Please excuse my definitional rustiness, as I am well out of practice in this kind of thing. So, I will use wordy explanations in place of proper mathematical notation.
Essentially, I am using as a definition of "/" the division process, inverse multiplication "*" such that:
a * b = (a + a + ... + a) [with b items]
Domains:
a: Real numbers
b: Integer numbers
By using a linear interpolation between integer values of b, it is possible to extend the domain of b to the real numbers without requiring "/" in the definition.
From there I am defining "/" as the inverse process of "*" so that the following will apply:
a/b = c if a = b*c
With the exception of b = 0, because if zero can be applied to the "/" operator, then the following problem will occur:
a/0 = c
Therefore, by the definitional relation above (a/b = c if a = b*c):
a = c*0
Which is only true if a = 0, therefore, 0 is excluded from the "/" operator.
So for a shorter version of my definition:
a/b = c if a = b.c
With a domain of
a: Real numbers
b: Real numbers, excluding zero
That seems messy and incomplete, could you point to the part of the definition that clashes with your own.
Edit: The exclusion of zero in the "/" operation does have an equivalent exclusion in the "*" process, because it is the same as multiplying by infinity. However, as infinity is not a part of the real number set, it is not apparent in the domains.
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a/x = a* 1/x, but you can't do that for 0; cause 0 is the e (neutraal element, don't know the english word) for "+". 0/0 does NOT equal 0* 1/0 !!!
I agree that you cannot apply the "/" operator to 0, but I don't understand your reasoning. Why does the identity operator for "+" become an unacceptable choice for the inverse of "*"? Clearly it is not the case in reverse, as the identity of "*" is well suited to the "-" operator, and is an essential part of basic mathematical definition.
Also, I believe that
0/0 = 0 * 1/0
However, the 1/0 term should be treated as infinity, as the limit of 1/x as x approaches zero is +/- infinity. Zero times infinity is undefined, which is consistent with 0/0 being undefined.