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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