Talk:Multiplicity (mathematics)

Categories

[This article] is not just about multisets. It also and mostly talks about the multiplicity of roots of a function. So it should not just be in the Category:Set theory. It should be in another category as well. I guessed that it should be in Category:Numerical analysis. If I was wrong, then what other category would you put it in? JRSpriggs 06:04, 1 May 2006 (UTC)

Fair point, I was a bit rash in removing Category:Numerical analysis. My point was that multiplicity of roots is used more widely than numerical analysis. Hence, I should have put it in a more general category. I have now added Category:Mathematical analysis. Does that make sense? -- Jitse Niesen (talk) 12:51, 1 May 2006 (UTC)

Category:Mathematical analysis is the good one I think. Oleg Alexandrov (talk) 15:13, 1 May 2006 (UTC)

OK. That looks like a better category. But this raises the question of whether "Numerical analysis" should be a subcategory of "Mathematical analysis". It seems like it should be, but it is not now. JRSpriggs 07:12, 2 May 2006 (UTC)

At the moment, Category:Numerical analysis is a subcategory of Category:Analysis, which also contains Category:Mathematical analysis and Category:Musical analysis. I don't see the rationale for that, so I changed it to make num. analysis a subcategory of math. analysis as JRSpriggs suggest. -- Jitse Niesen (talk) 06:54, 4 May 2006 (UTC)

Multiplicity of a root of a polynomial

Hey guys, I appreciate the precise mathematical definition of the multiplicity of the zeros of a function, but how about a simple one line explanation that the multiplicity of a root is the number of times that root is repeated? Followed by a simple example, like the root of f(x) = (x-1)^3 is 1, with a multiplicity of 3, because it occurs 3 times. —Preceding unsigned comment added by Nedunuri (talk • contribs) 18:59, 20 Oct 2006 (UTC)

I second that. Myers6609 (talk) 01:15, 7 January 2009 (UTC)

I added- Multiplicity can be thought of as "How many times does the solution appear in the original equation?". hopefully making it explicit and more clear.Nickalh50 (talk) 19:09, 2 May 2011 (UTC)

The problem is that this is just informal language; there is no such thing as "occurring as a solution" in general; a value either is a solution of a problem, or it isn't. The reason that one can define multiplicity in the case of a polynomial equation is that there is a method to "remove" a solution (namely division of the polynomial by a factor corresponding to the root), and that only if the root still persists as a solution of te new problem does one consider it to occur more than once. The text already says this, and the lead section already says that multiplicity in general means being present multiple times, when a precise meaning can be given to that. So what does the sentence really add? Marc van Leeuwen (talk) 03:39, 3 May 2011 (UTC)

Root / zeros

The article should be consistent in using root or zeros. Root is the more common in scientific use. Thomas Nygreen (talk) 22:52, 5 December 2007 (UTC)

Then edit it... However, in analysis, I think that the zeroes of a function is more common than roots. mattbuck (talk) 23:02, 5 December 2007 (UTC)

In my experience, root, zero, & solution are used more or less interchangeably in this context. We need links to Root_of_a_function and I added a few.Nickalh50 (talk) 20:42, 2 May 2011 (UTC)

Multiplicity of a zero section needs an added explanation of why it won't overestimate the multiplicity

This could be something simple, like "if k is an underestimate, then the limit evaluates to 0, and if k is an overestimate, then the limit evaluates to ∞, which is not a real number." Or it could give a full-fledged derivation for why both of these is the case. Regardless, an explanation is needed, IMHO.

It could also be pointed out that k need not necessarily be an integer. Take f(X) = X^1.5 at 0, for a simple example.

I don't really have any real expertise on this topic; I'm just going by what I see on face value (plus a little, I guess), so if there's any mistaken impression here, feel free to correct me, and there will be no hard feelings. :) --69.91.95.139 (talk) 01:00, 15 April 2009 (UTC)

"None"?...

* A little knowledge is a dangerous thing.

• • A little Learning is a dangerous Thing;
    Drink deep, or taste not the Pierian Spring:
    There shallow Draughts intoxicate the Brain,
    And drinking largely sobers us again. ~ Alexander Pope
    --Jerzy•t 07:38, 13 December 2018 (UTC)
    

Multiplicity of a zero section should be rewritten for precision or removed

In its current state, this section is not correctly formulated, and is a very bad illustration of the general notion of multiplicity. Since nothing is required of the function (an indeed the interval is not even required to not be reduced to a point), it need not be continuous at its zero; if it isn't, I think none of the definitions given applies, so one can only conclude that the zero has no multiplicity at all (not even multiplicity zero). Other problematic points are infinite multiplicities (which are not allowed in the most common definition of multiset; even those who prefer to be more liberal make it a cardinal number, not just infinity) and multiplicities that are in other ways undefined (what is the multiplicity of the zero of $x\mapsto x^{\alpha }$ for any positive real number α?). I insist that while infinite multiplicities can be admitted in some cases (any "root" of the zero polynomial), ordinary use of "multiplicity" does not allow it to be just any real number (and why not quaternions for tomorrow's new application?) Marc van Leeuwen (talk) 08:51, 3 April 2011 (UTC)

Root

Just a suggestion, for easier reading and to be a bit more concise, to change the first 2 sentences of the section Multiplicity of a root of a polynomial,

from

Let F be a field and p(x) be a polynomial in one variable and coefficients in F. An element a ∈ F is a root of multiplicity k of p(x) if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x).

to

Let p(x) be a polynomial in one variable. A solution x = a of p(x) = 0 is a root of multiplicity k if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x).

--Bob K31416 (talk) 16:09, 4 November 2017 (UTC)

In the standard mathematical terminology, a root of the polynomial p(x) is, by definition, a solution of the equation p(x) = 0. Thus the formulation "a solution of ... is a root of multiplicity ..." may be confusing for some readers, as using two different words for the same concept. For this reason, I suggest the following formulation:

Let p(x) be a polynomial in one variable. A root a of p(x) (that is a solution of the equation p(x) = 0) has the multiplicity k if there is a polynomial s(x) such that s(a) ≠ 0 and p(x) = (x − a)^ks(x). One says also that a is a root of multiplicity k.

If you agree, feel free to insert this in the article.

By the way, if a is not a root, one may say that it is a root of multiplicity zero. This may be useful for simplifying some statements and proofs. IMO, it is not useful to mention this here. D.Lazard (talk) 16:43, 4 November 2017 (UTC)

Other senses: usage by Cantor not (evidently) the common-language meaning.

@D.Lazard: reverted an edit of mine to explain the usage of "multiplicity" by Cantor at Absolute infinite, saying that this is the common-language meaning. It seems to me however that neither a set, nor an ordered set (which he seems to have meant here), is covered by the definitions at wikt:multiplicity. I had added a section Other Senses to this page to explain that usage, as linking "multiplicity" to cardinal number seems to me to help no-one. I changed that to link to my new section, but since that has been removed, it just links to the entire page, which is even worse. What I added was this:

Some older or translated texts use this term differently:

Georg Cantor, as translated into English, appears to use it to mean an ordered set (text in square brackets not present in original):

A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a "sequence".

...

Now I envisage the system of all [ordinal] numbers and denote it Ω.

...

Therefore:

The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.

References

Can anyone suggest a better way of resolving this confusion? I do not now have time to look into this further, but perhaps we have somewhere else an article which includes a description of his ideas of "multiplicity", and we could link Absolute infinite there, and maybe add a hat note here.

PJTraill (talk) 12:06, 6 November 2022 (UTC)

I have added the template {{talkref}} in the previous post, and removed a duplicate signature.

It is clear that Cantor or its translator used "multiplicity" for what is presently called a set with several elements. This the common meaning of the term as expressed in Wikt:multiplicity ("A large indeterminate number"). Per WP:NOTDICT, the meanings of "multiplicity" that refer to other concepts than the one of this article do not belong to it. So, Wikipedia rules justify my revert. The only thing that you can do is to use {{wikt}}. In any case, the details of Cantor's terminology have no encyclopedic value. D.Lazard (talk) 13:03, 6 November 2022 (UTC)