be equal to 1,  so every Integer is a rational number.( 1/1, 2/1,3/1, etc )

                • Irrational numbers 

                  If Rational Number is sufficient then what is the need for more extension in integers, Think ...???   
                  if we solve the following equation:- 
                        x =  ????
                  In Rationals no solution Exists (possible), since the root of three does not belong to Rational Numbers, Therefore Extension in Rational is required.
                  As per the famous Proverb "Necessity is the mother of Invention", so Irrationals Discovered by Mathematicians to Solve Above Problems. Therefore Rationals along with their positive roots.
                  Irrational Numbers are generally expressed in the form root of x from ancient times, where x is non-negative and not a perfect square Number. 

                                               OR
                  If we express in Decimal Fractions then NeitherTerminating Or Not RECURRING / Periodic / Repeating are known as Irrational Numbers.
                  Now a few examples in decimal expansions are such as:
                        1.31323334..., 0.30313234785..., 
                        0.70700700070000 ( See the Patterns)..., etc

                    • Real numbers ( R) 

                  • Union of Rational and Irrational Number is generally known as Real Numbers and denoted by as usual conventions by 'R'.
                  •  These numbers are in daily life used for Measurements so also called Measuring Numbers.
                  • Every Number as discussed above Like- Natural Number, Whole Number, Integer, Rationals, Irrationals are all subsets of real numbers.
                  • Every Real number can be represented on Number Lines with a unique representation.
                  (B) Classification of Numbers - OTHER STANDARD TYPES OF NUMBERS

                        • Even Numbers 

                    A whole number that can be exactly divisible by 2 ( Two) is known as Even Numbers or in other words, we divide by 2 then leaves remainder always zero.

                  • Even numbers always end with the last digits 0, 2, 4, 6, 8.
                  • Some examples are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ...etc
                  • Generally, we can say multiples of 2 are always even numbers.

                        • Odd Numbers 
                    A whole number that can be exactly not divisible by 2 ( Two) is known as Odd Numbers or in other words, we divide by 2 then leaves remainder always one (1).

                  • Odd numbers always end with last digits 1, 3, 5, 7, 9.
                  • Some examples are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 ...etc
                  • Generally, we can say multiples of 2 are cannot be odd numbers.
                  •  A number may be either Even or Odd not Both, so a natural number can be again redefined as a "Collection of all even and odd numbers".

                        • Prime Numbers 

                    Any number is said to be a prime number 'if it is a positive integer that has exactly two positive integral factors', 1 and itself.

                  • For example, if we list the factors of 14, we have 1, 2, 7, and 14. That's more than two (four) factors. So we say that 14 is not a prime number.
                  • If we consider another example 19, we only have 1 and 19. That's two factors. So we say that 19 is a prime number.
                  • we can also have a property that prime number is a positive integer that is not the product of two smaller positive integers.
                  • According to the definition of prime number 1 is not a prime number since it has only one factor.
                  • When a number can not be divided up exactly called Prime Numbers. For Example, 5 can not be divided into two equal groups so also Prime numbers.
                  • we can write a few prime numbers are such as:-
                    2 (Two), 3 (Three), 5 (five), 7 (Seven), 11 (Eleven), 13 ( Thirteen), 17, 19, 23, 29, 31, 37, 43, 41, 47, 53, 59, 61, 67, 73, 71, 79 (seventy Nine), 83, 89, 97, 101, 103, 107, 109, 113, 127, 137, 131, 139, 149, 151 (One hundred Fifty One), 157, 163, 167, 173, 179, 181, 191, 193, 197 ( One hundred Ninety seven), 199, 203, 207, 211, 213... etc.

                        • Composite Numbers

                    If any number has more than two factors then it is called a composite number. 

                  • For example, if we list the factors of 14, we have 1, 2, 7, and 14. That's more than two (four) factors. So we say that 14 is a composite number.
                  • If we consider another example 19, we only have 1 and 19. That's two factors. So we say that 19 is not a composite number.

                  • When a number can be divided up exactly called composite Numbers. For Example, 6 can be divided exactly into two groups of three's (6 = 3 + 3)  as well as two's ( 6 = 2+ 2 + 2) equal groups so 6 is composite numbers.
                  • One is neither Prime nor composite Numbers.
                  • We can write the few composite numbers are such as:- 4 (four), 6 (six) , 8 ( Eight), 9 (Nine), 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46,  49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70 (Seventy), 72, 74, 75, 76, 77, 78, 80, 81, 84, 82, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110 (One hundred Ten), 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128,  130, 132, 134, 136, 135, 133, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150 ( One Hundred Fifty), 151, 152, 154, 155, 156, 157... etc

                        • Perfect square Numbers 

                    The squares of the whole numbers are called perfect squares numbers.

                  •  We can write the few perfect square numbers are such as:- 0(Zero), 1 (One), 4 (four),  9 (Nine),16 (Sixteen), 25, 36, 49, 64, 81, 100, 121, 144 (One Hundred Forty Four), 169, 196, 225, 256, 289, …etc

                      • Perfect cubic Numbers 
                    The cubes of the whole numbers are called perfect cubic numbers.

                  •  We can write the few cubic numbers are such as- 0(Zero), 1 (One),   9 (Nine), 27 ( Twenty Seven), 64, 125, 216, 343 (Three Hundred Forty-Three), …etc

                        • Perfect  Number

                    A positive integer is said to be the perfect number that some of the proper divisor (Excluding given Number) equal to the number itself.

                  • Proper divisors of 6 are 1, 2, and 3, Now sum is 1 + 2 + 3 = 6 (Given number 6), So 6 is perfect Numbers.

                  • We can write a few perfect numbers are such as- 6, 28, 496, 8128...
                        • Co-Prime Number
                    Those numbers having only one common factor 1 (HCF = 1) are known as Co-Prime Numbers or Relatively Prime. 
                  • It is not essential that the Co-prime number must be Prime. 
                  • if we consider few examples like (1, 2), (1, 10), (1, 100), (5, 2), (3, 4), (5, 6), (19, 20) all are Co-Primes since their HCF = 1, although some of them are not Prime.

                  • Since by the definition of the co-prime number we know that "HCF of co-prime numbers is 1". Therefore LCM of given co-prime numbers is equal to the product of the co-prime numbers.
                  • We have an important property about co-prime numbers that "Co-prime Numbers LCM  = Product Of The co-prime Numbers".

                        • Discrete or continuous Numbers

                  • After the above discussion, all types of numbers can be split up into two groups discrete Numbers or continuous numbers.
                  • The first four of the above basis tyes of the number are Natural Numbers (N), Whole Numbers (W), Integers Numbers (I or Z), and Rational Numbers (Q) are considered as discrete numbers.
                  • It means that they are separate and distinct entities. Also, each of these sets is countable (those that can be counted are known as countable).
                  • The set of Irrational numbers (R) is also a discrete number but they can not be counted (Uncountable). 
                  • The set of Real numbers (R) is continuous so they can not be counted. 
                  We know that between any two real numbers,  there may be infinitely many real numbers that may be very close or dense.
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