MAIN WRITING PART 3-C

                              MAIN WRITING  PART 3-C


Point 4.- Case of Roots

  An implication that zero multiplied by zero does not result in zero is that zero does not have any numbers as a square root for there is no number with which, multiplied by another number equal to it, the multiplication results in zero. Zero does not have any numbers as a square root, as a fourth root, as a sixth root, etc, but does have a number as a cube root, as a fifth root, as a seventh root, etc for an odd quantity of factors zero does result in zero whereas an even quantity of factors zero results in every number except zero. 
  0x0x0x0 = = The set of all numbers except zero.
  0x0x0 = = 0
                                                                      
  0x0x0x0 = (0x0)x(0x0) = the set of all numbers except zero multiplied by the set of all numbers except zero = the set of all numbers except zero
                                      OR
0x0x0x0 = (0x0x0)x0 = 0x0 = the set of all numbers except zero.
  0x0x0 = (0x0)x0 = (the set of all numbers except zero)x0 = 0


Point 5.- Another important exception

  =  (for b or c different from zero).
 = but  is different from .

Point 6.-   Reasoning
 =  = 0
 =   = 0x0 = the set of all numbers except zero.


                           1/3 = 2/6
                           2/3 = 4/6 = 1/3+2/6
so a)  =  =  = . As we've already seen, zero does not have any numbers as a square root, as a fourth root, as a sixth root, etc.  =  =  empty set.  =  =  = = empty set multiplied by empty set.    ( symbol of empty set) So results in , which is equal to zero.  = . So = 0.
  b)=======.
                                 So=
                                      == 0x0 = the set of all numbers except zero.
So=  the set of all numbers different from zero.
  c)====.
                               =
                                    ==
So=.


THEREFORE empty set multiplied by empty set results in empty set as well as it results in every number, INCLUDING zero.
  ASTONISHED?? I think you shouldn't. This reasoning is only a theory for part b is based on the assumption that the associative law is applicable to factors which are empty set BUT there is a fact that seems to support this part (note that parts a and c do not need any additional support). In conventional Mathematics itself, there are the so called Imaginary Numbers. The product of two real numbers of the same sign is a positive number (+2 x +3 = +6, -2 x -3 = +6) but the product of two imaginary numbers of the same sign is a negative one (+2i x +3i = -6). These so called imaginary numbers are not real numbers (so they are not numbers at all). Therefore, couldn't they be some particular manifestations of empty set?