Talk:Square (algebra)

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 Field: Algebra


The section Square number#Uses is about squaring numbers, so the definition of squares do not belongs here. 12:34, 19 March 2006 (UTC) — Preceding unsigned comment added by Melchoir (talkcontribs)

Well, I've done the merge, as well as a bit in the other direction. If you'd like to expand on squares of integers, that material is accumulating at Square number, not here. 11:32, 24 March 2006 (UTC)

— Preceding unsigned comment added by Melchoir (talkcontribs)


Square (algebra) but Cube (arithmetic). I'd prefer cube (algebra). googl t 21:31, 14 June 2006 (UTC)

Square of a vector[edit]

Should we perhaps note that the "square" of a vector as commonly denoted by is usually understood to be ? Shinobu 09:42, 13 November 2007 (UTC)

state of this article[edit]

All I've done is clean up what was here, but the article needs to get organized. Maybe I'll be back. Michael Hardy (talk) 18:49, 16 March 2010 (UTC)

The phrase after the sigma/recursive formula (which I have just added) is redundant. It explains in words what the preceding symbolic representation should already convey. 2N+1 from N=0 to 2N+1 == sum of the first N odd numbers. (R-1)^2 = last square (= sum of odd numbers) + 2N-1 (= next odd number). The verbal explanation should precede the sum, if anything. If somebody more patient and artful with prose has time, could you please reword it to integrate them, so it doesn't look premature? Thanks. -- Qt-Q!U (talk) 13:22, 15 August 2010 (UTC)

Page structure[edit]

This page has become a mess. The lead is too long, there aren't nearly enough sections, the "facts" section is trivia. Here are my proposals to clean this page up:

  1. the lead consists of just the bare definition. No formula for the sum, no properties, etc.
  2. the "facts" would either be converted to a list format on another list page
  3. most of what is in the lead would go in a section entitled "Properties", as that's what they are

Any opposition, or feedback? -- (talk) 06:38, 24 September 2010 (UTC)


what about e.g. -2², does it mean (-2)² = +4 or -(2²) = -4 ? (talk) 10:51, 8 June 2011 (UTC)

See Order of operations#Gaps in the standard. PrimeHunter (talk) 11:11, 8 June 2011 (UTC)

Mistake in identity[edit]

I believe that there is a mistake right there in the intro. Currently, the article says that square (perfect) can be obtained using the identity: 1 + 3 + 5 + ... + 2N + 1. This is incorrect. It should be 2N - 1. I am going to correct this but feel free to correct me if I am wrong. -M.N. — Preceding unsigned comment added by Murtazanisar (talkcontribs) 23:11, 12 July 2011 (UTC)


There has been occasional discussion on Talk:Square number of merging these two articles; this is obviously appropriate (and most people who have commented there agree). I think further edits should mostly focus on this task. The other article (square number) is among the top 500 most-viewed articles; I suggest this article be converted to a redirect once merging takes place. -Joel B. Lewis (talk) 21:14, 31 July 2011 (UTC)

The merger (BTW announced improperly) was repelled, and these arguments were refuted. Nevertheless, 2012 article is markedly different from 2011 one. Thanks to all users who helped to make a correct, good article, at last. Incnis Mrsi (talk) 18:43, 3 September 2012 (UTC)
See here. This is getting unnecessarily heated when it was announced. We have been though all this. Again - where the hell are the links saying that they (Robo37, Froogle1099, Physics is all gnomes) "disagreed": not you writing the "refutes" for them (and therefore lying about the situation, if they didn't actually refute themselves), but their comments typed from their own hands and minds? Maschen (talk) 20:57, 3 September 2012 (UTC)
The argument of Physics is all gnomes was obsoleted by a new article (also, he probably overlooked a section about uses). Froogle1099’s "the same grade" itself does not, actually, imply the direction Square_(algebra)→Square_number; but in the context of this direction such a merger argument definitely was refuted. Joel B. Lewis did not introduce his own argument, he only referred to other 3 guys. It is only Robo37’s argument what was not refuted, but it constitutes mostly of WP: I don't like it. Incnis Mrsi (talk) 21:53, 3 September 2012 (UTC)

Even function[edit]

I consider this property of the function an important one, and it is not confined to real numbers. This is true both in rings and inner product spaces. Incnis Mrsi (talk) 15:33, 4 September 2012 (UTC)

The property that is important and true in most structures where the square function is defined. This does not implies that one may say that the square function is even in these structures. As far as I know this extension of the definition of "even function" is WP:OR. Can you find a source which qualifies the complex square function or the norm of an Euclidean vector space of "even functions"? D.Lazard (talk) 20:50, 4 September 2012 (UTC)
I do not think this trivial generalization is WP:OR-worthy. French Wikipedia says that complex numbers and vector spaces are eligible. Maybe, I search for reliable sources later. Incnis Mrsi (talk) 08:23, 5 September 2012 (UTC)
A bilinear map, you said? It has to mean: a bilinear map over the ring Z, i.e. only the distributive property holds. In a Hilbert space, "multiplication" is not bilinear over complex numbers, but the "square" is still "even". Actually, a "multiplication" only has to commute with additive inverse on both sides, it is sufficient indeed. It is inconceivably that, for a structure which already is an abelian group over addition, a term "multiplication" was applied for something which does not commute with additive inverse. Incnis Mrsi (talk) 08:23, 5 September 2012 (UTC)
I am not proud about my formulation, but I have not found a better one that is acceptable in a lead. In fact the minimal condition for the identity is that the addition has a left or a right inverse and that the multiplication is distributive with respect to the addition.
About "even": English WP defines only "even" for real functions. This article is not the place to introduce a more general definition, even if it is well sourced. Thus it is better to introduce first the identity and then to interpret it as characterizing an even function in the case of real numbers. This has the advantage to be directly understandable by the layman which does not know what is an even function.
About the place of this sentence. By MOS:MATH, the technicalities have to be avoided in the lead. Thus I suggest to replace the paragraph by the following one, and move the technicalities in a section: "In most cases (see below) the square function satisfies the identity . This means that, as a function of a real variable, it is an even function."
D.Lazard (talk) 09:59, 5 September 2012 (UTC)
I edited the article on even function to make it clear that it applies to more than real functions of a real variable. I don't have a reference in front of me, but I know I've seen the terminology used in the context of complex analytic functions. - Virginia-American (talk) 22:10, 6 September 2012 (UTC)

The functions from a vector space to itself whose squares are the identity function…[edit]

Sure, I understood what D.Lazard intended to say. But this is confusing: either the article must at least mention function composition, or this topic should be bypassed at all. BTW:

  • Only endomorphisms (linear maps) usually considered for vector spaces.
Not true: For example, the diffeomorphisms of Rn are widely considered. More specific: the affine transformations of a vector space are not linear and are implicitly considered, when talking of reflections. D.Lazard (talk) 17:01, 5 September 2012 (UTC)
  • An element of virtually any group such that c2 = 1 may be called and involution, especially in the context of group representations.
True, but too much generality would make the article too difficult for most readers. D.Lazard (talk) 17:01, 5 September 2012 (UTC)

Incnis Mrsi (talk) 18:32, 4 September 2012 (UTC)

By the way, I'll make exxplicit the fact that, in this paragraph, one considers the square of the composition. D.Lazard (talk) 17:01, 5 September 2012 (UTC)

Still a mess[edit]

I recently went through this article to improve the English, but besides correcting some obvious errors I did not do anything to the mathematical content. This article lacks a focus, it does not know what it is about. The title indicates that we should be talking about squares as they are used in algebra, but clearly the content of the page is concerned with the squaring function. This by itself is okay, but as every student is supposed to know, a function is not defined by a "rule", you must indicate the domain and this article gets tripped up on that point. If the domain admits a single binary operation, then the squaring function and the doubling function are the same function – just written in different ways (multiplicative vs. additive notation). You won't see that mentioned here because the doubling function is not written with an exponential "2". If you are going to call composition of a function with itself "squaring" then my point is germane since there is only one binary operation involved in that case. Calling the inner product of a vector with itself "squaring" in an inner product space is not appropriate. Multiplication is generally understood to be a binary operation - the result is another object of the same type, and this is not the case with the inner product form. Using an exponential "2" in this case is just abuse of notation. I could go on, but won't. As written this article should be titled Square (typography) as it appears to be about using the exponential "2" rather than anything algebraic. Bill Cherowitzo (talk) 16:43, 6 September 2012 (UTC)

Thanks for feedback, but, given a size of the article and importance of the topic, something more elaborated was expected. Composition actually has nothing to do with squaring; this is a (late) additions from D.Lazard which I do not agree with. Dot product is perfectly a binary operation – read the dedicated article, please. Generally, Bill Cherowitzo’s quibbles here are not more substantiated than this could be about multiplication or subtraction. Incnis Mrsi (talk) 20:14, 28 September 2012 (UTC)
The article dot product refers several times to a product and never calls this a square. Bill Cherowitzo's comment is about the fact that the dot product is a map from , i.e., the target set is different than the domain from which the operands come. (Note that the binary operation article is actually very helpful in figuring out his intention, since in its introduction it mentions that "[u]sually, but not always, the case of binary relations defined on a single set is of interest." Indeed this is the only example in the article of a "square" in which it is not possible to associate a "cube" or higher power. I support Bill Cherowitzo's suggestion that the phrase "dot product square" be removed. --JBL (talk) 22:18, 28 September 2012 (UTC)
[T]he only example? Note that the article's title is "square", not "power 2". Very good for the notability, indeed. Exact wording "dot product square" is, of course, poor, but terms like "dot square", "scalar square", or just "square [of a vector]" are not unseen (although not very common). – Why square of vector is scalar? – |a|² is called a scalar square.
You can find more with Find sources: "dot" square vector – news · newspapers · books · scholar · HighBeam · JSTOR · free images · free news sources · The Wikipedia Library · NYT · WP reference and similar queries. Incnis Mrsi (talk) 10:33, 29 September 2012 (UTC)

Just two points. First of all, if you would read the article on binary operation you will discover that there is a difference between a binary operation and a binary relation (even though one is redirected to the other). The dot product of two vectors is clearly a binary relation which is not a binary operation. A binary operation on set S is a special binary relation of the form S × S → S. The result of performing the operation on two elements of S is another element of the set S. As the article goes on to say, it is binary operations (not, I might add, binary relations) which are the cornerstone of algebra. Secondly, there is not a single reliable reference to the usage of square vector in the three you gave. In the first, you should note that the answer to the question never repeats the questionable phrase "square of vector". In the study guide, which is the second one, the inner product is not called the "square of a vector", it is correctly called the "square of a scalar" with a slight Russian twist in terminology. Finally, even the Darwin computer code manual says that " the square of a vector we mean the inner product...", showing me that they understand that the phrase by itself is meaningless.

I have actually not made any recommendations about this article, I have only tried to point out some weaknesses. Once a "focused" approach to the topic has been made, meaning, deciding what the article is about, it will be easy to fix it up. Bill Cherowitzo (talk) 20:00, 29 September 2012 (UTC)

Perhaps, Bill Cherowitzo will now meet such discoveries as:
The assertion that the inner product… is correctly called the "square of a scalar" indicates either inattention or some trouble with a grammar (virtually, of any language, not specifically English grammar). According to William Rowan Hamilton, "the correct term" for dot product (up to sign) is "scalar of the product", not "… of a scalar". Bill Cherowitzo apparently confused a word order, which makes his insinuations on a slight Russian twist particularly funny. Incnis Mrsi (talk) 21:13, 29 September 2012 (UTC)

My error in terms of the redirect, sorry! Your interpretation of "relation" is not within the context of this discussion. To quote from the article you linked to,

Definition 1. A relation L over the sets X1, …, Xk is a subset of their Cartesian product, written LX1 × … × Xk.

No talk here of a proposition-valued implication. As to my use of the term, I was quoting from the binary operation article – this is not the terminology I would use. I do agree with your conclusion ... an algebraic operation is not a binary relation, but it can be written as a ternary relation; a good reason to fix that article which has been incorrect since Oct. 2010. Hamilton has been dead for quite some time now, bringing up his dated and idiosyncratic usage does not make much sense to me. The study guide author, a Russian, states that a dot product of two vectors is "a scalar square" which in better English should be rendered as "the square of a scalar". The emphasis here is clearly that this is not the square of a vector, which is precisely the point I was trying to make. Bill Cherowitzo (talk) 02:43, 30 September 2012 (UTC)

I can mistake, but the term "scalar [of a vector]" seems to be a deeply sub-standard one, if it exists at all. There are terms "magnitude" (an engineering-flavored) and "norm" (a mathematics-flavored). Wikipedia articles generally discuss concepts, not the usage of corresponding terms. One cannot dismiss squaring of a vector as a "idiosyncratic usage" if it is already discovered and published, even 160+ years ago. Of course, we know that a representation of an n-ary operationfunction as an n+1-ary relation does exist, but it can define the function only within the classical logic. But not in: intuitionistic logic, constructive mathematics, and, last but not least, numerical analysis. n+1-ary relation can be used to define a function of n arguments, but it is not a function. A multiplication (of whatever arguments and whatever-valued) is a 2-argument function (binary operationfunction), not a ternary relation, whatever do you think about it. In any case, I still do not see the connection to our topic. Why Bill Cherowitzo diverts the dispute from algebra to mathematical logic? Incnis Mrsi (talk) 08:34, 30 September 2012 (UTC)

After certain events (Bill Cherowitzo knows what I mean) I withdraw my claim that any multiplication is a "binary operation". But it is a binary function anyway and has nothing to do with (ternary) relations. If a particular multiplication gives its two arguments from the same set (for example, dot product does, but multiplication of a vector by a scalar does not), then we can speak about the square function. Incnis Mrsi (talk) 14:50, 16 January 2013 (UTC)


Within the article, this concept is discussed in two places. First, in the Square (algebra) #In rings in general section. Second (under composition), in the Square (algebra) #In geometry and linear algebra section. Meanwhile, it is only tangentially related to the square function: I2 = I is only one form of its definition. The same expresses also as I(1 − I) = 0 (which relates it both to quadratic polynomial and zero divisor), and other definition is kN: Ik = I (which relates it to exponentiation, hence the name). How many people think that extensive considerations about idempotents in this article constitute an WP: undue weight? They have their own articles Idempotent element and Idempotence, at last. Incnis Mrsi (talk) 15:41, 16 January 2013 (UTC)


The following text was removed as being off-topic for this article

Another, more well known, function is the square of the absolute value | z |2 = zz, which is real-valued. It is very important for quantum mechanics: see probability amplitude and Born rule. Complex numbers form one of four possible Euclidean Hurwitz algebras that are defined with a real quadratic form q; here q(z) = | z |2. In a Euclidean Hurwitz algebra this q equals to the square of the distance to 0 discussed above, and the absolute value | z | can be defined as the (arithmetical) square root of q(z). Multiplicativity of q in these algebras explains (or relies upon) certain algebraic identities (see below).

The squaring function is a primordial quadratic form used to generate composition algebras, as indicated now in the edited text. — Rgdboer (talk) 23:39, 19 May 2016 (UTC)