Consider the following two functions, that we can define for any field F:
• Our first function f(x, y) is defined for all x in F, and all nonzero y in F. That is, the domain of this function f is (F × F\{0}). The definition of this function f is:
f(x, y) = x times the multiplicative inverse of y
(Note that in the definition of f(x,y) we could say "if y≠0", but whether we say it or not the meaning is the same, because the domain of f already requires that y≠0.)
• Our second function g(x, y) is defined for all x in F, and all y in F. That is, the domain of this function g is (F × F). To define this function g, we pick a constant C in F, say C=0 or C=1 (or any C∈F, e.g. C=17 if 17 is an element of our field). And having picked C, the definition of this function g is:
• g(x, y) = x times the multiplicative inverse of y, if y ≠ 0, and C otherwise [that is, g(x, 0) = C].
Note that this function is defined for all (x, y) in F×F, and when y≠0 it agrees with f, i.e. g(x,y) = f(x,y) when y≠0.
Would you agree that both of these are functions, on different domains?
Next, we have the notation x/y. To assign meaning to this notation (and to the word “division”), there are two conventions we could adopt:
• Convention 1: When we say "x/y", we will mean f(x,y) (as defined above) — that is, x * y^{-1}.
• Convention 2: When we say "x/y", we will mean g(x,y) (as defined above).
The point of the post is that we can well adopt Convention 2: with such a definition of "division", all the properties that were true of f (on the domain F×F\{0}) continue to be true of g, except that g is defined on a wider domain.
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Now, maybe Convention 2 offends you. Maybe you think there is something very sacred about Convention 1. In all your posts in this thread, you seem to be insisting that "division" or "/" necessarily have to mean Convention 1, and the provided justification for preferring Convention 1 seems circular to me — here are some of your relevant comments, with my comments in [square brackets]:
> Since there is no multiplicative inverse of 0, division by 0 is undefined behavior [This seems to be saying: Because of Convention 1, we cannot adopt Convention 2.]
> Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication [Here, your reasoning is "because we adopt Convention 1"]
> in algebraic fields division by x is equivalent to multiplication by 1/x. This is precisely why you cannot have a field that admits division by 0: because 0 has no multiplicative inverse. [Again, you're stating that Convention 1 has to hold, by fiat.]
> If F is a field with elements x, y, then the quotient x/y is equivalent to the product x(1/y). If 0 has no multiplicative inverse, there is no division by 0. The two concepts are one and the same [This is just stating repeatedly that we have to adopt Convention 1.]
> A multiplicative inverse is a division. [This is merely insisting that Convention 1 has to be adopted, not Convention 2]
> divisor cannot exist unless it is a multiplicative inverse [Again, stating Convention 1.]
> you can't look at the field axioms, observe that 0 has no multiplicative inverse, then proceed to define a special, one-off division rule that doesn't involve multiplicative inverses for that one element. [Why not?] ...you've introduced a division rule which is just a syntactical sugar [yes the same is true of Convention 1; what's the problem?]
All these comments, which emphatically insist on Convention 1, seem to ignore the point of the article, which is that Convention 2 has no more mathematical problems than Convention 1, because the function g is no less a “valid” function than the function f.
In mathematics when we use words like “obvious” or insist that something is true because it just has to be true, that's usually a hint that we may need to step back and consider whether what we're saying is really mathematically justified. What we have here is a case of multiple valid definitions that we can adopt, and there's no mathematical reason for not adopting one over the other. (There's a non-mathematical reason, namely “it breaks convention”, but the entire point is that we can break this convention.)