Talking about division or the verb divides without context is like discussing infinity without context. Division/divides arises in at least two situations: 1 a partially defined binary operation and 2 a relation. If one conflates these problems arise. 1. In a field we define only for b != 0, b^(-1) so that b b^(-1) = 1 and then for any a we have the partially defined binary operation a/b = ab^(-1). So in field theory a/0 is not defined. 2. In number theory we define the relation a | b if and only if b = ac for some c (in whatever ring we are dealing with). Leading to dealing with principal ideas, etc.) So we have a | 0 for all a and 0 | 0. Also in number theory we have the Theorem(Division Algorithm): For integers a and b with b > 0 there are integers q and r such that a = bq + r, with 0 <= r < b. In this case "division" by 0 is ruled out. Of course in analysis we have the usual business of "indeterminate forms" 0/0 which is yet another matter. One of my favorite remarks to students is that in mathematics to define a word we need a context. My favorite example is the word infinity. We can define what we mean by an infinite set. And we can define what we mean by the limit of a sequence f(n) is as n approaches infinity, and so forth. But who knows what infinity standing alone means. I think philosophy gets into a lot of trouble by picking up words and trying to imbue them meaning outside of a well define context. ----As in arguing over the mind-body problem. BTW how do you ask Mathematica, "does 0 divide 1"? WEC On Thu, Feb 12, 2015 at 1:35 AM, James Propp <jamespropp@gmail.com> wrote:
Actually, rereading Tom's earlier post I see that he's arguing for the truth status of 0|0 to be regarded as undefined, not false. Sorry for sowing confusion.
Jim
On Thursday, February 12, 2015, James Propp <jamespropp@gmail.com> wrote:
Tom's point seems to be that if we assume that the binary relation "divides" is supposed to have a direct relationship to the binary operation "divided by", with no exceptions, and 0/0 is undefined, then the truth value of "0 divides 0" should be FALSE.
But I don't want to get caught up in linguistic issues stemming from the fact that we use the word "divides" for the binary relation. For me the question is, if one is going to define a binary relation "|" (pronounced "gazinta", perhap!) that has the usual behavior when non-zero numbers appear to the left, what is the mathematically "nicest" way to extend it to zero? Or is it nicest not to extend it at all?
Jim Propp
On Wednesday, February 11, 2015, Tom Rokicki <rokicki@gmail.com <javascript:_e(%7B%7D,'cvml','rokicki@gmail.com');>> wrote:
If 0 divides 0, then it would logically follow that a=0/0 has a value.
If we assign any fixed value to 0/0, we can easily prove 1=2 and manner of other things.
If 0 divides 0, but 0/0 does *not* have a fixed value, then I fail to see what "divides" means.
On Wed, Feb 11, 2015 at 4:14 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
First of all, one must specify the integral domain R. a divides b (a|b) when there exists c in R with ac=b. If b=0, let c=0, so a|b. If b is nonzero and a=0, for no c is ac=b, so a does not divide b. This works uniformly for all elements in R, so there seems to be no reason to exclude 0. Does pi divide sqrt(2)? No, in the smallest domain containing these two numbers, but yes, in the real numbers. -- Gene
From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, February 11, 2015 3:51 PM Subject: [math-fun] Does zero divide zero?
Mathematica refuses to answer the question, and I've seen textbooks that duck the issue as well ("Suppose *a* and *b* are integers, with *a* nonzero. We say *a* divides *b* if and only if ...").
In fact, Mathematica also refuses to answer the question "Does 0 divide 1?"
Is it a standard convention that *a*-divides-*b* is a relation on (*Z*\{0}) x *Z*, so that asking whether 0 divides 1 is no more sensible than asking whether pi divides the square root of 2?
I would've been naively inclined to the view that 0 divides 0 is TRUE, while 0 divides 1 is FALSE.
I'm afraid to post this question to MathOverflow, lest I be reprimanded for asking such an inappropriate question ("Is this a homework problem that your professor assigned you?"). You guys are nicer, and more to the point, you all know me.
Jim Propp
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