Divide seems subordinate to multiply. In that we say "a divides b" for integers a, b) if there is an integer c such that b = c*a. Going by that definition, a = 0 divides all integers b, including itself. When b <> 0 itself, the only possible value of c is c = 0, of course. In the unique case of 0 dividing b for b = 0, the integer c is not unique. This is a problem for defining a single-valued binary operation of divide, needless to say, when a = b = 0. But it should not affect the answer to the question of whether 0 divides 0. --Dan
On Feb 11, 2015, at 3:51 PM, James Propp <jamespropp@gmail.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun