On 2/11/2015 6:51 PM, James Propp 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.
(I have a strongly-held opinion about this, but I'll try to avoid raving too much.) The refusal to define "a divides b" when a = 0 is one symptom of a common phobia of operations on zero, operations on the empty set, and vacuous statements. Sufferers typically hesitate to define 0^0, to allow functions with empty domain, or to compute empty products or intersections. I have sometimes called this "nullophobia" for lack of a better name. It is an unfortunate habit of thought that makes many problems harder by requiring a more difficult base step for induction or by omitting an early entry in a sequence. To answer Jim's question, I believe "a divides b" is an exact synonym of "b is a multiple of a"; that is, there exists some c for which b = ac. This definition is simple and avoids needless exceptions. So, yes, zero divides zero and zero does not divide one (except in the zero ring, where 1 = 0). Opponents sometimes object, "But you can't divide by zero!" The answer is that I don't; division is never mentioned in the definition of "divides", only multiplication is. I think Knuth agrees on this, and therefore allows a == b (mod m) to be defined for any integer m, including zero (when congruence reduces to equality). He has also staunchly defended 0^0 = 1, pointing out that it is necessary to avoid exceptions to the binomial theorem. My favorite example of how embracing empty operations makes things simpler is the definition of a topology on a space X, which is just "a collection of subsets of X closed under arbitrary unions and finite intersections". Because arbitrary unions include the union of no sets, the empty set is in the topology; because finite intersections include the intersection of no subsets of X (which is X itself), X is in the topology; these two special cases do not need to be mentioned separately in the definition as they usually are. -- Fred W. Helenius fredh@ix.netcom.com