Talk:Algebra

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A vital article.

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Cut from article pending reorganization:

An algebra is based on a 2→1 morphism having 2 inputs (multiplicator and multiplicand) and one output (product). It can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra named sometimes also cogebra.

I moved this paragraph to associative algebra. AxelBoldt 04:09 26 May 2003 (UTC)

Contents 1 Bablyonian algebra inaccuracies 2 Rewrite and/or move the following? 3 Blanking 4 Article is horrible 5 Non-English text removed from article 6 a question 7 is this true 8 "Rarely applicable" 9 COTW 10 On closure 11 Just be a disambig and move content to sub articles 12 Integers modulo? 13 al-jaladi 14 History 15 Elementary algebra 16 Not Related to Wikipedia Per Se 17 GA nomination 18 My revert from "Islamic" to "Persian" 19 Inverse Element or Inverse Operator? 20 (,−) has identity? 21 Algebra does not equal arithmetic algebra

[edit] Bablyonian algebra inaccuracies

The text claims that "sed an early type of algebra to solve linear equations, quadratic equations, and indeterminate linear equations in the second millenium BC."

However I see no evidence that the Babylonians had a knowlege of operations and axioms that would be necessary to call the calculations they did an algebra rather than arithmetical calculations for particular problems with inferred formula. Nor do I know of any evidence that they could take two formula and equate them using an agreed set of operations to solve new problems; i.e. they did not have algebraic equations, they knew some formula, could solve for unknowns, but did not have a general understanding of operations or a general system for solving equations. Correct me if I am wrong.Mrdthree 16:03, 6 April 2006 (UTC)

[edit] Rewrite and/or move the following?

The following text was moved from the article, since it seems out of place (also could stand a rewrite):

"The process of "balancing and restoration" is important in algebra. It is the process of equaling both sides of any given equation. Basically, "whatever you do to one side of the equation, you do to the other". Also in Algebra: the distributive law which is used to properly factor out and solve equations."

Paul August 16:29, Aug 26, 2004 (UTC)

You're certainly right that it "could stand a rewrite"! "To factor out and solve equations"?? Please! One factors polynomials or numbers or perhaps various other things; one does not factor equations. To "factor out A" is to pull out A when A is a factor shared in common by all terms; whoever wrote that failed to understand that "to factor out" does not mean the same thing as "to factor". I'm commenting on this here just so that anyone tempted to put this material back into the article will not make the same mistakes. Michael Hardy 22:00, 26 Aug 2004 (UTC)

It seems to me an article about factoring polynomials could be started, and this section (which is awful) eliminated. What do people think?

I put a mention of algebra over a field in front in addition to the one at the end, to help the confused person wanting a definition of the word "algebra" not attached to a qualifier such as "Boolean" which they might possibly find in some phrase such as "Let A be an algebra over Q".

user: Gene Ward Smith

I think an article about fatoring polynomials would be good idea. Paul August 02:08, Oct 18, 2004 (UTC)

[edit] Blanking

Please discuss before blanking large and long-standing sections of this article that you believe are wrong. -Casito⇝Talk 04:20, 21 Apr 2005 (UTC)

that section is absolute nonsense . First of all Al-Khwarizmi was a persian mathematician , who was a zorastrian . Origins of Algebra do not stem from Islam as Zereshk claims.

Then the list of individuals is irrelevant and unsubstantiated propaganda to create some sort of appearance that Islam is the source of Algebra that is why it was removed.--65.144.45.232 04:32, 21 Apr 2005 (UTC)

The removed content was added by Zereshk on Apr 14, 2005, hardly "long-standing". Paul August ☎ 05:26, Apr 21, 2005 (UTC)

The trigonometry page was also blanked out.

I'll add an entire section about Arab contributions to algebra, trigonometry, and mathematics in general later. Presently, I'm working on a comprehensive list of mathematician biographies which will take a week or more.--Zereshk 02:30, 22 Apr 2005 (UTC)

But probably putting back again the very long list of Arab mathematicians in the middle of the text is not advisable. Especially that most of them were red links anyway. Oleg Alexandrov 03:08, 22 Apr 2005 (UTC)

That's what Im working on: Turning red into blue links. It'll take a while to get all of them. The list is massive. At the end, I'll make a page for the list, and give the link here and on other pages.--Zereshk 03:50, 22 Apr 2005 (UTC)

[edit] Article is horrible

This article is horrible. --69.18.22.139 19:58, 27 May 2005 (UTC)

What are its problems? How can we make it better? Paul August ☎ 20:05, May 27, 2005 (UTC)

Yes, this article is pretty awful, particularly the "Algebraic equations" part. I started to rewrite it earlier today, but then gave up (I reverted my edits) and expanded Linear equation instead. Someone needs to rewite the aforementioned section taking into account that most of the information should really go/stay in the "main articles" Linear equation, Quadratic equation, etc. This article should limit itself to why these kinds of equations are important in algebra (or some other such "big-picture" discussion). - dcljr (talk) 29 June 2005 05:06 (UTC)

Probably one thing that would be good is to remove the sections on "equations", "factoring", and "symbolic method", and merge them into Elementary algebra. Paul August ☎ July 4, 2005 15:21 (UTC)

Having looked a little closer, I now think that:

1. The subsections on "Linear equations", "Quadratic equations" and "Cubic equations" should be removed and merged into Linear equation, Quadratic equation and Cubic equation resp. — I've now done this. Paul August ☎ July 4, 2005 23:04 (UTC)
2. The subsection on "Exponential equations" should be moved to its own article: Exponential equation. — I've removed this section. Paul August ☎ July 4, 2005 23:04 (UTC)
3. The section "Factoring trinomials should be removed and merged into either Factoring and/or Elementary algebra and/or perhaps its own article Factorization techniques. — I've now remvoved this section, however nothing was merged because, this material is better covered in the other articles.Paul August ☎ July 5, 2005 19:22 (UTC)
4. The section on "Symbolic method" removed and (perhaps) merged into Elementary algebra. — I've now remvoved this section, however nothing was merged because, this material is better covered in the other articles. Paul August ☎ July 5, 2005 19:22 (UTC)

I'd like the article to take a bigger-picture view of algebra, kind of like the Mathematics article does with all of mathematics. But if I knew exactly what to do with it, I probably would have done so already. I like where you're going with this, though. - dcljr (talk) 5 July 2005 09:21 (UTC)

I agree. All the problematic sections mentioned above have now been removed and where appropriate merged into other articles. Such topics are inappropriate for this article, which as dcljr says should be about the "bigger-picture view". I think the reason those sections were added is that editors see this article and think it is about elementary algebra. The intro needs to make it more clear what the article is about. I added the dab message at the top, to help in this regard. Also the history section needs to be expanded to include "modern algebra", which would also help. It would be good if an algebraist or two got involved. Paul August ☎ July 5, 2005 19:35 (UTC)

I've now had a go at expanding and rewording the intro. Comments/criticisms welcome. Paul August ☎ July 5, 2005 21:18 (UTC)

van der Waerden's book "A History of Algebra" would be a very useful reference for fleshing out the history section. It covers from al-Khwarizmi to Emmy Noether (in fact, that's the subtitle). Unfortunately, I don't have an easy way to get at the book, but maybe someone who does can take a look. The question is, how detailed do you want the history section to be? Also, it may be worth saying that abstract algebra is the study of algebraic structure in an abstract and general setting, rather than simply saying that one studies groups, rings, and fields; moreover, abstract algebra's use is pervasive throughout modern mathematics, and I'm not sure that comes across in the article. nparikh 23:58, July 9, 2005 (UTC)

And while we're at it, we shouldn't forget that we already have an Abstract algebra article that, it seems to me, needs just as much work as this one... - dcljr (talk) 07:00, 10 July 2005 (UTC)

[edit] Non-English text removed from article

I don't know what the following says, but it doesn't belong in the article unless it's translated -- and relevant, of course. - dcljr (talk) 21:36, 8 September 2005 (UTC)

Algebra (fra Arabisk "al-djebr") er en gren af matematikken der kan beskrives som en genralisering og udvidelse af aritmetikken.

Ved algebra forstås også "bogstavregning" og "læren om matematiske operationer".

Man kan lave en grov inddeling af algebra i disse felter:

Elementær algebra hvor man ser på egenskaberne ved de reelle tal, hvor man regner symbolsk med bogstaver som repræsentere tal, og hvor reglerne omkring matematiske udtryk og ligninger studeres.

Abstrakt algebra hvor man ser på strukturer som legemer, grupper og ringe.

Universel algebra hvor man ser på egenskaber der er fælles for alle algebraiske strukturer.

Computeralgebra hvor man ser på algoritmer til symbolsk manipulation af matematiske elementer.

Hentet fra "http://da.wikipedia.org/wiki/Algebra"

[edit] a question

i have a question some how my friend made 1=5 the method he used is: 0x5=0 substitute a as 0 ax5=a divide both side by a 5=1 how could this be possible?

Division by a=0 is not allowed, since x/0 is not defined. Paul August ☎ 16:45, 19 October 2005 (UTC)

[edit] is this true

I remember reading somewhere that an algebra was a set of digits, operators, and a few other things. So I want to know --> is this factually true, and if so, should this be mentioned in the article?

I think that you are thinking of algebras in the sense of universal algebra. That article doesn't have the nice basic definition it deserves, but it gives a sense of the definition you mention. I don't think it belongs in this article, as it is linked to. Maybe that link deserves a better description, though. Smmurphy 02:26, 22 October 2005 (UTC)

[edit] "Rarely applicable"

I removed this sentence because it seemed out of place in the introduction and POV in its assertional tone: "The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world." Maybe a separate section or article about this debate is called for if people are interested? - Gauge 08:44, 15 December 2005 (UTC)

I'm taking a high school Algebra course and agree, there seems to be little use for most of what we are taught. - Anon 11:47, 6 March 2006

Haha. Now that's funny. - grubber 18:21, 6 March 2006 (UTC)

[edit] COTW

I had nominated this for Wikipedia:Mathematics Collaboration of the Week, but the timing is bad. I have to go away for a week and won't be able to help as much as I'd like. Maurreen 19:05, 19 February 2006 (UTC)

• Don't worry, on COTW week has been redefined to mean anything from a week to a month.
• A few thoughts on how the article could improve:

1. generally give an overview of whole topic, and sub fields, keeping it at accessible as possible
2. Extend Algebraic structures to cover rings, fields and vector space, possibly also mention isomorphism (on my todo list)
3. mention some interesting examples of such, non solvable groups
4. Free groups polynomials and sequences and series
5. linear equations and linear algebra
6. include '(from the Arabic "al-djebr" meaning "reunion", "connection" or "completion")' somewhere appropriate.
7. List some of the key results.
   1. Galois theory - links polynomials to groups
   2. Reinmann-rock (now there's a task)
8. Applications of algebra, discuss how algebra is used to address problems in geometry, topology, number theory, Fermat?

• I'm sure theres more. How much we wish to distinguish algebra for abstract algebra is an important question. --Salix alba (talk) 19:54, 19 February 2006 (UTC)

I've slightly expanded and cleaned up the algebraic structures section. I think using groups as an example and giving a few specific groups is a good idea, to give the reader the flavor of the material, but probably we don't need to go into significant detail about many different algebraic structures here. This material is better suited for a page specifically devoted to algebraic structures than a general overview of algebra. We should probably link to that page, but the current version seems pretty obtuse and probably wouldn't serve as a good introduction for the type of people who find the basic material on this page useful. -- Zarvok | Talk 07:23, 20 February 2006 (UTC)

[edit] On closure

• JA: Closure is usually not counted as one of the 3 axioms for a group, but is counted to "come with the territory" of the binary operation: X x X -> X. Jon Awbrey 21:32, 20 February 2006 (UTC)

I agree that it is implicit, but I think it's helpful (although redundant) to mention it. I tried to use my edit to make that mention quick and short so as not to get in the way of the text. I'd like to hear any other ideas on how/whether to inlude a mention of it. - grubber 23:12, 20 February 2006 (UTC)

Looks good to me. --Salix alba (talk) 00:40, 21 February 2006 (UTC)

[edit] Just be a disambig and move content to sub articles

I noticed that the Mathematics#Major themes in mathematics just points to Abstract algebra rather than here. So it seems that we will be duplicating much of the material in that sub article. I propose that we make this a simple disambig page and move the relavant content to the four sub pages and so that this article becomes not much more than is in Algebra (disambiguation). This would allow readers to quickly get to the topic they are interestend in. In many cases (say Chain rule) the links really should be to Abstract algebra, but thats too much heavy lifting, as so many pages link here. This would also help school pupils, who are say looking for why -1×-1 = 1 (a very common question in the the college I've worked at), to get to their topic quickly with out having a lot of obscure mathematics thrown at them.

Just Curious: what does Chain Rule have to do with Abstract Algebra anyway? Skiperson 19:57, 6 October 2006 (UTC)

So the basic question, is how much do we want to go here, how much in the sub pages, and how much we wish to duplicate material? --Salix alba (talk) 01:11, 21 February 2006 (UTC)

[edit] Integers modulo?

I am not a mathematician. The end of the following sentence confuses me: "Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2+bx+c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n."

I am familiar with "integers" but not "integers modulo n." Maurreen 21:04, 20 March 2006 (UTC)

This is explained in the modular arithmetic article. I've added a link. --Zundark 22:07, 20 March 2006 (UTC)

Thanks. Maurreen 22:25, 20 March 2006 (UTC)

[edit] al-jaladi

I was reading an article on yemen and noticed this:- "And in Zabid, almost a millennium ago, the scholar Ahmad abu Musa al-Jaladi dazzled students from the new enquiring world of Islam with the resurrected intricacies of "al-Jabr" or algebra." It is from an old travel article on yemen in the Guardian newspaper and as i am from yemeni origin i have been to zabid before and remember being told that a grave which we visited was of the founder of algebra. i have also heard the story many other times. The article is from this link :- http://www.guardian.co.uk/Archive/Article/0,4273,4248384,00.html wonder if anyone can do anthing to the article?

I think it might be Al-Khwarizmi you are refering to, the full name (Abu Ja'afar Abdullah Muhammad Ibnu Musa Al-Khawarizmi) seems close, different spellings might be due to translation problems. I don't know enough about this period to do more than speculate. --Salix alba (talk) 22:07, 23 March 2006 (UTC)

[edit] History

There seems to be quite a bit of duplication in the history section at the moment. Material in the first few paragraphs appears again in the time line, almost with identicle copy. --Salix alba (talk) 22:09, 23 March 2006 (UTC)

[edit] Elementary algebra

The article says: "Algebra may be roughly divided into the following categories: elementary algebra, in which the properties of operations on the real number system are recorded using symbols as 'place holders' to denote constants and variables ..."

1. Is it safe to lose "the properties of" and just make it "... in which the operations on the real number system ..."?

2. Also, this statement implies that in other types of algebra, symbols are not used "as 'place holders' to denote constants and variables". Is that correct? Maurreen 04:45, 24 March 2006 (UTC)

[edit] Not Related to Wikipedia Per Se

Hi, I'm beginning to self-teach myself mathematics. Basic algebra was the last stuff I forgot so I'm starting there. My question is, what is the difference between basic algebra, intermediate, and advanced? What gets discussed in each of these that doesn't get discussed in the others? I understand that this is a kind of sophistical division, but I am curious anyways. Thanks for the help. 134.53.26.113 17:34, 28 March 2006 (UTC)

• Basic algebra (or elementary algebra) deals with numbers and polynomials. At a basic level, you learn a few rules and learn some properties of finding roots of polynomials and the like, but at a more sophisticated level, you start to explore the structure of these objects. The rules you learn in elementary algebra are special cases of more general rules, and the numbers and polynomials are simple examples of more general objects (elements of fields and polynomials over those fields). And by considering more general cases, you study the properties of fields and rings and groups and all their cousins (monoids, semirings, semigroups, etc.) And that's where it gets fun :) - grubber 19:39, 28 March 2006 (UTC)

• • Maybe it's too late for the question, but I'm wondering whether #134 was talking about the division between Algebra I and Algebra II as they are generally taught in high school mathematics. --Math Teacher 15:13, 27 June 2006 (UTC)

[edit] GA nomination

This article was nominated for good article status, but I don't think it meets the criteria - first of all in the intro, the sentence Algebra is much broader than elementary algebra and can be generalized is quite unclear. More significantly, the huge timeline list under the history section really needs to be made into prose. There appear to be two reference sections, and generally, see also sections should not be necessary - related topics should be mentioned in the text if they're significant. Worldtraveller 21:22, 24 April 2006 (UTC)

I actually quite like the timeline as a timeline though perhaps it could be broken up into an "early/middle/late" period. However, I do agree that this does not seem to meet good article standard. --Richard Clegg 22:46, 24 April 2006 (UTC)

[edit] My revert from "Islamic" to "Persian"

I changed it to follow the suit of the rest of the article. In the rest of the artcile, the other mathematicians are references as Indian, Japanese, etc. and not Hindu, Shinto, etc. Thus, instead of referring to these mathematicians as Islamic, they should be referred to by their ethnicity, Persian.--ĶĩřβȳŤįɱéØ 06:49, 6 May 2006 (UTC)

This is a common mistake. Islamic is not a relgious term, its is as I said before a chronological term, as in Islamic Civilization.Its like using the label Hellenistic mathematician. This label is also used in major encyclopedia's like Britanica. Check out Al-Khwarizmi there: [1], the word "persian" doesnt appear once. All Islamic mathematicians have in common that they wrote in arabic, studied in arabic, and did their most works in islamic capitals like baghdad, cairo, or cordoba, unlike the Indian or Japanese mathematicians who lived and did their work in their own countries(India or Japan), and wrote in their own languages. By the way; dont think I am doing this only becasue they are persians, I have done this also to arabs. jidan 09:26, 6 May 2006 (UTC)

Your example of Hellenistic only exlemplifies my point. Should Egyptian mathematicians be called Hellenistic instead of Egyptian? Of course not, because calling them Hellenistic makes someone assume that they were Greek. And just because they wrote and studied in Arabic doesn't imply Islamic; that was done simply because Arabic was the lingua franca. Many scholars wrote in Latin, but are called German, not Italian. Not writing in their own language does not mean they should not be classified according to their own language. Now I understand your viewpoint, so how about a compromise then? Persian Islamic mathematician? Or Islamic Persian mathematician? --ĶĩřβȳŤįɱéØ 16:32, 6 May 2006 (UTC)

Ok, choose the one you think most fits, but please leave Al-Khwarizmi as it is. There have been a huge ethnic-war over him as you can see from the size of his talk page [[2]]. I was hoping we could label all Islamic Scientist without ethnitices, since this totally irrelvant to this article, and since this is what I have been doing all time, and IMHO the one that is most accurate, as once George Sarton, a Belgian-American polymath and historian of science, in his book "Introduction to the History of Science" said:

On 8 June, A.D. 632, the Prophet Mohammed (Peace and Prayers be upon Him) died, having accomplished the marvelous task of uniting the tribes of Arabia into a homogeneous and powerful nation. ...In the interval, Persia, Asia Minor, Syria, Palestine, Egypt, the whole North Africa, Gibraltar and Spain had been submitted to the Islamic State, and a new civilization had been established. The Arabs quickly assimilated the culture and knowledge of the peoples they ruled, while the latter in turn - Persians, Syrians, Copts, Berbers, and others - adopted the Arabic language. The nationality of the Muslim thus became submerged, and the term Arab acquired a linguistic sense rather than a strictly ethnological one.

And the number of arab scientist are not as few as you might have thought( see List of Arab scientists and scholars - still incomplete). jidan 16:48, 6 May 2006 (UTC)

Yes, you are right. This is entirely irrelevent to the article, which is why I decided to compromiseon the issue. However, with Al-Khwarizmi, the issue is different. We can talk about him on his talk page. I'll choose Persian Islamic mathematician.--ĶĩřβȳŤįɱéØ 17:54, 6 May 2006 (UTC)

I am not persian and I am not arab, but I heard Al khwarizmi was persian and zoroastrian. Mrdthree 18:06, 6 May 2006 (UTC)

I agree the pushing of Islamic in front of Persian is a childish and looks ridiculous thus I have removed it.--SilverSurf 03:41, 16 May 2006 (UTC)

[edit] Inverse Element or Inverse Operator?

Does the concept "inverse" rightly apply to elements/operands? ... or to operations/operators? Would it be more correct, or less correct, to explain as follows:

For a general binary operator (binop) and identity element e, an inverse operator (invop) must satisfy

 a (binop) [e (invop) a] = e

For example

--Lonestarnot 17:08, 29 June 2006 (UTC)

The axioms of group theory require that every element must have a unique inverse; this inherently suggests there is a (bijective) unary operator "inverse" that maps elements to their inverses. So, they're really saying the same thing. I'm not sure if I said that clearly, or if that was your question.. haha - grubber 00:01, 30 June 2006 (UTC)

[edit] (,−) has identity?

In the table of the Groups section it is said so, and that the identity is 0. Well, it is true that for every a in a−0 = a But it is not true that 0−a=a. Instead, 0−a=−a I think the table is wrong, because in non-conmutative magmas (in any magma, really) the identity element should be neutral both by the right side (a*0) and by the left side (0*a). Otherwise, simply there is not identity element, and the table should say NA... Am I wrong?

However, it is true that (,−) is a quasigroup, which is the essential conclusion.

I will change the table as soon as I feel sure... please, help me! --Vivero 17:45, 29 September 2006 (UTC)

You are right: 0 is a right identity of (Q,−), but not a left identity. --Zundark 18:44, 29 September 2006 (UTC)

Thank you, Zundark. I have changed the table Vivero 15:43, 1 October 2006 (UTC)

[edit] Algebra does not equal arithmetic algebra

This whole article is written around arithmetic algebra. But an algebra is something more. Arithmetic Algebra is just an algebra, but not THE algebra. (unsigned, but added by Keikoforever)

I'm not sure what you mean. Very little of the article even mentions arithmetic. - grubber 20:26, 2 November 2006 (UTC)