[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
Considered as a combinatorics question, zero divides zero because the empty set can be formed from copies of the empty set. It's much the same as the question of whether 0^0 = 1 (yes---there's the identity function from the empty set to itself). On Wed, 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
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I would say that division is an operation restricted to a non-zero second argument. As such, asking for a boolean answer to "does zero divide x" is nonsense, with neither a "true" nor a "false" result. Even when x is zero. Maybe it's like asking if the square root of -2 is even. On Wed, 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
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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
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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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> 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 <javascript:;>> 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 <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> 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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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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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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I actually withdraw my statement; I like the argument that "divides" is unrelated to "divide" (no matter how linguistically fraught this may be). Once we let this go, "0 divides 0" can probably be true. I'll let those with more breadth of knowledge illuminate this issue. On Wed, Feb 11, 2015 at 10:29 PM, 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> 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 <javascript:;>> 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 <javascript:;>> To: math-fun <math-fun@mailman.xmission.com <javascript:;>> 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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----- Once we let this go, "0 divides 0" can probably be true. I'll let those with more breadth of knowledge illuminate this issue. ----- Statements like this are impossible to interpret unless one starts from a definition of what "0 divides 0" *means*. We assume everything is in Z. If "a divides b" means b = a*c for a unique c, then 0 does not divide 0. If it just means b = a*c for at least one c, then 0 does divide 0. I think the authority in this case should be how algebraists define the term. But I'm not stuck on that, as long as you define the term before discussing it. --Dan
I suppose this question is subjective, but I would personally define 'a divides b' as: `the ideal <b> is a subset of the ideal <a>' in which case 0|0, but 0 doesn't divide any other integer. This seems to agree with the general intuition on this discussion thread, and also generalises to arbitrary integral domains.
I'm afraid to post this question to MathOverflow, lest I be reprimanded for asking such an inappropriate question
THIS QUESTION DOES NOT APPEAR TO BE ABOUT RESEARCH LEVEL MATHEMATICS WITHIN THE SCOPE DEFINED IN THE HELP CENTER!!!
* Adam P. Goucher <apgoucher@gmx.com> [Feb 12. 2015 14:08]:
I suppose this question is subjective, but I would personally define 'a divides b' as:
`the ideal <b> is a subset of the ideal <a>'
in which case 0|0, but 0 doesn't divide any other integer. This seems to agree with the general intuition on this discussion thread, and also generalises to arbitrary integral domains.
In the context of GCD one sets GCD(a,0) = a. Now the GCD is the intersection of the multi-sets of exponents in the canonical prime factorization, so 0 has to be 0 = prod(k>=1, prime(k)^\infty ). This gives the same conclusion. Running off infinitely fast, jj
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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
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
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> 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
Only 0 divides 0
0 divides only 0 , perhaps? WFL On 2/12/15, 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> 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
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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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think it's perfectly clear (except for resolution) what the answer is, just by examining this division fact sheet (top center): < http://edupress.com/edupress/searchNav/division/Math-in-a-Flashtrade-Noteboo... >. --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
Would that be "flashing" as in exposing? (Ignorance that is, rather than anything else better kept private ...) WFL On 2/13/15, Daniel Asimov <asimov@msri.org> wrote:
I think it's perfectly clear (except for resolution) what the answer is, just by examining this division fact sheet (top center): < http://edupress.com/edupress/searchNav/division/Math-in-a-Flashtrade-Noteboo...
.
--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
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participants (11)
-
Adam P. Goucher -
Dan Asimov -
Daniel Asimov -
Eugene Salamin -
Fred Lunnon -
Fred W. Helenius -
James Propp -
Joerg Arndt -
Mike Stay -
Tom Rokicki -
W. Edwin Clark