Talk:Algebra/Archive 1

Archive 1

Archive 2

Untitled

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)

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)

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)

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.--[[User:65.144.45.232|GLBNG,FBFGNJHCJSVBJHVBHSJAKKDHFJUROQWPBCJVsection 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)

anything is ununderstandable to me, i donnot think that algebra is a good thing, it is only a thing to waste out time, for me, i have never faced the problems of the algebra in my real life.

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)

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)

"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)

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)

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)

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)

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)

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)

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)

New Question .... In history why is the indian influence on Khwarizmi not mentioned? It is well documented that the Arab mathameticians had translations of the works of Aryabhata and Bhramagupta and further developed them. The name of Aryabhata is not even mentioned in history ! This needs correcting.

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)

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)

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)

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)

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

 $a+[0-a]=0\$ 
 $a\times [1\div a]=1\$

--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)

($\mathbb {Q}$,−) 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 $\mathbb {Q}$ 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 ($\mathbb {Q}$,−) 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)

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)

Editors should not unilaterally start converting eras from BC to BCE (or vice versa)

The policy on this is in the Manual of Style, at [3]. I'm quoting two points that apply to this case:

• "Both the BCE/CE era names and the BC/AD era names are acceptable, but should be consistent within an article."
• "When either of two styles are acceptable it is inappropriate for a Wikipedia editor to change from one style to another unless there is some substantial reason for the change..." EdJohnston 15:50, 12 December 2006 (UTC)

It's just a link, not a text.By text, it must be consistent;by link, it would be valid.--Kilva 00:20, 13 December 2006 (UTC)

OK, you mean it's part of the URL of the link, not in the visible text. That's reasonable. EdJohnston 00:38, 13 December 2006 (UTC)

History section wrong

The history section is currently wrong. By 1535 Fontana and several others had the complete solution to the general cubic equation. By 1545 Cardano had published a method also due to Fontana that solved the general quartic equation. The section claims that these solutions are only for specific equations, but they are actually general. The reason for this confusion is probably that the solutions relied on reducing the equations to ones of a simpler form through various transformations. I am not as familiar with the work of Kowa Seki, however I believe there is a good chance that the line on him is also incorrect in a similar way. I am fixing what I can and adding some references, but can anyone comment on Kowa Seki? Grokmoo 17:54, 11 February 2007 (UTC)

Algebra is derivative from the word Algebr not Algabr.TylerJarHead

I couldn't find any books on Seki, so I had to use a website as reference, but I think I have made the blurb on him factually correct as it now stands. Grokmoo 17:55, 19 February 2007 (UTC)

What is the car?

I told my friend that you could apply algebra to almost anything. So he told me to solve "The car is _____" with algebra. Any suggestions?--Maier 03 23:57, 1 March 2007 (UTC)

• the key word would be that you said "almost" everything.

68.192.50.109 (talk) 10:43, 24 March 2008 (UTC)

Origin of word 'algebra'

My concern is that the person below making the statement starts with "um". The writer's assertion that "Al" is a traditional prefix meaning "of" or "from" is incorrect. "Al" is the definitive article "the". [Unsigned]

Um, in Arabic, 'Al' is a traditional prefix meaning 'of' or 'from', so having the entire thing mean 'reunion' or something of the like seems incorrect. A little searching reveals this gem of a web-page-nugget: algebraquoted here as follows:

Origin of the Word Algebra The word algebra is a Latin variant of the Arabic word al-jabr. This came from the title of a book, Hidab al-jabr wal-muqubala, written in Baghdad about 825 A.D. by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi.

The words jabr (JAH-ber) and muqubalah (moo-KAH-ba-lah) were used by al-Khowarizmi to designate two basic operations in solving equations. Jabr was to transpose subtracted terms to the other side of the equation. Muqubalah was to cancel like terms on opposite sides of the equation. In fact, the title has been translated to mean "science of restoration (or reunion) and opposition" or "science of transposition and cancellation" and "The Book of Completion and Cancellation" or "The Book of Restoration and Balancing."

Jabr is used in the step where x - 2 = 12 becomes x = 14. The left-side of the first equation, where x is lessened by 2, is "restored" or "completed" back to x in the second equation.

Muqabalah takes us from x + y = y + 7 to x = 7 by "cancelling" or "balancing" the two sides of the equation.

Eventually the muqabalah was left behind, and this type of math became known as algebra in many languages.

It is interesting to note that the word al-jabr used non-mathematically made its way into Europe through the Moors of Spain. There an algebrista is a bonesetter, or "restorer" of bones. A barber of medieval times called himself an algebrista since barbers often did bone-setting and bloodletting on the side. Hence the red and white striped barber poles of today.

I think it would be nice if the word origin was robusted up a bit, and actually made reference back to the document that was the source of the name: Hidab al-jabr wal-muqubala ":) --Fish-man 00:01, 12 October 2005 (UTC)

2 different definitions of the source for the word algebra

In the first paragraph, the origin of the word algebra is described as

"Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing")"

And in the History section, first paragraph as

"The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction"

The two arabic sentences differ, and (presumably because of this) so do the translations. —The preceding unsigned comment was added by 82.46.78.227 (talk) 23:05, 4 March 2007 (UTC).

SCIENCE Is Related To ALGEBRA.

You can see how ALGEBRA is related to SCIENCE. such in which is "PERCENTAGE" sometimes even Variables, For Example: You use alot of FORMULA'S in science especially CHEMISTRY./Kulse Du Silphe/[3/14/07] —The preceding unsigned comment was added by 205.252.70.2 (talk) 14:33, 14 March 2007 (UTC).

This is already explained in the article. --Ķĩřβȳ♥ŤįɱéØ 05:28, 15 March 2007 (UTC)

On the Redudancy of Closure...

The present version of the article states, following the requirement that a group must be closed under its binary operation: "In fact, it is redundant to mention this property, for every binary operation must be closed. So, the statement "a group is a combination of a set S and a binary operation '*'" is already saying that the operation is closed. However, closure is frequently emphasized repeating it as a group property."

I disagree that an explicit mention of closure is mere redundancy. Saying that a "binary operation must be closed" without specifying the set in which it is closed leads to ambiguity.

A subset of a group, such as the kernel of a homomorphism, inherits the operation from its parent group. The binary operation is closed with respect to the parent group in which it was defined, but it still needs to be verified that the operation is also closed in the subset S under consideration. Example: Addition is a binary operation defined on the integers, and subsets of the integers inherit that operation. If S ⊂ Z is the set {0, 1, 3}, 1 + 3 inherits an unambiguous meaning, and it is not an element of S. On the other hand, if S is the set 2Z = {..., -4, -2, 0, 2, 4, 6, ...}, then the inherited operation from Z is also closed under S, so S is a subgroup of Z.

One may argue that the operation "+_Z" defined on Z and the operation "+_2Z" defined on 2Z are technically different binary operations, since one is a function from Z × Z → Z and the other is a function from 2Z × 2Z → 2Z, but in practice we hardly ever make this distinction. -- Heath 24.127.52.67 14:10, 23 March 2007 (UTC)

Strictly speaking it is redundant, because, as you have said, a binary operation is defined on specific set S, and must be closed. However, this is interesting to think about. Strictly speaking, I think the operation "+" on Z and "+" on 2Z really are the same operation, since the elements in 2Z really are elements in Z. It is not the case that Z contains an isomorphic copy of 2Z, where then the two "+" operations would really be different. So then I think that if you have a subgroup, it actually isn't redundant to specify it must be closed under the given binary operation, since it really is the same operation as that of the larger group. In any event, I think the little blurb about closure is wordy and I would not object to its removal. Grokmoo 16:58, 28 March 2007 (UTC)

Differential Equations in history of algebra?

I noticed in the history timeline that there are several mentions of differential equations and solutions to differential equations, particularly when being attributed to Indian mathematicians. While I am skeptical of many of these claims, I think that the subject matter (differential equations) is not really related to the core of this article. So, as a way of shortening the rather long timeline, I think these references should be removed from this page, regardless of their veracity. I am removing them for now, but if someone has objections, we can put them back in until a consensus is reached here. Grokmoo 17:06, 28 March 2007 (UTC)

Proposed link

My students have had great success with my Precalculus course notes, which emphasize algebra. They have enjoyed my section on graphs of high-degree polynomial functions. I believe the Precalculus article is barely maintained by anybody. Would anyone be interested in posting an external link to them? http://www.kkuniyuk.com/Notes Thanks! (Sorry for my earlier link; I'm a new user) Ken Kuniyuki 00:25, 1 April 2007 (UTC)

I put a reply at Talk:precalculus. Cheers, Doctormatt 02:46, 1 April 2007 (UTC)

Proposal to reorganize the references

This article now has TWO reference sections, which seems excessive. In the first of the two, numbered citations are mixed in with normal references. I'd like to tidy this up. One idea is to put all numbered citations in a 'Notes' section, and the others in References. I'd also like to prune the external links using a strict application of the WP:EL policy. Any comments on this plan? EdJohnston 16:47, 1 April 2007 (UTC)

Well, I'm not sure. The numbered references are really references, but they are being used in the page, whereas the other references are just for the general subject, I guess. I notice now that a number of the references don't seem (from the title) to pertain much to algebra; it is likely that information from these sources is confined to only a small section of the article, especially the history section. I think we should either get some inline citations for these references or get rid of them.

Also, I noticed that #2 and #3 in the numbered references are from the same source; shouldn't they just be one numbered reference? I'm not all that experienced with Wikipedia's policies on references, so I'll defer to your opinion. Grokmoo 17:15, 5 April 2007 (UTC)

vandalism reverted

Jddphd 04:23, 11 June 2007 (UTC)

reference clarification

The first reference in this article is this:

Toomer, Hogendijk 1997, Oaks

I don't know what this means. It seems there is a Hogendijk who is a student of Toomer, so this is perhaps a joint paper. I have no idea what "Oaks" means here, and according to MathSciNet, Hogendijk published nothing in 1997. Can someone please fix this reference, by adding journal name, article title, etc.? Thanks. Doctormatt 04:57, 30 October 2007 (UTC)

Okay, Lambian has improved this, but should that be Hogendijk 1998? There is no 1997 citation for Hogendijk. Doctormatt 07:31, 30 October 2007 (UTC)

That should indeed be 1998: the 1998 article has the sentence (in Dutch) Hij behoorde tot een Perzische familie, which must mean something like "He came from a Persian family". Here is a page on Hogendijk establishing that he is an authoritative source. The error was copied from the al-Khwārizmī page; I've corrected it both there and here.  --Lambiam 11:32, 30 October 2007 (UTC)

Failed GA

I have failed this GA, because in addition to renovations and polishing, the article also hasquite a few large holes that need to be filled in.

• this seems like a difficult topic to cover since there is so much
• Housekeeping stuff.
• There are not that many inline citations for what is there and for those that are there, they need to be done in a consistent manner etc. There is supposed to be a booklist. This needs to be in alphabetical otder, and the format should be the same. {{cite book}} is a good place to start. Use last name, first name and so forth, full ISBNs/ At the moment not all the references are in the same style. Then for the inline cites there is a separate section with “Boyer, p. 228.” And so forth. Alternatively source in the daughter article per WP:SS – the daughter articles are not currently inlined.
• See also is usually before the refs.
• Timelines and dot point lists are not preferable in articles. The timeline list should be turned into narrative form. The history section would benefit since an intertwining of the developments would make the evolution of algebra more natural to the reader.
• The other thing that is unusual is that the list of algebraic developments stops at 1832 which seems unusual. What about things like Lie groups and the Sylow theorems, among others. But the end of development in 1832 seems a bit unusual.
• The section about structures known as algebras is not properly developed and is only a list of dot points. The reader who has not studied pure maths at uni will not know what “over a field” or “over a ring” is.
• If the concept of integral domain is discussed, then the element of unity needs to be discussed. Simply saying additional properties does not shed any extra light.
• ”it does not need to have identity, or inverse, so division is not allowed” is definitely wrong. Fields are rings and in fields, division is possible because the inverse exists. It should be “division is not needed” right?
• Monoid and semigroup stuff should go before the table of examples.
• You should say what a simply group is. Then that would require people to define a normal group, whjch might complicate things for the lay person. Perhaps an analogy with finite groups being broken into a composition/principal series being akin to a number being broken into primes, so that finite simple groups can be regarded as “prime groups” and basic building blocks and so forth.
• At the moment however, the article is not complete. The start of the article mentions the different subdivisions of algebra, but only elementary and abstract algebra are discussed. Vector spaces, alegraic geometry and so forth are not discussed. So there is a large gap in the coverage of the topics. So those parts need to be covered.
• We also need a motivation on why algebra is studied. Eg, abstract algebra allows us to abstract the structures of the algebra and study them at large and so forth. Other uses, such as Lie algebras being studied in physics to study symmetry of transformations and so forth, or Lie algegras being used as a handiwork in differential geometry for studying localised neighbourhoods and so forth. It doesn’t really say how algebra links in with other maths or theoretical physics.
• At the moment the article is a half which is a manual of definitions and another half is a general history. There needs to be more in the middle ground as well I feel. It’s difficult but at the moment the first half of the article doesn’t seem particularly useful except as a definitions manual and doesn’t really tell anything about the world of maths and how things fit together.
• Sorry if the last few comments aren’t really useful in a direct sense.

Best regards, Blnguyen (bananabucket) 01:25, 6 November 2007 (UTC)

The Classification

I don't really think the classification given is appropriate. For one it is in contradiction with the introduction to this article: the intro mentions number theory, geometry and combinatorics as other branches of math, thus algebraic number theory/geometry/combinatorics are rightfully part of these other branches that happen to use methods from this branch of mathematics or ask questions in an algebraic vein. To me, a better classification would be along the lines of: Elementary Algebra, Linear and multilinear Algebra, Ring Theory, Group theory, Homological algebra, Topological algebra (to deal with analytic stuff), and something about other "exotic" structures (e.g. Jordan algebras, Clifford algebras). More thought should be put into this, but I'll wait to see if people agree with me in principle first. Any comments? RobHar (talk) 09:41, 26 November 2007 (UTC)

QUESTION

A Certain car covers12km at a certain average speed.if this average speed is increased by 10km/h.the car takes the same amount of time to cover a distance of 20km.find the speed of the car in the first part of the journey? —Preceding unsigned comment added by 196.3.61.4 (talk) 15:17, 23 January 2008 (UTC)

This page is for discussing improvements to Wikipedia's Algebra article, and not for discussing algebra problems. Try Wikipedia:Reference desk/Mathematics instead.  --Lambiam 22:19, 23 January 2008 (UTC)

Problem with word problems

Question? Suppose that a market research company finds that a price of p=$20, they would sell x=42 tiles each month. If they lower the price tom p=$10, then more people would purchase the tile, and they can expect to sell x=52 tiles in a month's time.Find the equation of the line for the demand equation.Write your answer in the form p=mx+b(hint: write an equation using two points in the form (x,p)) I have more put I want to see if i can finish it. please help —Preceding unsigned comment added by Lighteyes22003 (talk • contribs) 15:51, 19 March 2008 (UTC)

Question has been reposted at the Maths section of the Reference desk.  --Lambiam 11:51, 20 March 2008 (UTC)

Reason why Algebra formed

Khwarizmi contributions on top of the Indian mathematics was based on a determination to predict the future. In Islam, there are two consepts of "Algaber" and "Alghader". Algaber means everything has a destination and God almighty is aware of all destinations. Alghader means all beings have the power to change their own faith and destiney. As a Muslim, he was interested to know if a change in "controlling" variables can determine destination of "unknown" variable(s). In his book, Algabre val moghabelah, he devised techniques to resolve and predict destination of unknown varibles. During crusades between Christians and Muslims, Jacobson broders tried to tranlsate his book in english and since they were not able to find a good word to do the name justice, they settled on the name "Algebra".—Preceding unsigned comment added by Msalehisedeh (talk • contribs) 21:02, 31 May 2009 (UTC)

Algebra Comes From Arabic

I changed the unsourced opening sentence, with a more factual, sourced one. To my knowledge, the word Algebra comes from the Arabic Al Jabr. It might have found its way into Farsi/Persian after the Islamic conquest of that region, along with many other words. I'm sure it can be found somewhere in the Quran/Early Arab/Islamic texts, which is proof it was originally an Arabic word before contact with the Persians. I am currently researching that with people who actually do read religious texts, unlike myself.

Despite being of Persian heritage, Mohammad ibn-Musa al-Khwarizmi spent most his life, and implimented most his studies in Baghdad and Cairo - Arabic countries where he undoubtably had to adopt the Arabic language in order to communicate, maybe even becoming Arabised later on. Thus his book's title, where Algebra came from, was Arabic not Persian.

Either way, I have replaced the unsourced material with a sourced info. Apologies I don't know how to put sources in with those little numbers, I'm learning how to do it now. The sources are:

- http://www.und.nodak.edu/instruct/lgeller/algebra.html

- http://mathworld.wolfram.com/Algebra.html

- http://wiki.answers.com/Q/From_where_did_the_Word_algebra_come_from

- http://83.223.102.16/index.cfm?fuseaction=main.viewBlogEntry&intMTEntryID=2740

- also, further down the article it says: "The word "algebra" is named after the Arabic word "al-jabr , الجبر" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala , الكتاب المختصر في حساب الجبر والمقابلة, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Islamic mathematician Muhammad ibn Mūsā al-khwārizmī in 820." The first sentence is contradictory to that - as well as virtually every reliable internet source.

Also, I'm researching claims that Algebra might have been developed further back in time than al-Khwarizmi by someone whose name is Al-Jabr - which to this day is a common Arabic name.

Please feel free to contact me of you are doing similar research, or willing to help. Take care. Pink Princess (talk) 17:52, 28 August 2008 (UTC)

Algebraic combinatorics

Resolved

{{editsemiprotected}} I found an error. Algebraic_combinatorics points to http://en.wikipedia.org/wiki/Combinatorics instead of pointing directly to http://en.wikipedia.org/wiki/Algebraic_combinatorics. Good day.—Preceding unsigned comment added by 84.108.53.132 (talk • contribs)

OK done. --Salix (talk): 11:00, 26 May 2009 (UTC)

should we mention

That this is one of the more toughest part in maths cause a lot of people seem to find it to be a very tough subject. user--Stephendwan (talk) 19:54, 14 December 2009 (UTC)

No. Besides not being relevant, it is also quite untrue. David spector (talk) 00:59, 16 December 2009 (UTC)

First paragraph of history section

Well I made a change and it was reverted, asking me to discuss it here first. I didn't realize what I said was controversial. I'm not sure what format this should be in since I don't edit too much. Here's what I was thinking:

"While the word algebra comes from the Arabic language (al-jabr, الجبر literally, restoration)" -- this is discussed more completely in the 4th paragraph so it seems redundant

"and much of its methods from Arabic/Islamic mathematics" -- Just reading over the rest of the algebra page, it's clear that "much" of it didn't come from one culture. Contributions from India, Greece, Babylon, Egypt, and Europe are mentioned, although the volume of work isn't detailed. Nowhere is Islam even mentioned so I have no idea why that was added. And since Persia isn't the same as Arabia, the Arabic part is also suspect. Fact is algebra evolved from many different cultures, it's a cumulative body of work with roots in ancient Babylon (as mentioned in the 2nd paragraph).

"its roots can be traced to earlier traditions, most notably ancient Indian mathematics" -- Why is ancient Indian mathematics the "most" notable early tradition? Ancient India certainly contributed a lot, but so did the Greeks, and the genesis of algebra in Babylon seems pretty notable too. I think this is just opinion on the part of the author so I removed it.

"which had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780-850)" -- this seems like too much of a surprise introduction. I think the treatment in the 4th paragraph is simply better worded and flows better, plus there's no question the information is redundant with the 4th paragraph, so I deleted it.

"He later wrote al-Kitāb al-muḫtaṣar fī ḥisāb al-jabr wa-l-muqābala (The Compendious Book on Calculation by Completion and Balancing)" -- the translation of the book's title differs from that in the 4th paragraph. I know translations are subjective, but shouldn't we stick to one, or state both at the same time? The way it's currently worded just looks like an error. Still redundant either way.

Okay that's what I deleted in the first paragraph, leaving basically nothing, so I wrote another paragraph to serve as more of an introduction with less detail and less redundancy.

This is what I added:

"Algebra as we know it today does not have a fixed origin in any one culture or region." -- This is just patently true. Algebra "as it's known today" didn't spring up in finished form in one culture. As an example, many of the familiar symbols we use today such as the equals sign just weren't around 1000 years ago. In terms of algebraic concepts, well, since we have major concepts coming from more than one culture (example: cubic roots from Persia, integer exponent equations from Greece) it's obvious they can't all have come from one culture.

"Instead, algebra is a body of knowledge that accumulated over time with additions from Babylonian mathematics, Egyptian mathematics, Azn mathmatics, Indian mathematics, Arabic mathematics, and European mathematics" -- This is basically a summary of what the rest of the history section is about. Highlights from each of these cultures is discussed in a relatively chronological fashion.

"as well as strains that developed more independently such as Chinese mathematics." -- I added this but it's true, just check the Chinese mathematics page.

Anyway, that's it. What do you think?

Stdarg (talk) 16:45, 1 February 2010 (UTC)

classification

Well, I am quite astonished: the following appear to me to be part (or subbranches) of abstract (or modern) algebra, and not distinct branches!

1. Linear algebra
2. Universal algebra
3. Algebraic number theory
4. Algebraic combinatorics

of course, together with

1. group therory
2. ring and field theory,

probably also together with

1. Algebraic geometry

--78.15.195.181 (talk) 21:32, 27 March 2010 (UTC)

Elementary... algebra?

The problem to me seems to be that elementary algebra and abstract algebra are two completely different things with no special connection. They just happen to have the same surname, like Dave Smith and Harry Smith or something. Classifying mathematical subjects according to their names seems to me to be no more informative than classifying people according to their names, when two subjects/people can have the same name by coincidence.

When students do elementary algebra they are doing integer arithmetic or (naive) real number arithmetic, except with variables as well as specific numbers. It starts being called algebra when one moves from specific integers, to arbitrary/variable integers. The same goes for real numbers. It is the move from just having constants, to having constants and variables. But this has nothing to do with any particular branch of mathematics. All branches of mathematics do this. What on earth does it have to do with abstract algebra that it doesn't have to do with combinatorics or analysis, say?

A better name for elementary algebra would be "elementary arithmetic/number theory" or "elementary (naive) real analysis".

We can all think of better names for things, but what we cannot do is change what things are called. Elementary algebra is much more like abstract algebra than it is like advanced arithmetic or number theory, so I don't understand your objection. In any case, many more people understand "algebra" to mean "using letters to represent numbers" than any other sense of the word. Dbfirs 12:22, 17 June 2009 (UTC)

Elementary algebra has something in common with advanced algebra. Both are the study of the rules of operations. In elementary algebra, these rules are taken from real numbers, or some subset thereof. We think about what rules hold if we know that x is a number and use those rules, for example to solve an equation. Advanced algebra is just the study of what happens when those rules change.

Actually, warming to my theme, I would say that something along the above lines should be stated in the article. Yaris678 (talk) 12:08, 5 August 2009 (UTC)

Second paragraph of history section

I have made a change to the first introduction to Muhammad ibn Mūsā al-Khwārizmī; I have added that he is Persian, as the preceding words can cause misinterpretation that he is of Arabian origin. It is common in articles for notable figures to be distinguished by there nationality.

Can somebody edit the introduction to read this: "While the word algebra comes from the Arabic language (al-jabr, الجبر literally, restoration) and much of its methods from Arabic/Islamic mathematics, its roots can be traced to earlier traditions, most notably ancient Indian mathematics, which had a direct influence on the Persian scholar Muhammad ibn Mūsā al-Khwārizmī (c. 780-850)." —Preceding unsigned comment added by 203.161.88.18 (talk) 04:41, 6 October 2010 (UTC)

"father of algebra"

I changed this sentence:

 "a book written by the Islamic Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī (considered the "father of algebra"), in 820."

and got rid of the (considered the "father of algebra") part. The next few sentences in the paragraph dispute it, so why is it there? Diophantus has been considered the father of algebra for a long time, and now there is debate over whether al-Khwarizmi would be a better candidate. It's certainly not already decided that he is considered the father of algebra. This parenthetical should be removed, leaving the more in depth discussion in the subsequent sentences to explain the situation.

Stdarg (talk) 16:45, 1 February 2010 (UTC)

Then you should remove the sentence: " Diophantus (3rd century AD), sometimes called "the father of algebra"..." Because, there is an ambiguity who is "father of algebra". —Preceding unsigned comment added by 188.41.10.205 (talk) 10:23, 17 August 2010 (UTC)

History

Should we unify the Latin transliteration of the arabic الجبر, currently sometimes al-ğabr but sometimes also al-jabr? (maybe al-ğabr should be preferred) --Ernsts (talk) 20:07, 6 March 2011 (UTC) The Mother Tongue for Mr.Kharazmi was Persian so one has to goback in time and find the equivalent of algebra. In times of Kharazmi Persia was occupied by Arabs. so the books hdtobe written in Arabic or they would be burned. Kharazmi's name shows the city where he was born an that was in Persia (currently part of Uzbekistan) so w must refr to him as Kharazmi and not Al-kharazmi to show him as an Arab. It maybe ok to use the word algebra for s work but his name should be usd correctly. — Preceding unsigned comment added by 220.178.50.227 (talk) 17:01, 20 December 2011 (UTC)

5th paragraph

I assume that the last two sentences, which now looks like «His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."» (the actual ref-link has the number '12'), should look like this: «His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".», because the quotations are not entire sentences (Wikipedia:Manual of Style→Punctuation). I think also that for the former quotation, its reference should be inserted, too. If it is the same as of the latter, well, it'll be the same as for the latter all right. JLincoln (talk) 11:30, 2 September 2011 (UTC)

Bt i think it all started in India. All these INDIAN topics were then taken to Arab nations n then it got universally accepted with those arabic names. — Preceding unsigned comment added by S.R.Aniruddha (talk • contribs) 16:57, 1 January 2012 (UTC)

Lagrange

I noticed that the link in the 7th paragraph of History section doesn't point to Joseph-Louis Lagrange, as it should, but to Count Joseph Lagrange. I didn't know how to correct it myself.

93.96.143.64 (talk) 19:26, 3 December 2011 (UTC)

Someone had created a new page at Joseph Lagrange replacing the redirect. I've moved that page elsewhere and have restored the redirect, to fix this and a few other pages that used it, as the mathematician is the most well known claimant on the name.--JohnBlackburne^words_deeds 20:03, 3 December 2011 (UTC)

Rings and Fields

I looked under the Rings and Fields section and found that the editor used "1" as the multiplicative identity and "0" as the additive identity. Should these identities be generalized to include fields besides those of the complex numbers? Fraqtive42 (talk) 03:25, 14 September 2011 (UTC)

Edit request from Cal-linux, 21 September 2011

All of the occurrences of Abelian should be replaced by abelian (except of course if starting a sentence).

See http://en.wikipedia.org/wiki/Abelian_group#A_note_on_the_typography (though they say "often written", from all of the math teachers I've had on related areas, they emphasize and note it as a remarkable fact, that abelian is written with lowercase a --- not "often written", but just "written").

Cal-linux (talk) 00:22, 21 September 2011 (UTC)

Done, thanks. A simple search on Google scholar also confirms lower case is the more common typography among reliable sources. Adrian J. Hunter^{(talk•contribs)} 11:16, 22 September 2011 (UTC)

Please add the following sourced information

At the end of the 15th century the German mathematician Johannes Widmann held the first known university lectures on algebra at the University of Leipzig.
Source: Moritz Cantor, "Johannes Widmann", In: Allgemeine Deutsche Biographie, published by the Bavarian Academy of Sciences and Humanities, Volume 42 (1897), p. 355. — Preceding unsigned comment added by 79.230.171.53 (talk) 15:50, 12 July 2012 (UTC)

Edit request on 29 July 2012

The first section makes the assertion "Algebra has numerous usages in daily life". The page gives no examples of these usages. There should be a section addressing this assertion, or it should not be made. 71.250.39.46 (talk) 16:17, 29 July 2012 (UTC)

Not done: Changes have to be specified in an X to Y format. Floating Boat A boat that can float! 18:18, 29 July 2012 (UTC)

History

Is it just me or is the history section a bit weird? While there is no need for it to be exactly chronological, I find it odd that it starts in the middle (Greeks), goes to a later stage (Arabs), then jumps to the very beginning (Babylonians). I will re-arrange it in a more coherent format, but will be happy to take suggestions.

Cheers!

Λuα ^{(Operibus anteire)} 04:59, 30 July 2012 (UTC)

Edit request on 2nd of August 2012

Please add the following sourced information (cf. request above):
At the end of the 15th century the German mathematician Johannes Widmann held the first known university lectures on algebra at the University of Leipzig.
Source: Moritz Cantor, "Johannes Widmann", In: Allgemeine Deutsche Biographie, published by the Bavarian Academy of Sciences and Humanities, Volume 42 (1897), p. 355. 87.150.251.113 (talk) 18:45, 2 August 2012 (UTC)

Reason of the revert in oct 2012

I reverted much of the last edits by User:JimsMaher, (but incorporated good parts by Jims,, and improved the article a bit more) for the following reasons.

• In the Jims version, the first sentence is an abrupt assertion, saying that algebra means the abstract algebra. But, Jims retained the second paragraph which says that the bare word "algebra" has three meanings, so it's a contradiction. Furthermore, although it's up to the Wikipedia community what the scope of this page should be, and it can be changed later, the current page treats both (the history of) elmentary algebra and abstract algebra, and also contains a list including algebra over a field, F-algebra, etc.
  • (Less important) In the 4th paragraph, Jims says "algebraic branch of mathematics is...", but here, the use of the word "algebraic" is confusing, in the above meaning, so it should be avoided.
• The section "classification" is also better in my revision. None of elementary algebra, linear algebra, algebraic geometry, at least, are branches of "abstract algebra". The appearances of the word "algebra" there mean "(polynomial) equation (solving)". Algebraic combinatorics is more combinatorics than algebra, no? (As math is always so, algebraic combinatorics contributes to algebra, but that's less relevant.) (See also algebra (disambiguation) and, for example, Linear Algebra in MathWorld; the latter reads: Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra". In fact, I think this sentence lacks accuracy, too. "Linear algebra" means "the study of linear equation", so it's "algebra" in the classical sense.)

"Linear Algebra" is itself ambiguous. The study of vector spaces over an arbitrary field, linear transformations, subspaces, quotient spaces, etc. is part of abstract algebra. The usual course in matrix manipulation is not. As far as I could tell from my Commutative Algebra course, algebraic geometry involves a lot of abstract algebra.

• The sentence Algebra has numerous usages in daily life is not a hoax, but I'm afraid most examples are group theory, and a bit of finite fields, so I think it's not so accurate. Then it follows as and is commonly taught in public schools. Well, this refers to elementary algebra, not abstract algebra, no? This confusion was introduced because clear distinction of elementary algebra and abstract algebra was lacking.
• (Not so much important) The link to the article algebraic constructions at an early stage is not so much relevant to the context, thus nor so useful. That page is quite advanced. ("Elementary" topics like quotient rings and vector bundles are included, but they may already be too advanced for the lead parts of this overview article.)
• "father of algebra" reference at the lead is out of context, and it's meaningless without proper context. (And I think it's part of the silly "which is greater?" dispute. Only the facts count. That's science.)

I wrote in my previous edit that online etymology is not a good reference, but I withdraw it. But the reference may not be necessary, since it's explained in the "history" section.--Teika kazura (talk) 05:18, 15 October 2012 (UTC)

To be clear, the edits of mine you refer to were: copy edit, some reverted edits of Teika (mostly for clarity), and a few added links. For now, I am just restoring some deleted links and references, and leaving you with some questions ... Why would you delete a reference you added (I think I know why, but I'm restoring it)? Why would you include an advanced topic in the leading section, but consider accompanying related links as "not so much relevant"? Why do you insist on weasal words in section titles ("Important" topics)? And why did you need to bring these edits to the talk page exactly? JimsMaher (talk) 20:47, 18 October 2012 (UTC)

Hi. Thanks for your contributions.

Why would you delete a reference you added?: Etymonline says it's "al-jebr", and pocket oxford "al-jabr". Notice the difference. You cited both for "al-jebr", but it's wrong, and I don't know how to make both coexist. It may be better to delete it altogether from the lead, since the history section explains it, and Wikipedia is not a dictionary, so leaving it to wiktionary. (If you write al-jabr<ref>... OR al-jabr<ref>..., then I'm afraid that "OR" be an original research which Wikipedia should avoid. But adopting only one may also be wrong. If you write "according to A, al-jabr, and to B, al-jebr" it's correct but seems too clumsy to me.)

Why do you insist on weasal words in section titles ("Important" topics)?: I had to correct the word "classification", as I wrote above. Ok, you already deleted "important". Thanks.

Why would you include an advanced topic in the leading section, but consider accompanying related links as "not so much relevant"? (It must refer to "algebraic constrution" part.) Sorry, I don't understand the question. Anyway, leaving aside if it's advanced or not, the current contents of algebraic constructions has little to do with the current lead of algebra. I don't know the exact meaning of "algebraic construction", but that page seems to mean "applying algebraic (here, meaning abstract algebra, of course) concepts to existing notions to get another structure", while the lead of algebra means "recognizing abstract notions fundamental to abstract algebra such as groups, rings, etc." Now you put back the link to algebraic constructions, I wonder if you could fix the definition of that page to resolve the inconsistency.

And why did you need to bring these edits to the talk page exactly? Please understand you often have to communicate, since Wikipedia is open to anyone. Without open talk, it'd be an uncontrolled mish-mash. See also Wikipedia:Edit warring.--Teika kazura (talk) 00:35, 19 October 2012 (UTC)

I would've most likely made the same responding edits with or without your comment. As you've probably noticed by now ... most edits on Wikipedia occur without comment. For sake of communication: I'm not sure what you mean by "hoax", but vague statements without citation get deleted all the time. Hoaxes are not required for deletion.

And regarding 'constructions', I'll have to leave that as is, for someone more knowledgeable to can clarify. If you'd like, add the tag {{clarification needed|date=October 2012}} = ... or for an expert to review the page WP:EXPREV, except there is no such expert review system on Wikipedia. Hmm ... try an RFC instead if it's that much a concern. You should start a new section on this talk page for that, see WP:RFC. Take care. JimsMaher (talk) 01:52, 20 October 2012 (UTC)

I reverted the last three edits by Jims. Please read the talk between Jims and me (Section #algebra) for the reasons.

Thanks User:Quondum, for restoring the texts I wrongly deleted in this talk page. Sorry. --Teika kazura (talk) 04:47, 21 October 2012 (UTC)

'Algebra' has another meaning in measure theory. (A nonempty set of subsets of the given set closed under complement, finite union and finite intersection.)

Elementary algebra

I am not an expert but I think the explanation for "Algebraic_equation_notation.svg" is incorrect. "5" is undefined and should be labeled as a "constant". I think "c" is therefore not a variable. I can't change this page as it is locked to me. Will someone confirm and make any necessary changes? — Preceding unsigned comment added by 205.250.213.179 (talk) 00:45, 8 January 2013 (UTC)

First, there is no big difference between constants and variables. Sometimes it makes sense to declare a variable to be constant during some argument, or to make a constant variable. There is no strict rule against using c for a variable or z for a constant, although some teachers may impose such rules in the belief that that's helpful.

In an earlier version of the image, the 5 was a 3, and that was correct. Even a single constant (or variable) is also a term. I have added "5 - constant term" for clarity. Hans Adler 15:03, 14 January 2013 (UTC)

Table of examples of algebraic structures

The table is a little bit ugly, though it does give a lot of good information.

1. Rather than N/A (which might be interpreted as "not available"), I suggest using DNE for values which do not exist (i.e. the identity element for subtraction DNE in the reals, rationals, or complex numbers ).
2. The title of the table should be more descriptive, "Examples of algebraic structures" instead of merely "Examples." Or get rid of this row entirely.
3. The table should not be in the section marked groups, because it describes some structures which are not groups, such as the natural numbers. For this reason, it may be wise to move the table to the article on Algebraic Structures.

 — Preceding unsigned comment added by 71.20.83.90 (talk) 02:05, 31 January 2013 (UTC) 

Teaching (elementary) algebra

That sections needs some work - in particular the part:

It has been suggested that elementary algebra should be taught as young as eleven years old,[22] though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.[23]

The first source is article from is an article from 1904, which (under the assumption we generally talking about current education) is almost ridiculous as a source. The second source is at least recent but merely a newspaper article, which may use the term algebra us slightly different manner that the Wikipedia section (basic arithmetic/laws govern math operations and partially variables are taught way before 8th grade, I believe the second source refers to algebra as in working with (simple) equations & Variables). In any there is wealth of (somewhat recent) authoritative Literature in math didactics which should be consulted here and used as a source.--Kmhkmh (talk) 22:59, 24 February 2013 (UTC)

Complicated starter.

I think the opening is a mite on the complex side. In "Algebra as a branch of mathematics" it immediately uses a polynomial as an example. Considering the fact that the folks diving into algebra in school are at middle-school age, or are adults who are now going into college, leaping straight into a polynomial may be a bit of a shock - I can understand that to a person who already has a knack for math may find this a perfectly logical explanation, but not so much for the novice. A simpler equation not using any powers or multiple variables may be a little better, like 1+X=3. It's a simple equation that doesn't require Algebra to solve, but can be solved with algebra and the novice will know the correct intended answer from the beginning.--Varkman (talk) 22:34, 18 August 2013 (UTC)

Upgrading the article

I have edited algebra to clarify the function of this article (WP:DABCONCEPT article and article about a mathematics area). I have also transformed algebra (disambiguation) into a redirect, moved History of elementary algebra to history of algebra, and adapted the first lines of Elementary algebra and abstract algebra. The main parts of my edits are:

• A complete rewriting of the lead to make it clearly a WP:DABCONCEPT
• Inserting a new section "Algebra as branch of mathematics" to explain what is algebra and how the answer to this question has evolved.
• Splitting the history section into "prehistory" and "history".

As these edits are very far to be minor and this article is a vital one, some reviews of my edits would be helpful.

Moreover this article needs some more work implying some decisions that may be controversial.

• The section history is a stub that should be expanded. My competence in history of mathematics is not enough to do it myself.
• There are sections dedicated on two among the sub-areas (elementary algebra and abstract algebra), and almost nothing a bout the remainder of algebra, outside links without explanation in the new lead and in the section Topics containing the word "algebra".

My opinion is that

• The lists in section Topics containing the word "algebra" should be moved at the end and transformed in a kind of section "See also". In particular, the list of the sub-areas should be completed by the sub-areas whose names do not contain the word "algebra".
• The sections "Elementary algebra" and "Abstract algebra" contain material that duplicate unnecessarily part of the content of the corresponding article and that have to be replaced by a description of what are these areas and how they interact with other sub-areas of algebra. For example, describing what is a polynomial is not useful here, but it is important to mention that polynomials are considered in elementary algebra, and that the theory of polynomials is much wider and has numerous aspects that are far to be elementary (Gröbner basis and Resultant, for example)
• The other main sub-areas of algebra (see the lists in the lead and in section "topic") need to appear at the same level as "Elementary algebra" and "Abstract algebra" (WP:NPOV) in sections describing their subject and their interrelations.

This is a wide program that I am unable to do alone. But I think that the vital importance of the article deserve it.

D.Lazard (talk) 16:24, 24 February 2013 (UTC)

Hmm... I don't think this idea is workable. To be more precise, I think that elementary algebra and abstract algebra fit within WP:DABCONCEPT, they are different ways (elementary and abstract) of looking at the same concept (algebra). Things like algebra (ring theory) don't fit within WP:DABCONCEPT. they are different meanings of the word algebra and should be linked to from Algebra (disambiguation), which should be a separate page.

I think the problems started with this edit by Teika kazura. (This was tidied up by JimsMaher to give this a combined diff.)

I think we should restore Algebra (disambiguation) and make the lead more like it was in this version. That lead wasn't perfect but it would be a better starting point.

Yaris678 (talk) 13:36, 4 December 2013 (UTC)

WP:DABCONCEPT is clear: when two concepts are related, they should be disambiguated in a broad concept article. We are in this case here, because the study of algebras (structure) is a part of algebra (part of mathematics). Moreover, the distinction between these two meanings is too subtle to be explained in a dab page. Also, the old state of Algebra (disambiguation) was, in fact a dab page with only two broad concepts, Algebra (part of mathematics) and Algebra (algebraic structure). The other items of the page are special cases of one of these broad concepts. It follows that they should not appear in this dab page. It follows that we are in a WP:TWODAB case. Thus the manual of style recommend to replace the dab page by two interlinked articles by hatnotes, which, here could be Algebra and Algebra (structure). I agree that the latter is a lacking article. Are you willing to write it? Even when this article will exist, the subtle distinction between algebra with article and algebra without article should be somewhere. I do not see any better place than in Algebra. I agree that this is an unusual structure for the lead, but the situation is quite unusual. IMO, it is a case for WP:IAR. D.Lazard (talk) 16:17, 4 December 2013 (UTC)

It sound like we mostly agree. I think we both agree that, under WP:DABCONCEPT, there should be an article that describes both elementary and abstract algebra. I think we both agree that Algebra over a field is a different meaning of the word algebra. I think we both agree that this second meaning should ultimately be treated separately and doesn't fall under WP:DABCONCEPT. Am I right in thinking that you think that putting Algebra over a field into this article is a good stop-gap until an appropriate WP:DABCONCEPT article can be created that includes Algebra over a field and related meanings of the word "algebra"? Can you explain why you think this?

I think this approach is difficult because it results in a confused and confusing article. If we are not ready yet to create an article about this meaning of the word "algebra" then we should just use Algebra (disambiguation) as best we can. That disambig page will help people find what they are after without confusing the Algebra page, which I think should be about the meaning that includes elementary algebra and abstract algebra, because this is by far the most common meaning of the word.

I think that Algebra over a field is actually the general term for the other meaning. e.g. Lie algebra and Associative algebra are types of Algebra over a field. I stand to be corrected on this. But if it is true then we already have a WP:DABCONCEPT article for this meaning. It is Algebra over a field.

Yaris678 (talk) 18:53, 6 December 2013 (UTC)

You forget Algebra over a ring, Boolean algebra (structure), Algebra over a monad, algebras of universal algebra (I am not sure that this list is complete). None of them are algebras over a field. It is an important property of algebra (part of mathematics) that many structures that are considered in this area are called "algebra". This deserves an explanation. As dab pages should not contain explanations, this explanation should be in an article. My opinion is that this does not deserve a separate article. It could be in a section of Algebra, but, as this may be very confusing, and as some readers may even not know that they are confused, my opinion is that the clarification deserves to appear in the lead. However, maybe, you could find a better clarification of the numerous meanings of "algebra" than that I have written. D.Lazard (talk) 22:36, 6 December 2013 (UTC)

I don't know if forget is the right word. I did say I stand to be corrected.

Thanks for those links. They were very helpful. I particularly appreciated Universal algebra. It says "From the point of view of universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A." I think this is the best definition of the other meaning of the word algebra. This implies that we already have an article for that other meaning: Algebraic structure.

Yaris678 (talk) 13:56, 7 December 2013 (UTC)

Some care is needed: Universal algebra is a very narrow part of algebra, which is rarely referred to by algebraists who are not specialized in it. Moreover, in universal algebra, a group is a particular algebra, belonging to the variety of groups. The assertion that a group is an algebra does not make sense in the main stream of mathematics, outside universal algebra. Therefore, linking to this article for a general definition of the word is, at least, confusing. Thus this article is not the right place for a sentence such as "In algebra (part of mathematics), various mathematical structures are called "algebra". The right place for this sentence is in the lead of algebra. This suggests me to move the description of the different meanings in a section and to replace them by this sentence. I'll do that, hoping that the result will be fine for you. D.Lazard (talk) 14:45, 7 December 2013 (UTC)

Yes. I think a sentence like this in the lead would be a good idea. We could say "In abstract algebra, the word algebra is also used to mean an algebraic structure, especially an algebra over a field or a more general algebra over a ring."

Yaris678 (talk) 15:17, 7 December 2013 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Algebra/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs more examples. Poor coverage of polynomials, and linear algebra. Salix alba (talk) 10:57, 22 April 2006 (UTC)

Last edited at 19:05, 15 April 2007 (UTC). Substituted at 20:12, 2 May 2016 (UTC)

Al Khwarazmi name means he is originally from Khorezm. Khorezm is very Uzbek region where authentic Uzbeks resided since being. Why Khwarazmi referred to Persian

Persian language was popular at those times but Al Khwarazmi was born in current Uzbek territory. His name suggests he is authentic Khoresmian meaning his ethnisity Khoresmian tribe. Why wiki suggest he was Persian 2A02:6B68:5447:0:D415:D011:C991:5A2A (talk) 07:15, 20 January 2022 (UTC)

Aryabhata

In the whole article the mathematician Aryabhata doesn't get mentioned. Which is somewhat strange since fe. al-Khwārizmīs book is based on his works. Some even argue that the term Algebra actually is derived from his name via arabization.--46.125.249.20 (talk) 21:26, 11 September 2021 (UTC)

Being the earliest Indian mathematician and astronomer, Aryabhata certainly flourished and opened the door for many others in his culture when it comes to math and astronomy. Although he didn't have a lot of discoveries like other known mathematicians, I definitely believe he deserves a bit more recognition. Pumpkinlover03 (talk) 15:23, 18 September 2023 (UTC)

Sorry but we can't add aryabhatta article since he only derived a single formula and has less source but brahmagupta has more sources to back. Obiwana (talk) 14:02, 13 December 2022 (UTC)

What can we do to nudge this article into a "good article" status?

What are the things that need improvements? E.g. copyediting, citations, more links, etc. What stride can we make to making this article better to the point of becoming "good"? - S L A Y T H E - (talk) 19:20, 6 January 2023 (UTC)

Please bear in mind that the things which will improve the article in the mind of a Wiki editor (citations, etc.), have little to do with making it better for the user. Such improvements include, but are not limited to, making the content understandable to anyone who is not a Maths nerd.

Nuttyskin (talk) 15:35, 26 October 2023 (UTC)

Maths

Law of exponents 41.114.202.100 (talk) 12:52, 5 November 2023 (UTC)

Changes to the article

I was thinking about implementing changes to this article with the hope of moving it in the direction of GA status. There is still a lot to do. Major parts lack references, and there are a few maintenance tags, such as "citation needed" and "better source needed". The section "Areas of mathematics with the word algebra in their name" is dubious since it is mainly a collection of links that were selected by whether they contain the string "algebra". Maybe this could be converted into a standalone list article but it probably does not meet the notability guidelines, see WP:LISTN. Some of the topics in it are quite relevant (like linear algebra and universal algebra) and should probably be discussed somewhere in the prose of another section rather than in the form of disconnected list items. The most important remaining links could be moved to the "See also" section.

I'm also not sure that it's a good idea to dedicate the whole section "Different meanings of 'algebra'" to disambiguation since this type of discussion belongs more to dictionary entries than to encyclopedia entries (see WP:NOTDICT). The better solution would probably be to convert this section into a Definition section that explains the main meaning of the topic of the article (roughly as a branch of mathematics that covers areas like elementary algebra, linear algebra, abstract algebra, and universal algebra) and then clarifies to the reader that there are also other ways how the term is used (for example, for specific subareas or for specific algebraic structures). This way, we have the disambiguation and hopefully do not violate WP:NOTDICT.

It might further be a good idea to expand the article in a few directions (while still staying within the limits of WP:SIZERULE). Several high-quality overview sources I checked present linear algebra and universal algebra as major subfields of algebra (for example, [4], [5], [6], and the algebra entries of [7] and [8]). The discussion of the application of algebra to other fields should probably also get some more details, like the algebraization of mathematics (such as geometry, number theory, and topology) and logic. Various smaller adjustments would be needed for the different topics discussed in the article but they can be addressed later. I was hoping to get some feedback on these ideas and possibly other suggestions. I still have to do some research to work out the details. After that, I would start implementing them one at a time but it will probably take a while to address all the points. Phlsph7 (talk) 14:03, 11 January 2024 (UTC)

This sounds very well, and I am waiting to see the result. However, there is a probelem that you do not mention: section § History.

This section is very detailed on algebra before the 18th century, at a time where algebra did not yet exist. Most of the results presented are not presently considered as belonging to algebra, except if one consider that "algebra" is a synonym of "theory of equations". For example, among many, Diophantus's work is presented as algebra without saying that Diophantine equation belongs presently to number theory.

On the other hand, almost nothing is said about the evolution of algebra during the 19th and 20th centuries, and fundamental results are not even mentioned, such as Gaussian elimination, Hilbert's work on polynomials, McCaulay, who provided the first method fof reducing the solution of polynomial systems to the univariate case, Kummer's work, etc,

I ignore whether there exist reliable sources for the history of algebra during this period (maybe Dieudonné) but an history section must not be empty on this period. D.Lazard (talk) 12:11, 12 January 2024 (UTC)

Thanks, that's a good point! The history section seems to be overly detailed on early developments while having very little on what came afterward. The solution would probably be to summarize a lot of what is currently there and then add more on the evolution of algebra during the 19th and 20th centuries. I haven't had the time to dig into the reliable sources on the history but I hope that a lack of sources won't be a problem here. Phlsph7 (talk) 13:32, 12 January 2024 (UTC)

Definition and etymology

I suggest to add at the end of § Definition and etymology something like the following. I do not add it boldly myself because I am not very happy with the formulation, and a bettter formulation would be easier to source:

The etymology relates algebra to transformation of equations. Indeed, until the 19th century, algebra was essentially the theory of equations; that is, the art of manipulating equations in view of solving them. This explain why some earlier work such as that of Diophantus were considered as belonging to algebra (presently Diophantine equations belong to number theory). During the 19th century, most of the older problems of algebra were solved by the proof of the fundamental theorem of algebra (existence of complex solutions of polynomial equations) and the introduction of Galois theory (characterisation of the equations that are solvable in radicals). So, the scope of algebra has evolved toward what has been described above, and many texts about the history of algebra refer implicitly to a different definition of algebra. D.Lazard (talk) 15:34, 27 January 2024 (UTC)

That's a good idea. I adjusted your text to better fit the sources and to dodge the controversial topic of whether Diophantus as the supposed father of algebra actually engaged in algebra. Phlsph7 (talk) 17:04, 27 January 2024 (UTC)

Semi-protected edit request on 2 February 2024

Under the Linear algebra section the example coefficient matrix has an error on the third column second row, where it is entered as 9 but the system of linear equations above states that the corresponding element should be 7, as that is the coefficient of x3 in the second equation, a bit of a nitpick but a nit none the lesser. Hubblez (talk) 19:54, 2 February 2024 (UTC)

 Done Thanks. Liu1126 (talk) 20:36, 2 February 2024 (UTC)