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Laws valid for the positive rationals

   The positive rational numbers (numbers given by n+a/b where n, a 
   and b are positive integers) were the numbers used for centuries, 
   up to the time of the invention of 0.  What are the algebraic 
   rules that are followed by the positive rationals?  They obey 
   the following laws: 

   Let S be the set of all positive rational numbers.  For every 
   a,b,c in set S:

         1.  Closed under Addition. 
                 a,b in S implies a+b in S

         2.  Associative Law under Addition.
                  a+(b+c) = (a+b)+c
         5.  Commutative Law under Addition.
                  a+b = b+a

         6.  Cancellation Law under Addition.
                  a+x = b+x implies a=b

         7.  Closed under Multiplication.
                  a,b in S implies ab in S

         8.  Associative Law under Multiplication.
                  a(bc) = (ab)c

         9.  Identity Element under Multiplication.
                  1a =a1 = a

        10.  Inverse exists for every element under Multiplication.
                  aa' = a'a = 1    where a' is the inverse of a

        11.  Commutative Law under Multiplication.
                  ab = ba

        12.  Cancellation Law under Multiplication.
                   ax = bx implies a=b

        13.  Left Distributive Law -- Multiplication over Addition
                    a(b+c) = ab + ac

        14.  Right Distributive Law -- Multiplication over Addition
                    (a+b)c = ac + bc

   Note that there is no identity element under addition.  And no 
   inverse elements exist under addition.  To get an identity 
   element, zero had to be appended to the positive rationals.  To 
   get inverse elements, negative numbers had to be appended. 

   The reason the most of these laws hold is intuitively obvious 
   when viewed in the right way.  In the case of addition we can 
   think in terms of adding sets of objects.  In the case of 
   multiplication we can view the product "ab" as an area and 
   the product "abc" as a volume to confirm that the associative, 
   commutative, and distributive laws hold. 

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