Talk:Scalar (mathematics)

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[edit] Nonsense definition

"In mathematics, scalars are components of vector spaces..."

Sigh.............

The above is of course nonsense. A scalar may be a component of a vector, but to say that a vector space is a thing composed of components, each of which is a scalar, is absurd, and to define "scalar" that way, in an article that should treat the concept in a more general way that that which appears at scalar (physics) does not make sense. Michael Hardy 02:46, 3 January 2006 (UTC)

How about the following:
In mathematics, objects such as modules, vector spaces, polynomial rings, matrices and tensors can be defined in terms of lists or arrays of elements taken from an underlying field or ring. Within the context of a specific object of this type, the term scalar refers to a member of the underlying field or ring (see, for example, scalar multiplication). More loosely, the term scalar is also used to refer to a member of a sub-set of the defined object that is isomorphic to (i.e. behaves in the same way as) the underlying field or ring (see, for example, scalar matrix).
Gandalf61 10:31, 3 January 2006 (UTC)

OOps, that is not what I meant! The sense is: a vector space has three components (K,V,*) where K is a field (the scalars), V is an abelian group(?) (the vectors), and * is an operation from K x V to V satisfying...

Granted the word "component" was a very bad choice. Perhaps "ingredient"? I do not think that one should mention "components" in the other sense, except perhaps in the context of the "coordinate space" example. Jorge Stolfi 17:40, 3 January 2006 (UTC)

[edit] Reasons for reverting

It is not sufficient to simply revert the page and state that you did so, Jorge. You are obliged to provide reasons for doing so. Dysprosia 22:42, 3 January 2006 (UTC)

Sorry, you are right, I owe an explanation. From the above comments and the subsequent edits, I realized that my use of the word "component" to describe the scalar field was very unfortuante, and it led User:Michael Hardy and other think the article was defining "scalars" as the coordinates of vectors. That would indeed have been utter nonsense, and Hardy's reaction was quite understandable. Since the purpose of the subsequent edits seemed to to be to patch up the defintion, but without removing that misunderstanding, it seemed best to revert and fix the real root of the problem. Sorry if that meant wasting the other editor's time. Jorge Stolfi 23:45, 3 January 2006 (UTC)
You also removed a lot of structural and generalizing edits as well. If you have no objection to these edits, I'd like to replace them. Dysprosia 00:23, 4 January 2006 (UTC)
Um, let me have another look at those edits. Hang on a minute... Jorge Stolfi 00:31, 4 January 2006 (UTC)
  • As for moving the "Etymology" section up: fine with me (done).
  • As for the changes to the head section: let me defend my version. The term "constant" has no absolute sense in mathematics, it is very context dependent; and anyway, in a scalar multiplication αv, the α may be variable and the v constant. Also, it is not appropriate to say that a scalar "could be a nonvector element of a vector space", since "element of a vector space" is always understood to mean one of its vectors, never one of its scalars. Finally, there are six (at least) quite distinct senses in which the word "scalar" occurs in mathematics, and they must all be listed in the head paragraph; so it seems best to keep each sense as succint as possible, to be amplified and elaborated in the article body. Jorge Stolfi 00:58, 4 January 2006 (UTC)
Yeah, constant isn't the best term, but if one can be found...? The introduction should aim to be as general as possible. Dysprosia 03:01, 4 January 2006 (UTC)
There is already an informal and general "definition" of the term in the scalar article, that tries to cover its uses in math, physics, and computing, and is pretty much along the lines of "simple value, without direction". I am assuming that readers who get to scalar (mathematics) instead of scalar expect to see a more mathematically oriented (and hence precise) definition.
Jorge Stolfi 12:54, 4 January 2006 (UTC)
Not quite; scalar's introduction should deal with the term in its most general form, scalar (mathematics)'s introduction should deal with the term in its mathematics context in its most general form also. Dysprosia
  • As for getting rid of the "Scalars in vector spaces" section: for the above reasons, the first three head senses (at least) need to be expanded in the body of the article; and anyway there are some non-trivial things that can be said about the nature of the scalars. For instance, the additional constraints that they must satisfy when the space has a inner product or a norm (see the invisible comment in the article). Also here one may add that, as a consequence of the basis theorem, every vector space (K, V, *) is isomorphic to a space of functions from some set S to the scalar field K. (Is this right?) In other words, vectors are collections of scalars, after all—if one does not mind an isomorphism.
On an (personally, admittedly) cursory inspection, the section appeared only to deal with a definition of scalar multiplication which is better handled by the scalar multiplication article. Perhaps we can deal with a discussion of coordinate vectors and its relationships to "collections of scalars", which would be a much more fruitful use of the article. Dysprosia 03:01, 4 January 2006 (UTC)
Coordinate vector spaces are treated in the article coordinate space and summarized in examples of vector spaces. (I just added a redirect from coordinate vector space to coordinate space). There is also a coordinate vector article with perhaps should be merged into coordinate space.
Jorge Stolfi 12:54, 4 January 2006 (UTC)
Thus similarly shouldn't an indepth discussion of the mechanics of scalar multiplication be done at the scalar multiplication article? I mention coordinate vectors since the relationship to scalars here is close. Dysprosia
  • As for the separate "Scalars in modules" section: I don't know if there is much else to say about the "scalars" of modules, other that they are just a ring. If that is the case, perhaps we can squeeze that material into a paragraph, and merge the two sections into one.
I didn't really touch that section, it looked OK to me. Dysprosia 03:01, 4 January 2006 (UTC)
Did I miss anything else? All the best, Jorge Stolfi 01:19, 4 January 2006 (UTC)
Anyway, this line of discussion is appearing to move towards a rather extensive examination of these details, which really isn't warranted given the nature of the points of contention here. I'm not going to really pursue this further, since there are obviously better uses of both our time :) Dysprosia 13:06, 4 January 2006 (UTC)

[edit] Main sense should be the (precise) linear algebra sense

A recent edit expanded the definition as follows:

"In mathematics and especially in linear algebra, the term scalar is frequently used to denote an ordinary real or complex number in order to clearly distinguish such numbers from other numerical objects such as vectors and matrices. Central to the subject is the notion of a vector space, which relates scalars to these objects through the operation of scalar multiplication. Multiplying a vector by a scalar has the effect of scaling it (changing its length), hence the choice of term.
More generally, the set of scalars associated with a vector space may be any field. In abstract algebra the notion of vector space is itself generalised by that of a module, in which the scalars may come from any ring."

I mostly reverted these two paragraphs to the original form. It seems inappropriate for a mathematics article to give priority to the informal and fuzzy sense of the term (which, by the way, was already listed further down in the head section) over the precise sense (which got largely lost in the rewrite). Note that the informal sense is almost surely derived from the precise one (namely, 1x1 matrices are informally called scalars because they behave like elements of the scalar field), not the other way around.
Also, the details on what fields can be scalars are better left for the appropriate section.
Jorge Stolfi 16:33, 9 January 2006 (UTC)

I'd say your current introductory paragraph is considerably less useful than the ones you reverted and quoted above. It provides no insight at all as to the significance or rationale of the concept and is bound to be all but meaningless to anyone who is not a mathematician (to mathematicians it's merely a very poorly phrased tautology - do we multiply scalars "into" things? do we call the elements of modules "vectors"?). I'll leave it to other objective parties to compare the two, though.  — merge 04:05, 12 January 2006 (UTC)
The rationale, if needed, should be added in the article body, as in other methematical articles. As for the style, and vector in modules, I have rewritten it again; please check. (BTW, what was wrong with "multiply into")? Jorge Stolfi 08:33, 14 January 2006 (UTC)

[edit] Etymology of scalar

The suggestion that the "scalars" have that name because they "scale" (i.e. "change the scale") of vectors sounds plausible, but does not seem consistent with Hamilton's quote; so it may be just folk etymology. It may well be that the verb "to scale (up/down)" and the noun/adjective "scalar" evolved from the noun "scale" by separate routes: the former possibly among draftsmen and geometers, from the "scale of a drawing", the latter in algebra as per Hamilton's justification.
Jorge Stolfi 16:33, 9 January 2006 (UTC)

[edit] Is "scalar" a number?

The introduction says : "...a scalar is a number..." I'm not very comfortable with that statement. A scalar need not be a number in any ususal (however general) sense of the word. I think it should be reworded somehow. --Meni Rosenfeld 07:03, 12 January 2006 (UTC)

I agree and hence I edited the intro slightly. -- Jitse Niesen (talk) 13:43, 12 January 2006 (UTC)
You are right. The only fields I have ever seen used as scalars were either subfields of the complex numbers, or the integer numbers modulo a prime power. However, ther are weirder fields around (such as rational functions of a variable X).
I have reworked the head paragraph again. See if it is better now.
All the best, --Jorge Stolfi 08:29, 14 January 2006 (UTC)

[edit] Basis theorem

I have added a parag about every vector space being isomorphic to a coordinate space over its scalar field. Did I get it right?
Thanks, --Jorge Stolfi 08:29, 14 January 2006 (UTC)

[edit] Refactoring

Hi. I've significantly refactored this article to improve organisation and tried to clarify some of the text, as well as simplify the intro per a suggestion from Jitse Niesen. If I inadvertently removed an important piece of information that someone added, I'd be grateful if you'd add it back in. Thanks!  — merge 15:55, 20 February 2006 (UTC)