TYPES OF NUMBERS

NATURAL NUMBERS :- The natural numbers are the counting numbers.
                                          { N = 1,2,3,...... }
  NOTE :-  Zero is not a natural number.

WHOLE NUMBERS :- The whole numbers are the counting numbers including zero(0).
                                      Thus  0,1,2,3,4,........ are whole numbers.
  NOTE :- 1. Zero is a whole number.
                 2. Every natural number is a whole number.

INTEGERS :- Integers are defined as -
                       All counting numbers together with negative of counting numbers.
                OR, set of whole numbers and their opposites are called integers.

 (A.)  POSITIVE INTEGERS :-              {1,2,3,.......}    are positive numbers.
 
 (B.)  NEGATIVE INTEGERS :-             {-1,-2,-3,.....}  are negative integers.

      *  Zero is neither positive nor negative, but in both.
          NON NEGATIVE INTEGERS :-          {0,1,2,3,4,.......}

          NON POSITIVE INTEGERS :-            {0,-1,-2,-3,-4,.......}

EVEN NUMBERS :- Even numbers are the integers that can be exactly divided by 2.
  For eg.   0,2,4,6,8,...  are even numbers.

ODD NUMBERS :- Any integer that cannot be exactly divided by 2 are called odd numbers.
                                For eg.   1,3,5,7,9,.......   are odd numbers.

PERFECT NUMBERS :- If  the sum of all the factors of a number is two times the number then the                                                      number is called a perfect number.
  For eg. (1.)  The factor of 6 are -    1,2,3 & 6
                      1+2+3+6 = 12                                   (sum all factors)
                      2 * 6 = 12                                          (two times the given number)
                     Hence 6 is a perfect number.

              (2.)  The factor of 28 are -    1,2,4,7,14 & 28
                      1+2+4+7+14+28 = 56
                      2 * 28 = 56
                     Hence 28 is a perfect number.

PRIME NUMBERS :- A prime number can be divided only by 1 and itself, and the number must be greater
                                    then 1.
  NOTE :- 2 is the only even number which is prime.

METHOD FOR TESTING WHETHER A GIVEN NUMBER IS PRIME  :-
                    Suppose q is a give number,

                    Find a whole number x such that  $x > \sqrt{q}$
                    Take all prime numbers less than or equal to x.
                    If no one of these divides q exactly, then we say that q is prime number,
                    otherwise q is not prime number.
       For eg.  Test whether 101 is prime or not.
             sol.                $x > \sqrt{101}$               {Find  x }
                                  $11 > \sqrt{101}$             {Find prime numbers upto 11 }
                    prime numbers upto 11 are 2,3,5,7,11
                    No one of them divides 101 exactly.
               $\therefore $ 101 is prime number.

COMPOSITE NUMBERS :- A composite number is any positive integer, greater then 1 that is not prime number.
  For eg.  4,6,8,9,........

CO - PRIMES:- A pair of natural numbers a and b, are said to be co-prime, if their HCF is 1.
  For eg.  (2,3), (4,5), (7,9) etc.

RATIONAL NUMBERS :-  Rational number is a number that can be put in the form $\frac{p}{q}$, where p & q are integers and q is not equal to 0.
                                             Here p is the numerator and q is the denominator.
  For eg.  $\frac{  3}{-4}$  is a rational number.
  NOTE :-  All natural numbers, whole numbers,integers and fractions are rational numbers.