Re: GCD(0,0)
- From: quasi <quasi@xxxxxxxx>
- Date: Sun, 01 Jan 2006 03:11:08 -0500
On 31 Dec 2005 21:12:16 GMT, Marc Olschok <invalid@xxxxxxxxxxx> wrote:
>quasi <quasi@xxxxxxxx> wrote:
>> On 31 Dec 2005 04:18:07 -0800, Rufus.Zee@xxxxxxxxx wrote:
>>
>> >> Rufus....@xxxxxxxxx wrote:
>> >> > It's fair to argue that GCD(0,0) shouldn't be defined because 0 doesn't
>> >> > divide anything.
>> >
>> >> Yes, it's fair, but it's also incorrect. The GCD(a,b) involves numbers
>> >> which divide into a and b, not numbers that a and b divide into.
>> >
>> >GCD stand for greatest common divisor; its generally agreed upon that
>> >if we are to define GCD(0,0) to be anything, we will define it to be
>> >zero;
>>
>> My choice would be to leave it undefined, but I agree -- if we are
>> forced to define a value for gcd(0,0), the only possible value that
>> makes any sense for it is 0. Well, alternatively, we could define
>> gcd(0,0) to be the symbol infinity.
>>
>> >so what I was saying that a fair objection to this was to object
>> >to calling 0 a divisor of something.
>>
>> Very fair.
>>
>> Objection sustained.
>>
>> >You thought I was getting my 0 from the a or b inputs, rather the 0 i was referring to was the output.
>>
>> Too many zeros -- there ought be a law.
>>
>> "Now, it's all been done before,
>> It's all been written in the book,
>> But when there's too much of nothing,
>> Nobody should look."
>>
>> [Bob Dylan]
>>
>> >> > >Rufus....@xxxxxxxxx wrote:
>> >> > >> It's fair to argue that GCD(0,0) shouldn't be defined because 0 doesn't
>> >> > >> divide anything.
>> >
>> >> > >Yes, it's fair, but it's also incorrect.
>> >
>> >> > References?
>> >
>> >> You really need a reference for "zero doesn't divide evenly into any
>> >> integer (except zero)" ? I guess you could call it "Heckman's Third
>> >> Theorem":
>> >
>> >i believe he asked for references because you seemed to be claiming
>> >that zero divided into something when you contradicted me; you
>> >responded to his call for a reference by claiming it divided into 0;
>> >which you could reasonably say it did, using your definition of
>> >dividing in the context of rings. but we could make a less permissive
>> >definition of "divides"
>> >
>> >compare the following
>> >a | b if and only if there exists at least one k such that a*k = b.
>> >a | b if and only if there exists a unique k such that a*k = b.
>> >
>> >In an integral domain the second definition and the first coincide
>> >except for that the first allows 0 | 0 and the second does not.
>>
>> I think the following is the standard:
>>
>> a|b iff a<>0 and b=a*x for some x.
>
>I always thought it was standard to leave out the part " a<>0 ".
That's just the point. For some of these concepts there is a clear
standard, nearly universal, while for others there are competing
alternatives.
For a|b, the choice as to whether or not to bar 0|0 is not so
universal, but it is barred in most of the texts I've looked at.
So if you think it's standard to leave out the restriction a<>0, how
about some references? I'm sure there are references that allow 0|0 as
well as many that bar it. But here too, the power of a reference
depends to some extent on whether the author's text is itself viewed
as a standard.
If it's clear that there is no real standard, then you can't insist
one is "right" -- it becomes author's choice. However, in such cases,
it's only fair for the author to warn the reader of competing
definitions or at least clearly declare the choice being made.
However if there is an overwhelming standard, then in that case, using
a nonstandard definition is surely confusing. It's still author's
choice, but if an author does too much of that without some very good
reasons explained in advance, the resulting confusion will typically
cause the text to be rejected by the target audience, Who wants to
constantly be doing mental translations between nonstandard and
standard?
>> The above definition works in any commutative ring, and while it
>> doesn't require x to be unique, it does bar 0|b.
>>
>> But even though I prefer 0 not to be regarded as a divisor of
>> anything, I can see how the case can be made for 0|0.
>
>By not imposing artificial conditions. While it is at least
>conceivable to bar 0 altogether from the natural numbers,
In fact, for many books (not all) N = Z+, that is, {natural numbers} =
{positive integers}. However this definition is far from universal,
the alternative being N = Z+ union {0}, that is, {natural numbers} =
{nonnegative integers}.
I think (but I'm not sure about this) that most number theory books
use N = Z+, while I know some computer science and discrete math texts
use N = Z+ union {0}.
>once it is accepted there, it should not be barred from divisibility.
Not necessarily. Many (and perhaps most?) abstract algebra and ring
theory texts define a|b in a ring theory context (where positive vs
nonnegative is not applicable), but still bar a=0 in the notation a|b.
>> After all 0=x*0 would seem to suggest that 0 is both a multiple of 0
>> and a factor of 0.
>>
>> I have a feeling there's less agreement by texts as to whether 0|0
>> than as to whether gcd(0,0)=0.
>
>Some people like maximal elements.
I think we all do, depending on what's being maximized.
>> Certainly any text which bars 0|0 would also make gcd(0,0) undefined.
>>
>> But conceivably, there are some texts where 0|0 is accepted but for
>> which gcd(0,0) is still undefined (because of the g in gcd).
>
>A common misconception, see <41levcF1fatb8U1@xxxxxxxxxxxxxx>.
I can't get that link -- it appears to be an email address, not a web
site.
>The definition of the gcd uses the divisibility preorder on
>a ring.
Perhaps a course in abstract algebra or ring theory would make that
choice, but most number theory texts that I've seen, when defining
greatest common divisor, use the word greatest without any such
qualification, implying that the intended meaning is the standard
ordering. But as I've pointed out, even abstract algebra and ring
theory text tend not to allow gcd(0,0).
To recap -- we appear to disagree on what's standard.
I claim:
(1) Most (but not all) reputable texts define the notation a|b as: a|b
iff a<>0 and b=a*x for some x.
(2) The overwhelming majority of reputable texts define gcd in a way
that precludes giving a value to gcd(0,0).
If you think otherwise, let's see some references.
quasi
.
- Follow-Ups:
- Re: GCD(0,0)
- From: Marc Olschok
- Re: GCD(0,0)
- Prev by Date: Re: Degree of a map S^1 -> S^1
- Next by Date: Re: GCD(0,0)
- Previous by thread: Re: GCD(0,0)
- Next by thread: Re: GCD(0,0)
- Index(es):
Relevant Pages
|
