I like the way Edwin (or should I say "W."?) brings gcd's into the picture in an illuminating way. One could argue that ideals provide the right way to think about all these questions, and that in particular one should say that a divides b iff the ideal generated by a contains the ideal generated by b. Jim Propp On Wednesday, February 11, 2015, W. Edwin Clark <wclark@mail.usf.edu> wrote:
Ask Mathematica to find the gcd of 0 and a or 0 and 0. I don't have Mathematica but I'll bet it gives gcd(0,0) = 0 and gcd(0,a) = a for any positive integer a.
A common definition of gcd(a,b) is that it is an integer d which divides a and b and for which there are integers x and y such that d = xa+yb. Thus if a = b = 0 then d = 0 and no other integer will do for d.
For a nice treatment of divisibility see Tom Apostol's treatment of divisibility in his book Introduction to Analytic Number Theory. A beautifully written book --- written after 25 years of teaching the subject.
As Apostol says: For integers a, b Definition: a | b if and only if b = ac for some integer c.
Thus Every integer divides 0 Only 0 divides 0
On Wed, Feb 11, 2015 at 6:51 PM, James Propp <jamespropp@gmail.com <javascript:;>> 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.
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