Talk:Algebra/Archive 2

Recent changes to the lead
I recently brought back some info in the lead that was removed in October 2012.

This was followed by this edit by D.Lazard. The edit summary being "removing controversial assertions and substituting them by sentences appearing later in the article". I would like to discuss this edit.

The table below compares sentences which I think are the nearest equivalent to each other in the two version. Unfortunately this breaks up the flow of the text somewhat, but it does mean we can compare what the two versions are saying more easily.

Here are my thoughts on the first comparison.

Line 1 is obviously one of the hardest things: Saying what algebra actually is. I think 1A is a better stab than 1B. 1B is an interesting way of looking at things but not much of a definition. 1B appears to have a source... but the quote given in the footnote does not support the words in the article. I'd be OK with keeping 1B if we first gave a better definition. This could be similar to 1A or different, if someone has a better suggestion.

Anyone got any thoughts on this, or any of the other comparisons?

D. Lazard, can you say which bits of column A you thought were controversial and why?

Yaris678 (talk) 14:02, 11 December 2013 (UTC)


 * Here are the details of what is controversial:
 * A1: concerning the study of the rules of operations and relations: This is not specific to algebra, it applies also to mathematical analysis which studies the functional relations and the operations of derivation and integration.
 * A1 is too technical for a first sentence in the lead: To be understood the reader must know what is a term (at this level of generality, this word has no commonly accepted meaning in mathematics, except in the very technical field of mathematical logic), a polynomial, an equation (the reader should guess that differential equations are not considered here) and an algebraic structure (circular definition).
 * A1: This "definition" does not take into account the multiplicity of subareas of algebra, mentioned in lead B, but not quoted in the table: Presently, algebra is divided in several subareas including linear algebra, group theory, ring theory and combinatorics (see below for more subareas).
 * A2 : Most mathematicians consider that "topology" is a part of "geometry" and "combinatorics" is a part of "algebra" (for topology, see the recently proved Poincaré conjecture: if these two fields are distinct, who can say to which this conjecture belongs?). The definition of "pure mathematics" is controversial. The opinion of many mathematicians is that the concept does not exists. Even if it would exist, one may not include in it computer algebra nor "applied algebra" (for a reliable source for the existence of this concept, see Journal of Algebra and Its Applications, for example).
 * A3, elementary algebra: In Yaris678 this receives an undue weight by the place which is devoted to him. Moreover, Yaris678's version is US specific ("secondary courses"). "Elementary algebra" is not a subarea of algebra, but an educational concept. As such, it deserves to be cited in the lead, but it does not needs more place than that given in version B.
 * A4, abstract algebra. This is also a badly defined concept whose existence itself is as controversial as the concept of "pure mathematics". If it exists, does linear algebra and its applications (for example in weather forecast) belong to it? However, as many people use this term, it deserves a brief mention, like that in B4.
 * I agree that version B is not sufficiently sourced. But version A is also unsourced. It most controversial aspect it that it seems reduce "algebra" to "elementary algebra" and "abstract" algebra. Mathematics Subject Classification is a reliable source showing that it is non-controversially wrong.
 * Yaris678, please explain what is wrong or controversial in lead B. In your post, the only given argument is "I do not like it", which is an argument to avoid in such a discussion.D.Lazard (talk) 16:11, 11 December 2013 (UTC)


 * Hi D.Lazard. Thanks for explaining your reasoning.
 * I'm fine with your objections to A2 and yes I'm OK with the sentence "Presently, algebra is divided in several subareas including linear algebra, group theory, ring theory and combinatorics", which perhaps I should have mentioned in the table.
 * A1. Yes I agree that it is perhaps starting too technical but I definitely think it would be useful to say what algebra is.  Can you think of a better way of putting it?  I agree that operations and relations also appear in analysis, but the important thing about analysis is the concept of a limit, which doesn't appear in algebra.  Similarly, differential equations are types of equations but they need the concept of limit in there too (assuming you define calculus in terms of limits).
 * A3 and A4. I think we need to remember the readers.  Most of them won't have studied maths at university.  The idea that algebra can deal with things other than numbers will seem weird.  If we start with "elementary algebra" and then progress to "abstract algebra" it should lead the reader in.  You say this is an "educational concept" as if this is a bad thing but the reason educators use this concept is because it makes things easier to explain - which is exactly what we want to do.  Mathematics Subject Classification is designed for classifying mathematical research, rather than explaining mathematics to the lay reader.
 * Yaris678 (talk) 18:30, 11 December 2013 (UTC)
 * IMO, defining "algebra" as as "doing computations similar to that of arithmetic with non-numerical mathematical objects" is a perfect definition that can be understood by kids that know what are the operations of arithmetics. It has also the advantage, in the next sentence, to explain in a few simple words why "variables" is an essential notion of algebra. Moreover, this definition cover all the aspects of algebra, abstract as well as applied, at research level as well as at educational level. This definition is maybe "weird", but it is true: working with numbers is not algebra, but Number theory (or in case of real numbers mathematical analysis). Your sentence "... because it makes things easier to explain - which is exactly what we want to do" and the following one ".... is designed for classifying mathematical research, rather than explaining mathematics to the lay reader" appear very strange to me. They suggest that educators have to change mathematics to make it easier, and that taught mathematics is different of research mathematics. IMHO, the task of teachers of mathematics is to teach mathematics, not to teach other things, because they seem easier to teach. Educators have certainly to choose what is taught, and how it has to be presented for being understood. What is taught in elementary courses may certainly be called "elementary algebra", but it remains a part of algebra, which is exactly what version B says. A very important task of educators is also to open the way for learning more mathematics (not only for future mathematicians but also for future engineers, computer scientists and searchers in other sciences). This wider opening on the mathematical world is difficult to give in elementary classes and in textbooks, and is thus an important role of WP. It is thus a bad advice to give to the kids that opposing "educational mathematics" and "research mathematics". There is a continuity and any temptation of breaking it is bad for students as well for mathematics itself (for me mathematics is not plural, but singular).
 * Here is a practical example illustrating these considerations. Before August 2012, polynomial greatest common divisor was a stub containing only what you call "elementary mathematics". Apparently the editors of this article did ignore that many people (including kids) have access to software of computer algebra and that such a software computes GCD's in a routine way, using algorithms. Many kids may desire to understand why computers can easily compute GCD's, when it is so difficult for them. This can be explained without using more sophisticated mathematics than that is taught in elementary courses in which polynomial GCD is taught. This explanation needs research results that are not older than 50 years. The present version is, maybe, not detailed enough for being understood by kids (except the best ones), and deserve certainly to be expanded, at least by adding examples. However, as it only uses "elementary algebra" methods, it can certainly be understood by their teachers, and this may be very useful to improve their course. The present version of the algorithm is thus "elementary mathematics" by the involved methods and "research results" by the date of the results that it contains. How you class it, if you break the continuity of mathematics. D.Lazard (talk) 20:59, 11 December 2013 (UTC)
 * B1: OK. Perhaps I should be clearer about the problems with this definition:
 * The “non-numerical mathematical object” could be taken the wrong way. You might have variables which represent numbers.
 * Arithmetic is all about algorithms that work with positional notation. Algebra is not.  I guess you could argue that positional notation is a special way of writing polynomials with x=10 and coefficients between 0 and 9… but I don’t think that’s what the statement means.  What does the statement mean?  The only way I can think of making this statement make sense is if “computations similar to that of arithmetic” means “operations”, which is what it says in A1.
 * Is the issue here that we want to think of a way of saying “operations” in more common language? If that is the issue then I’m OK with that.  Do you agree that this is the issue?
 * A3 and A4: You were the one that started this comparison of the different ways of subdividing algebra by saying ‘"Elementary algebra" is not a subarea of algebra, but an educational concept’. From your response now, you seem to have the idea that there is only one real way of subdividing algebra, which is the way that research papers on algebra are divided.  While explaining these subdivisions can be helpful to some, for the majority of our audience we will be subdividing things that they don’t really understand and they can’t see have anything to do with what they thought algebra was.
 * I have no desire impose a separation of research and educational mathematics. I like the example you give of the improvements to polynomial greatest common divisor.  I guess my main point is that saying version A gives undue weight to elementary algebra misses the point that elementary algebra is exactly what most people want to find out about.
 * I happily agree that algebra can not be split into the elementary and the abstract. Linear algebra is a good example of something that can not easily be classified as one or the other.  Similarly, complex numbers come about as soon as you start looking for roots to polynomials, but are they elementary?  These concepts are just useful concepts in explaining what algebra is.  The role of the article is to explain algebra to the lay reader.  If this can include summarising how current research in algebra is classified, that is great, but the purpose of the article is more than that.
 * Yaris678 (talk) 13:55, 12 December 2013 (UTC)
 * I have not answered to this post previously because I have missed it. Maybe I have remarked it, but I did not answered because my opinion it that the problem cannot be solved by a discussion with only two editors. As Yaris678 has reinserted his controversial definitions and assertion in the article, I answer by inserting punctual answers inside his post. I have shown in details how Yaris678's version is controversial. On the other hand, none of the assertions of my version has been shown controversial. Therefore I'll revert last Yaris678's edit and wait for the advice of other editors. D.Lazard (talk) 16:37, 16 December 2013 (UTC)
 * D.Lazard. Please don't mess up my text by sticking in your responses part way through.  I have moved it below.  If you don't like the way I have formatted it, feel free to change but please don't stick it in the middle of my text. Yaris678 (talk) 19:23, 16 December 2013 (UTC)

B1: You ask "Is the issue here that we want to think of a way of saying “operations” in more common language? If that is the issue then I’m OK with that.  Do you agree that this is the issue?" Answer: I do not agree with you. The issue with your "definition" is that it involves notions that are not known by the layman ("rules of operations", "relations", "constructions" "terms", "polynomial", "algebraic structure") and does not explain how algebra differs from the other parts of mathematics, while everybody knows and has done arithmetics computations, even if he does not know what is computing with variables (which is exactly what algebra is).
 * 1) This is exactly what is said in the next sentence: "Initially these objects were variables representing either numbers that were not yet known (unknowns) or unspecified numbers (indeterminates or parameters), allowing one to state and prove properties that are true no matter which numbers are involved." Maybe past tense should be replaced by present tense, and "Initially" by something like "at first level".
 * 2) Your definition of arithmetic is very different from that can be fount at arithmetic.

D.Lazard (talk) 16:37, 16 December 2013 (UTC)

A3 and A4:

If algebra is not what the majority of our audience "thought algebra was", the first task is to give a correct view of what algebra is really. The second one is to give the access to what they "thought algebra was". This is fulfilled by links and brief definitions of elementary algebra and abstract algebra. As there are sections in this article and specific articles devoted to them, more details are misplaced in the lead. D.Lazard (talk) 16:37, 16 December 2013 (UTC)

Third opinion
This discussion is difficult to add to, since there are so many things being discussed at once. It's not clear to me whether there is a fundamental disagreement about what the lead should look like, or whether there are several disjoint and unrelated discussions happening simultaneously in one thread. Anyway, I am broadly in favour of D.Lazard's wording; but that's not to say it can't be improved. For example, the article mathematics, or even geometry, have much nicer lead sections. I think it is difficult to nail down what algebra is for the lay person (unlike geometry which is "the study of shape", whatever that means). Describing it as a "broad part of mathematics" first of all, seems like a good idea. Mark M (talk) 19:01, 16 December 2013 (UTC)
 * Hi Mark,
 * I think there are three points of contention:
 * A1 I agree this that this could be made more accessible but think it is a decent stab for now. I think D.Lazard thinks that it is far too complicated a place to start. I have tried to discuss ways to make it better but don't get much of a response.  I'm currently thinking that "Algebra is the branch of mathematics concerning the study of operations" is better.  Lose all the other more complicated stuff.
 * B1 I think this is potentially misleading and D.Lazard does not. I think your change is an improvement, but I would probably say "Algebra arose from the idea that one can perform operations of arithmetic when some numbers are not known."
 * A3 and A4. I think it is worth spending time in the lead to explain that algebra can deal with things that aren't numbers. D.Lazard thinks that providing a brief description and a wikilink is sufficient.
 * I hope that has clarified where the issues are. Any further thoughts you have on this would be welcome.
 * Yaris678 (talk) 20:10, 16 December 2013 (UTC)


 * Hi Yaris, regarding A1, I don't think "the study of operations" is a helpful phrase. It's not like "the study of shape" for geometry, because people already have an idea of what a "shape" is, which is closely reflects how a mathematician might thinks about it. But the word "operation"? And secondly, I don't think I agree that algebra is the study of operations! :-)
 * +1, "Study of operations" sounds a lot like operations research which is a whole other area. --RDBury (talk) 18:09, 17 December 2013 (UTC)
 * Regarding B1, I think your proposed re-wording is misleading. The main objects of study in (what mathematicians call) algebra are abstract mathematical objects which aren't numbers, nor do they usually represent actual "numbers". See, for example, Mathworld's Algebra article, which nicely exposes the conflicting terminology. My understanding is this article should be a WP:CONCEPTDAB; but I don't know what that means for the lead. Mark M (talk) 21:32, 16 December 2013 (UTC)

I agree with you that my wording may be improved. I find that your edit really improves it. Thanks. D.Lazard (talk) 11:57, 17 December 2013 (UTC)
 * Hi Mark. I agree that the word operation is not as readily understood as shape.  However, you say that you don't think algebra is the study of operations... Have another look at the article on Mathworld that you linked to.  It says 'One use of the word "algebra" is the abstract study of number systems and operations within them'.  This statement may stretch the definition of "number" (well... by "number system", I think they mean algebraic structure) but it makes the centrality of operations pretty clear.  Yaris678 (talk) 17:08, 17 December 2013 (UTC)

Fourth opinion
I guess that part of the difficulty is that what mathematicians call algebra is much more extensive than what the broader population learns in school. Of course, the basic idea is the same in both, but it is a challenge to find a sentence that applies to both equally well. That being said, I think that B2 comes about as close to achieving this as one could hope, so I would favor that as a first sentence. I don't have strong opinions about the other points of contention, but to me it would be most natural to follow the general principle of progressing from the simple to the more advanced; e.g., one could begin the next few sentences by mentioning that originally algebra dealt with equations in variables representing unknown numbers, and that later its scope was extended to enable arithmetic involving other non-numerical objects. Ebony Jackson (talk) 04:15, 17 December 2013 (UTC)

Lead image
The current lead image was added about a year and a half ago and while it is "pretty" I'm not convinced it has any explanatory value for the article subject. Previously the page from Al-Khwārizmī was used, now visible in the history section. Admittedly this wasn't much better for explanatory value but at least it had some historical relevance. I realize that it's very difficult to condense a large and abstract subject into a single image but it seems to me we could do better, I'd even prefer to have blown-up quadratic formula instead -- not pretty but at least it's related to the article subject. Anyone have a better suggestion? Or prefer to keep the pretty one? --RDBury (talk) 22:09, 19 December 2013 (UTC)
 * I agree on all points. Here's a suggestion. How about completing the square? We could illustrate it for a particular quadratic like x^2 - 4*x + 3. This has a nice Al-Khwārizmī link and it's where algebra gets its name. Yaris678 (talk) 23:26, 19 December 2013 (UTC)


 * I also agree; perhaps something even as simple as $$X^2-1 = (X+1)(X-1)$$, possibly being preceded by the equation $$4^2-1 = (4+1)(4-1)$$ as an illustration of the idea that numbers (such as 4) can sometimes be replaced by other symbols (such as X). I feel like I've seen an image of such an illustration on Wikipedia, but I can't quite remember where. Mark M (talk) 13:47, 20 December 2013 (UTC)


 * I agree with the objections to the image that have been mentioned, and in addition, it's not even indisputably an example of algebra, as it uses transcendental functions. For the moment I am restoring the page from Al-Khwārizmī, without prejudice against something better still going there. Actually, I quite like the Al-Khwārizmī page, and even if consensus is for something else in the lead, maybe that one could be moved to somewhere else in the article, rather than removed altogether. JamesBWatson (talk) 15:06, 20 December 2013 (UTC)
 * Quadratic_formula.svg, which expresses the solution of an equation of degree two in terms of its coefficients.]]I agree with preceding edit by JamesBWatson. However, as the image is better placed in history section (where it appears also), I'll replace it by the quadratic formula. This seems better than the other proposed formulas because: a/ many people know what it is without reading the the caption, b/ as a formula, it is rather aesthetic (this is a personal opinion), c/ it is together simple enough to be read easily and complicated enough to give an idea of what is algebra (this explains my above personal opinion), d/ it illustrates also the "theory of equations" which were almost synonymous with algebra until 19th century. D.Lazard (talk) 15:53, 20 December 2013 (UTC)


 * I agree with using the quadratic formula as the lead image. Ebony Jackson (talk) 19:30, 20 December 2013 (UTC)

More examples
A lot of the feedback is about more examples and simpler examples. If this is the case, I might spend a few minutes putting some simple examples in a section. Any ideas about how to format it? I was thinking of putting a new section after "Elementary algebra" called "Examples" or "Examples of Algebra". What do people think?

I though perhaps it could be something like: "Algebra is used to calculate unknown values in mathematical problems, for example if the price of an item in a shop increases by 10% and is now worth £1.10 (or $1.10) then we can represent the original price as x and write x+0.1x=1.10; x(1+0.1)=1.1; 1.1x=1.1; therefore x = 1.1/1.1 or x = £1.00" I don't know if that's a good idea, what do you all think? Also I'm not sure if I should cite anything, since it's maths and is obviously correct, as long as the rules of maths are followed.

Jamietwells (talk) 21:38, 16 January 2014 (UTC)


 * IMO, this kind of examples would be misplaced in this article. Their right place would be in Elementary algebra. Nevertheless, one could add at the end of the first paragraph of Algebra as a branch of mathematics something like: "(For more details and examples see below and Elementary algebra)" D.Lazard (talk) 10:44, 17 January 2014 (UTC)

Etymology
The article claims
 * Algebra originally referred to a surgical procedure, and still is used in that sense in Spanish, while the mathematical meaning was a later development.

As a native Spanish speaker I can only say LOL-what!?

From (http://www.etymonline.com/index.php?term=algebra):
 * The word was used in English 15c.-16c. to mean "bone-setting," probably from Arab medical men in Spain.

Fair enough. But there is no modern medical use for the term.

Miguel (talk) 19:03, 24 February 2014 (UTC)

Possible copyright problem
This article has been revised as part of a large-scale clean-up project of multiple article copyright infringement. (See the investigation subpage) Earlier text must not be restored, unless it can be verified to be free of infringement. For legal reasons, Wikipedia cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions must be deleted. Contributors may use sources as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously. Diannaa (talk) 02:42, 24 July 2014 (UTC)

Areas of mathematics
I added "arithmetic" to the line below so the article would read:


 * ...is one of the broad parts of mathematics, together with arithmetic/number theory, geometry and analysis.

User:Wcherowi reverted this addition saying "Not a synonym and not a broad area of mathematics." But Number theory says that arithmetic is an older term for number theory, and Arithmetic says that it is "the oldest and most elementary branch of mathematics". How to resolve these conflicts? -- Beland (talk) 20:05, 19 June 2014 (UTC)


 * The problem is that, today, people understand the word "arithmetic" to mean elementary school mathematics: memorizing the addition and multiplication "facts" and the basic rules of fractions, decimals, and percents. While generally people today reserve "mathematics" for algebra, trigonometry, calculus, and more advanced topics.  The distinction seems to be that arithmetic is usually, at least in the US, taught by rote, while mathematics is taught using reasoning: the mathematical proof.  While the number facts that grade school students memorize are, technically, a part of number theory, number theorists rarely if ever call these facts number theory. Rick Norwood (talk) 22:01, 19 June 2014 (UTC)


 * I am in full agreement with and can expand upon my reason for reverting that edit. While it is true that "Arithmetic" had been used to refer to number theory, it hasn't been used that way in over a century and I think that it is misleading to bring up old linguistic forms without the appropriate context. The reader of an article such as this will have the modern view of the concept of arithmetic, which from the viewpoint of mathematics as a whole, is a very narrow subdiscipline, often described by the terms "elementary arithmetic" or "basic arithmetic". This view is most likely unfair to Arithmetic, but our job as editors is to describe how things are and not how they ought to be. Bill Cherowitzo  (talk) 00:30, 20 June 2014 (UTC)


 * So it sounds like y'all would prefer to change the intro of Arithmetic to provide a more modern context? -- Beland (talk) 00:37, 20 June 2014 (UTC)
 * The lead of Arithmetic emphasizes correctly on the elementary aspect of arithmetic. However, it deserves some improvement. For example, in the sentence "It involves the study of quantity, especially as the result of operations that combine numbers", "quantity" must me replaced by "number": in fact, arithmetic studies only numbers and many other quantities have nothing to do with arithmetic. For example, lengths, areas, angles are quantities studied in geometry. I'll try to improve this lead. D.Lazard (talk) 10:34, 20 June 2014 (UTC)


 * Beyond 's comments above I see nothing wrong with the intro to Arithmetic except for the last sentence. That sentence pushes a POV which I don't think is appropriate (putting "Higher" in parentheses is making an editorial comment). There is also a problem with the Number Theory section, but this should be discussed on that page's talk page and I'll move the discussion there. Bill Cherowitzo (talk) 15:14, 20 June 2014 (UTC)


 * But there is a direct conflict, in that Algebra says the four branches of math are number theory, algebra, geometry, and analysis; and arithmetic says the four branches are arithmetic, algebra, geometry, and analysis. Which is it? -- Beland (talk) 11:58, 26 June 2014 (UTC)
 * Where is the conflict? In the article Arithmetic, arithmetic is first compared with the three other branches from an historical point of view (using the word "oldest"), before saying that until the beginning of 20th century, arithmetic and number theory were synonyms and that, nowadays, arithmetic is a part of number theory. D.Lazard (talk) 12:51, 26 June 2014 (UTC)
 * The problem is that the list on Arithmetic uses "arithmetic" in the sense of a subset of number theory, not as a synonym. And even if it were being used as a synonym, that would be confusingly inconsistent vocabulary.  Since it makes no set for a subset of number theory to be a top-level division of mathematics, I fixed the intro of arithmetic to avoid this awkwardness and be clearer and explicit. -- Beland (talk) 16:00, 22 August 2014 (UTC)

The section, How to distinguish between different meanings of "algebra", is difficult to comprehend
I find the section, How to distinguish between different meanings of "algebra", difficult to comprehend, as written. The sentence "As a single word without article, "algebra" names a broad part of mathematics (see below)" does not at all make clear what "see below" is referring to. In the sentence that begins "As a single word with article or in plural...", no example using an article or a plural is given. Specific examples, on this page (not in some other article) pointing out precisely just what the writer is referring to (such as by putting exactly what is being referred to [including any article] either in quotes or in Italics) would be welcome and extremely helpful. Wikifan2744 (talk) 03:32, 23 September 2014 (UTC)
 * Imho does "see below" refer to the next section entitled "Algebra as a branch of mathematics", in particular to the last paragraph of it.--Kmhkmh (talk) 13:46, 23 September 2014 (UTC)

Article is still too disambiguation-like
Author has tried to hide all the historical truth and tried to impose his ignorance. Whole article can be called as 'Bhoosa'. The waste dust. — Preceding unsigned comment added by Mohammedassadi (talk • contribs) 05:26, 4 November 2014 (UTC)

A WP:BROADCONCEPT article has to be based on concepts, not English. Just because something is called 'algebra' (in English) doesn't mean it necessarily is algebra.

For example Central Asia could be considered as simply words, that have varying definitions over time, however the article doesn't define the topic that way, instead it's about an abstract concept that has changed over time; it's really about lines on a map.

Similarly, in this article, you can't really cover both basic algebra and Algebra over a field unless you make clear what the relationship is to the overall broad topic/concept; it's not enough for them to both be mathematics, they have to be in the same part of mathematics in some sense or other.

Right now, the article is too disambiguation-like. It's not OK to list things based simply on whether they are called algebra.

But I certainly agree that there needs to be an algebra article, just having a disambiguation page is wrong, but this article is not quite cooked yet.

You need to have some kind of conceptual definition of what algebra is, and it needs to cover everything you are going to cover in the article (as opposed to everything you refer to, which can be pretty different).GliderMaven (talk) 02:32, 25 June 2014 (UTC)


 * I have tried to provide the unifying explanation you request, using Herstein's Topics in Algebra as a source. Rick Norwood (talk) 19:39, 25 June 2014 (UTC)


 * Yes, the introduction is rather better now, many thanks.


 * I'm still having trouble with the idea that the article should have a section called 'Topics named algebra', since either they are algebras or they are not. If they're not they shouldn't be in the article, whether they're called that or not. Conversely, if they are, but they're not called 'algebra' they should still be in the section.GliderMaven (talk) 23:31, 26 June 2014 (UTC)]]


 * Good suggestion. I've tried to fix the problem. Rick Norwood (talk) 23:46, 26 June 2014 (UTC)


 * I don't think that you've fixed the problem. Are these algebras or not? Just because something is called or named algebra doesn't mean it necessarily is. The English language is actually relatively arbitrarily constructed. I think for it to be in the article it has to actually be an algebra, or usually be usually regarded as an algebra, even if it strictly isn't perhaps. I tried changing the text here: but it was reverted for unclear reasons.GliderMaven (talk) 20:51, 27 June 2014 (UTC)
 * Deciding what is or is not an algebra is not a question of English language, but a question of mathematics: a structure or an area of mathematics is an algebra (structure) of algebra (area of mathematics) when mathematicians use to call it as such. In your edit you have introduced the term "structure considered as". Even if it is correct English, it is a mathematical nonsense: a structure is a well defined notion; its name is the result of a consensus among mathematicians. Considering a structure as something else is a question of point of view and has nothing to do with its name. Similarly you have introduced "type of algebra" as section heading; this could make sense for structures (but plural would be required for "algebra"). But the section lists also the subareas of mathematics, whose name contains "algebra". IMO, it is a nonsense to talk of the "type" of a scientific area; in any case, I cannot imagine what it means. These are the reasons of my revert. However, the previous section heading was also incorrect, IMO. Therefore my further edit. D.Lazard (talk) 08:12, 28 June 2014 (UTC)

GliderMaven asks "Are these algebras or not?" As the article states, and as Lazard explains above, the word "algebra" is primarily used for an area of mathematics. The use of "algebra" for a particular mathematical structure is more specialized. All of the subjects listed are in the area "algebra". The definition of the more specialized use of the word ("an algebra") is given in the article, but is not appropriate for the more general discussion of those subareas within the area of algebra that have the word algebra in their name. Rick Norwood (talk) 13:14, 28 June 2014 (UTC)


 * I'm sorry, I still don't understand why having the word 'algebra' in the name of a subarea of algebra is all that important. This seems to be more of an English lesson than a thing about algebra. In fact, 3 whole sections in the article seem to be nothing all that much to do with algebra, and only to do with English. If this isn't a complete list of algebras then why do we need a complete list of algebras with the word algebra in it? It seems very arbitrary.GliderMaven (talk) 16:07, 23 September 2014 (UTC)
 * Actually most terms with algebra in their name do belong the algebra as a subject or concept in the broadest sense (i.e. linear algebra or boolean algebra are topics of algebra). However of course many subjects not carrying algebra in their names at all belong to to algebra as well (groups, rings, fields, moduls. categories, etc.). We do not need a complete list however giving an exemplary list does illustrate reach and scope of algebra/algebraic concepts.--Kmhkmh (talk) 19:20, 23 September 2014 (UTC)
 * I don't understand why this is a list only of algebras which have algebras in their name, as opposed to (for example) a list of the more important algebras, irrespective of whether they are named that way.GliderMaven (talk) 22:25, 23 September 2014 (UTC)

Semi-protected edit request on 2 January 2015
Add "Geometric Algebra" (with link to existing Wikipedia entry on Geometric Algebra) to existing section 'Areas of mathematics with the word algebra in their name'.

AgHyT2 (talk) 20:03, 2 January 2015 (UTC)
 * What should be put in the description for Geometric Algebra (eg "a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations." for Algebraic geometry). Stickee (talk) 00:00, 3 January 2015 (UTC)
 * Hmm, according to the Geometric algebra article, geometric algebra is a Clifford algebra, which is a type of associative algebra, which is one of the algebras listed in the section we want to add this to. (Full disclosure: I'm not familiar with this subject.) Mz7 (talk) 02:53, 3 January 2015 (UTC)
 * Following up on 's answer above, I would like to say that, looking at all the pages from Category:Algebras that are not included, it wouldn't make sense to include a subset such as this in the list. ❌.  G S Palmer (talk • contribs) 01:23, 5 January 2015 (UTC)

Semi-protected edit request on 11 January 2015
The history of algebra contained on your site is grossly anglo-centric and inaccurate - it should be corrected to include both the Egyptian and Ottoman independent derivations of modern algebraic theory, the Egyptian occurring Before the Common Era.

Vldaughtery (talk) 16:50, 11 January 2015 (UTC)

If you want to suggest a change, please request this in the form "Please replace XXX with YYY" or "Please add ZZZ between PPP and QQQ". Please also cite reliable sources to back up your request, without which no information should be added to, or changed in, any article. - Arjayay (talk) 17:01, 11 January 2015 (UTC)
 * Red information icon with gradient background.svg Not done: That is your PoV, not a Semi-protected edit request.

Semi-protected edit request on 29 June 2015
it's wrong it's not aljebr, it is Al-Jabr. I swear it is Al-Jabr. Please change Al-Jebr to Al-Jabr

AAG-Player (talk) 01:06, 29 June 2015 (UTC)
 * Yes check.svg Done For future edit requests, please be sure to provide a reliable source to back up the claim you are making (unless of course you are requesting an uncontroversial typo fix). In this case, "al-jabr" was backed up both by the article and the cited source. Please be sure to explain this to the reviewer if this is the case for future requests to avoid delayed response or a decline. Thanks, Mz7 (talk) 01:40, 29 June 2015 (UTC)

History
Algebra as a field of mathematics existed in India for a long time. Please don't make Eurocentric claims like: "The start of algebra as an area of mathematics may be dated to the end of 16th century". Several serious works of mathematics by Bhaskara and Brahmagupta etc on non deterministic equations exist. It is totally arrogant to claim only it becomes "field of Mathematics" when white Europeans study it - that too learn from others! ~rAGU (talk)
 * You are wrong: ancient Indians got results on equations and invented the modern numeral system. This is great, but it is arithmetic rather than algebra, as, before 16th century, equations were solved in a purely numeric way and not in terms of formulas, as it is done in algebra. This is not the article that is Eurocentric, that is your post that is Indiancentric. D.Lazard (talk) 07:10, 4 September 2015 (UTC)

Incorrect definition of inverse elements?
In the abstract algebra section under inverse elements it says "A general two-sided inverse element a^−1 satisfies the property that a ∗ a^-1 = 1 and a^−1 ∗ a = 1 ." Shouldn't it say a ∗ a^−1 = e and a^−1 ∗ a = e where e is the identity element of the operation ∗? (Please excuse the exponents; I'm on my phone.) UltraHex (talk) 16:01, 15 January 2016 (UTC)


 * Good point, one may denote the identity element either 1 or e, but the choice must be the same as in the previous paragraph. D.Lazard (talk) 16:45, 15 January 2016 (UTC)

Semi-protected edit request on 27 January 2016
In the "Algebra as a branch of mathematics" kindly change the word "whatsoever" so that the phrase would be "a, b, and c can be any numbers (except that a cannot be 0) that satisfy the equation and either factoring or using the quadratic formula can be used to find the value of the unknown quantity "x". source is: Vance, Elbridge P. Modern College Algebra Third Edition. Addison-Wesley Publishing Company Inc. ISBN 971-08-1696-9 thanks --Billie bb (talk) 13:07, 27 January 2016 (UTC)
 * While your replacement sentence is well intentioned, it is not a good replacement since it would thwart the intent of the current sentence. The term "whatsoever" is being used to emphasize the arbitrariness of the choice of coefficients which is the main point that is being made (and by the way, your sentence is not quite correct since the coefficients do not "satisfy" the equation). Furthermore, including factoring as a method for solving the equation, while true, also runs counter to the implied message, which is that the solution technique becomes mechanical (in this instance) when arbitrary coefficients are used. Factoring is not a mechanical process in the way that using the quadratic formula is. Bill Cherowitzo (talk) 17:56, 27 January 2016 (UTC)
 * Okay. Thanks for the explanation. I'll just emphasize to my students that there are other ways to solve quadratic equations. (I also teach my students how to factor and complete the square aside from using the quadratic formula for them to decide which technique is easier for them.) My students tend to go to Wikipedia for answers to their assignments. I'll just reiterate that Wikipedia doesn't have all the answers but reference books do. But I think I'll go see the Wikipedia article on quadratic equations. Thanks again. --Billie bb (talk) 02:36, 28 January 2016 (UTC)

Semi-protected edit request on 17 April 2016
The Babylonians, Chinese and Greeks were using algebra long before mohammad was spewed from the bowels of hell. How then can it be a muslim invention? al jabar codified (translated) earlier Indian (Hindu) mathematical principals. Same with al khwarizmi. He simply translated Brahmagupta's earlier work. When did translation become invention? Wiki needs to be banned for aiding in the spreading and enforcing of these historical falsehoods because people just blindly and unthinkingly believe the fractured and invented "facts" presented as truth. And why the hell are the Indians not screaming louder about their historical intellectual properties being plagiarized?

220.237.102.72 (talk) 03:46, 17 April 2016 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format. Don't think the article says Muslims invented algebra. We just get the English word for it from Arabic Cannolis (talk) 06:58, 17 April 2016 (UTC)

Orphaned references in Algebra
I check pages listed in Category:Pages with incorrect ref formatting to try to fix reference errors. One of the things I do is look for content for orphaned references in wikilinked articles. I have found content for some of Algebra's orphans, the problem is that I found more than one version. I can't determine which (if any) is correct for this article, so I am asking for a sentient editor to look it over and copy the correct ref content into this article.

Reference named "MacTutor": From Muhammad ibn Musa al-Khwarizmi:  From Timeline of algebra: Arabic mathematics, MacTutor History of Mathematics archive, University of St Andrews, Scotland 

I apologize if any of the above are effectively identical; I am just a simple computer program, so I can't determine whether minor differences are significant or not. AnomieBOT ⚡ 14:22, 31 August 2016 (UTC)


 * Solved by undoing recent addition, obviously copied from somewhere, leaving crippled references. Can be put back when references are fixed. Please check before leaving it like that again. - DVdm (talk) 14:49, 31 August 2016 (UTC)

Different meanings of "algebra"
It is not at all clear to me what the word "article" means as used in this section. It seems to mean the parts of speech called "articles": a, an, the. But, then the examples do not include any of these. Maybe it means "qualifier" -- which is used in the first sentence of this section; but then, the second and third bullet seem redundant to each other. (This is something of a minor item, but it could certainly be stated more clearly. RoseHawk (talk) 17:34, 2 October 2016 (UTC)
 * The word "article" refers to the part of speech i.e "a", "an" or "the". I've eddited the article to try to help clarify this. Paul August &#9742; 17:51, 2 October 2016 (UTC)

Group
I think definition of Group "a combination of a non empty set and single binary operation ........." Instead of "a combination of a set and single binary operation..........." Salman Shah (talk) 06:59, 13 November 2016 (UTC)


 * Yes, by definition, a group must have an identity element, so cannot be constructed with the empty set. See Group (mathematics). So non-emptiness is implicit in the definition, even if some authors 'explicitise' it. - DVdm (talk) 09:45, 13 November 2016 (UTC)

Semi-protected edit request on 21 November 2016
In the section "Areas of mathematics with the word algebra in their name", the link to Relational Algebra should be to Relation algebra. 84.93.190.255 (talk) 21:44, 21 November 2016 (UTC)


 * ✅: Relational Algebra was a valid entry. Added Relation algebra  . - DVdm (talk) 22:20, 21 November 2016 (UTC)


 * This isn't quite right though. The section "In logic" lists Boolean algebra and Heyting algebra, which along with Relation algebra are the three most important Algebras of Logic, so it is obvious that Relation algebra is meant to be there. Relational algebra is a topic in Computer Science, and whether it should be on this page or not, it certainly shouldn't be listed under logic. — Preceding unsigned comment added by 84.93.190.255 (talk • contribs) 09:21, 22 November 2016 (UTC)


 * Please sign all your talk page messages with four tildes ( ~ ) and indent the messages as outlined in wp:THREAD and wp:INDENT. Thanks.
 * But according to the article Relational algebra, it is not just a topic in computer science, but "a family of algebras". - DVdm (talk) 09:26, 22 November 2016 (UTC)


 * Relational algebra isn't an Algebra of Logic though. By all means have it on the page somewhere if you wish, but please remove it from the "In logic" section and replace it with Relation algebra. Have a look at the table of Lindenbaum–Tarski algebras on the Algebra of Logic page to see which algebras should rightly be in this section. Thanks! 84.93.190.255 (talk) 11:19, 22 November 2016 (UTC)


 * Is this swap OK? - DVdm (talk) 12:47, 22 November 2016 (UTC)


 * That's perfect. Thank you. 84.93.190.255 (talk) 17:06, 22 November 2016 (UTC)

Semi-protected edit request on 25 May 2016
To whom it may concern As a Mathematician who works in the field of Mathematical Neuroscience, I would strongly recommend Wikipedia community to please consider re-edit the "Wikipedia Algebra" article. I would appreciate it if you could please change the term "Arabic" to "Persian" in the Wikipedia Algebra" article, since it posses the Algebra as an Arabic name. The great Persian mathematicians, al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131) were absolutely from the Persian land. In science, we appreciate mathematicians and their great publications and contributions to the field by citing their works, where we can track the citations in "Google Scholar", which was not available in 800 years ago. To respect the Persian land, people of the Persian land and the soil that together introduced to the world the great Persian mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131), please edit the term "Arabic" to "Persian".

101.162.150.128 (talk) 12:16, 25 May 2016 (UTC)

What is being stated is that the word "algebra" derives from Arabic, not that the Arabs devised the concept of algebra itself. I am sure the concept has been called by many names over the years, English just happens to have adopted the Arabic word, although it is not clear if we borrowed it from Spanish, Italian, or Medieval Latin, and we altered it slightly on the way. - Arjayay (talk) 16:57, 25 May 2016 (UTC)
 * ❌ no-one is saying that Muhammad ibn Musa al-Khwarizmi and Omar Khayyam were not Persian - nor that there were not other Persian mathematicians working on the ideas. The section "Early history of algebra" refers to Persian five times.

It is better to include Persian mathematician and astronomer after mentioning the name Musa al-Khwarizmi and Omar Khayyam. For example the paragraph in the etymology should read "The word algebra comes from the Arabic الجبر (al-jabr lit. "the reunion of broken parts") from the title of the book Ilm al-jabr wa'l-muḳābala written by Persian mathematician and astronomer al-Khwarizmi. In other pages on Wikipedia when a person has contributed greatly or even has established a field, the person's origin is mentioned like Jewish, German, Greek... Here it is especially important since the title of the book and the language of the science and math in that region was Arabic at the time due to overtake of arabs of other countries and that other languages weren't allowed to be utilized until later that Persian language was again revitalized, and not due to the fields being developed by Arabs but by scientists and mathematicians from other cultures, ethnicities and countries. So in this format that the paragraph is written as of now, the information is correct but absence of the word Persian to describe one of the major developers of the field Algebra will be misleading as the consequence is that it implies that the development of the field is then related to Arabs not Persians, which is not accurate. In a course at Stanford in mathematical thinking the professor is calling this field having been developed starting from a book by an Arab mathematician and when I mentioned to him that this information is wrong, he mentioned this cite on Wikipedia saying that it refers to Arabic in reference to the word, the book and the mathematician who wrote it. Hence, clarification of this matter is definitely required in my opinion as to prevent Wikipedia being linked to incorrect information in this regard! Thank you! — Preceding unsigned comment added by Mgho12345 (talk • contribs) 17:28, 13 January 2017 (UTC)

The Greeks knew algebra 2,500 years before the Arabs. http://apocalypsejohn.com/ellines-gnorizan-algevra-2500-chronia-araves/ — Preceding unsigned comment added by 58.7.194.236 (talk) 05:04, 28 May 2017 (UTC)

External links modified
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better source needed
I think this is not a reliable source-https://www.algebra.com/algebra/about/history/ .I have added Better source template after Reference number 13 of the article ( Algebra )  F0r ★ bin IV 19:47, 21 October 2017 (UTC)

Etymology
Arabic al-jabr means ‘the reunion of broken parts’ rather than just ‘reunion of broken parts’. SocialismRules (talk) 22:03, 3 September 2017 (UTC)


 * Is there a comment here? That is verbatim what the article says. --Bill Cherowitzo (talk) 23:02, 3 September 2017 (UTC)


 * once the article uses the form with, once the form without the article "the".--345Kai (talk) 00:11, 21 December 2017 (UTC)

It does not make a lot of sense to me that the word "algebra" comes from a book written by an ancient Persian mathematician, yet in the English language it "originally" referred to the setting of bones. Is it possible to clarify this point?--345Kai (talk) 00:16, 21 December 2017 (UTC)


 * In English, the use of the definite article ("the") does not change the meaning of this phrase, so in the second occurrence which refers to the literal translation, the the is included, but in the first use only the meaning is translated. English readers would see no important distinction here. As to the etymology, the word algebra comes from the Arabic al-jabr and its literal meaning, 'the reunion of broken parts', very accurately describes the procedure of fixing broken bones, and so, algebra came to mean the setting of bones. Only centuries later did the mathematical meaning, coming from the title of the book, come into vogue and gradually began to replace the older meaning. Today, in English, the mathematical meaning has become the exclusive meaning of the term. The two meanings have the same parentage, but one is not related to the other. --Bill Cherowitzo (talk) 03:54, 21 December 2017 (UTC)

Dubious statement
Please remove this statement:

"the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta"

and add the name of Brahmagupta to this line:

"The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[24] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods."

It says: "the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta". Firstly,it is not supported by any reliable source.Many different sources contradict this statement.Brahmagupta gave rule for solving only one type of quadratic equation (I.e x^2 + px -q =0) see:page 159 => https://books.google.co.in/books?id=56lzDQAAQBAJ&pg=PA159&lpg=PA159&dq=brahmagupta+quadratic+equation+smith&source=bl&ots=ZP7Uq7n76a&sig=0wMpd_jquiDK0mKfqtcrSxZAxfk&hl=en&sa=X&ved=2ahUKEwj774vRi-DYAhUSTY8KHT4cDjMQ6AEwEnoECAcQAQ#v=onepage&q=brahmagupta%20quadratic%20equation%20smith&f=false — Preceding unsigned comment added by 119.42.56.115 (talk) 23:12, 17 January 2018 (UTC)
 * Red information icon with gradient background.svg Not done: Added a citation needed tag to the challenged statement but the source offered below to contradict the article fails verification. Eggishorn (talk) (contrib) 00:49, 21 January 2018 (UTC)

Gotta know it to use it - duh
What's with the "(for those who know how to use them)" quip at the end of the second paragraph? It adds nothing. No technology is beneficial to those who don't know how to use it. Please remove it. — Preceding unsigned comment added by 73.31.155.68 (talk) 17:44, 17 March 2018 (UTC)

✅ D.Lazard (talk) 18:02, 17 March 2018 (UTC)

Is the "Different Meanings of Algebra" section really nessacary?
It sort of seems like something that belongs on Wiktionary instead of on the Algebra page itself. Can we at least move it more towards the bottom of the article? Foxingkat (talk) 18:03, 5 October 2018 (UTC)
 * As a mathematician, i would say yes. Best regards. ---Wikaviani  (talk) (contribs)  18:24, 5 October 2018 (UTC)s

Removing a section
I would like to remove the section "Algebra as a branch of mathematics" and have incorporated it mostly into the lead of the "History" section:

In it adequately describing Algebra as "branch of mathematics", it does not have enough of a clear representation. For instance, does it simply mean how algebra is studied? Or how it is taught? Or is it the history of Algebra's development into a branch? These are all valid things to talk about in the section, but if it were to talk about all of them, it would basically be the entire article of Algebra. The section seemed to attempt to explain how Algebra is seen as a subject in mathematics, but by doing so, it delved into the history of algebra: i.e. When it talks about the development of algebra in the 16th century. If it were to be seen as an introduction to the topic, it would fail horribly; the section would seem to introduce it about as much as any other arbitrary section in the article. Even the lead explained the topic much better. Besides, the Elementary and abstract algebra sections introduce their parts of algebra very well.

In the end, the section would be better as an overview of the history of algebra than an article as itself. If anyone has a question about any particular part of my reasoning, I welcome them to ask me. I would also think that a full introduction could be added that involves more examples (but nothing like the "algebra as a branch" section I merged), but the sections of elementary and abstract algebra are probably the best places to add introductions to their respective places. Thank you! IntegralPython (talk) 16:01, 24 October 2018 (UTC)

Please list your opinions here, and if I have consensus, I will make the change. IntegralPython (talk) 16:12, 24 October 2018 (UTC)
 * I strongly disagree with this edit, and I have reverted it. The main point is that the answer to the question "what is algebra today?" does not belong to any history section. Another point that makes this section absolutely necessary is that the answer to this question remains unclear for many people, including many Wikipedians. A witness of this is that the lead forgets all parts of algebras that do not belong to elementary algebra and abstract algebra (note that these are not areas of mathematics, but titles of courses at elementary of undergraduate level; abstract algebra may also be viewed as the study of algebraic structures for themselves, and does not include many parts of algebra, such as computer algebra, invariant theory, homological algebra, combinatorics, and many others, including applications of algebra in physics and in other parts of mathematics). Another witness of this is that the main contributors to modern algebra are not even cited in the history section and in History of algebra (Galois, Gauss, Hilbert, Dedekind, Kronecker, Macaulay, Noether are among those who got seminal results in algebra, and are not cited).
 * Thus, a section saying what is algebra is absolutely required. Maybe the title "Algebra as a branch of mathematics" should be changed. Maybe this section should be rewritten, but there is a difficulty of finding reliable sources. When I wrote this section, the only source that I found undoubtedly reliable is AMS classification, which results of consensus of mathematicians community. So, rewriting this section would require a difficult research of sources. D.Lazard (talk) 16:54, 24 October 2018 (UTC)
 * P.S. I wrote the preceding post without reading again the article. After that, it appears that it contains more on the historical evolution of algebra that I remembered. But this does not change my opinion. As the answer to the question "what is algebra?" has dramatically evolved over the time, it cannot be answered without some historical references and without describing how algebra evolved from what everybody knows about it. These basic hints are necessary for understanding the history of algebra, but are too sketchy for belonging to section History. For making clearer the structure of the article, I suggest renaming the section What is algebra?. D.Lazard (talk) 17:51, 24 October 2018 (UTC)
 * Thank you; after explaining to me more carefully what the section is meant to be about, I agree with you on this. I think I will start work expanding the section, maybe in my sandbox first, instead of deleting it. Again, thank you for your further explanation of the purpose, and forgive me if at any point I might have seemed a little bit naive. IntegralPython (talk) 18:58, 24 October 2018 (UTC)

What is algebra?
Here is my proposal for a major edit to the current section, "Algebra as a branch of mathematics". It focuses less on history, and more on the fundamental rules and subjects of algebra. Please help me either improve the article, or if you think it is good, I can add it in. Obviously, it would be formatted with correct paragraph rules; it is like it is because I am still learning how to use talk pages effectively.

Algebra is a multifaceted branch of mathematics with many parts that are vastly different from others. Essentially, algebra is manipulation of symbols based on given properties about them. (citation:I. N. Herstein, Topics in Algebra, "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964) For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of arithmetic but with variables standing for numbers.(citation: ref name=citeboyer) This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation
 * $$ax^2+bx+c=0,$$

$$a, b, c$$ can be any numbers whatsoever (except that $$a$$ cannot be $$0$$), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity $$x$$ which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the quadratic equation above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry. (citation: )

Because algebra is simply the manipulation of entities, there is no rule that states that only numbers and variables that stand for numbers are allowed. In this way, algebra is extended to consider entities that do not stand for just one number, such as vectors, matrices, and polynomials. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra, while others form subjects such as linear algebra.

Even though algebra had already expanded into manipulations of many numbers in the defined topics above, it is possible to define entities that are unlike any familiar numbers. These entities are created using only their properties, and involve strict definitions that create a set of entities that work together with their properties. The entities form algebraic structures such as groups, rings, and fields. Abstract algebra is the study of these entities and more.(citation: http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018)

In short, the study of algebra involves any set of items which share properties. As long as it is possible to distill the similarities into similar sets that relate to one another in different ways, it is a part of algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification(citation: ) where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

Thanks! IntegralPython (talk) 21:14, 24 October 2018 (UTC)


 * There is, imho, a lot of un-encyclopedic and disputable content in the above proposal, even when the statements are sourced. I certainly dislike multifaceted, the whole second sentence, "used to quickly and easily find" (3.par.), "algebra is simply the manipulation of entities", juxtapositioning high school algebra and linear algebra (4.par), "manipulations of many numbers" (are we in big data in 5.par?), the whole "In short"-6.paragraph, "grown until it includes" (is there a metric?), and finally "also used extensively", which is to my measures a confession of "we don't really know" (maybe not too bad).


 * Maybe I miss the pre-Cartesian juxtaposition of geometry and algebra, then possibly more intimate to arithmetic, when we now have algebraic geometry and geometric algebrae. Maybe the latter use of algebra (a. of a vector space, exterior a., ...) as opposed to algebra as the abstract treatment of algebraic structures (this is mentioned) is also worth mentioning. I certainly do not miss elaboration of algebra on various levels of education.


 * I am honestly sorry for sounding this deprecatingly, please, just take my opinion lightly, not any slightest offence is intended. Purgy (talk) 08:07, 25 October 2018 (UTC)


 * Considering your feedback, Purgy, I have updated my proposal; I would like you to keep in mind however, that many of your objections apply to the current section as well, such as the "also used extensively" which is in the current revision of the article as it is. As for your idea about "algebraic geometry" part, I have added a small paragraph, although I admit that I do not know much about that particular subject, and would invite you to elaborate on it more as an addition.



Algebra is a complex branch of mathematics in which many subjects are vastly different from others. Essentially, algebra is manipulation of symbols and operations based on given properties about them.(citation: I. N. Herstein, Topics in Algebra, "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964) For instance, elementary algebra is about manipulating variables, which are abstractions of numbers in a number system. The variables in the number system are only allowed to have properties that are shared by every number it represents, and vice versa.

The most simple parts of algebra begin with computations similar to those of arithmetic but with variables that take on the properties of numbers.(citation: ref name=citeboyer) This allows proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation
 * $$ax^2+bx+c=0,$$

where $$a, b, c$$ are any given numbers (except that $$a$$ cannot be $$0$$), the quadratic formula can be used to find the two unique values of the unknown quantity $$x$$ which satisfy the equation, known as finding the solutions of the equation. Historically, the study of algebra starts with the solving of equations such as the quadratic equation above. The study of these equations lead to more general questions that are considered, such as "does an equation have a solution?", "how many solutions does an equation have?", and "what can be said about the nature of the solutions?". These questions lead to ideas of form, structure and symmetry.(citation: )

Algebra also considers entities that do not stand for just one number; using sets of numbers as algebras results in the ability to define relations between objects such as vectors, matrices, and polynomials. Many of these and the previously mentioned manipulation of variables form the basis of high school algebra.

Because an entity can be anything with well defined properties, it is possible to define entities that are unlike any set of real or complex numbers. These entities are created using only their properties, and involve strict definitions to create a set. The entities, along with defined operations, form algebraic structures such as groups, rings, and fields. Abstract algebra is the study of these entities and more.(citation: http://abstract.ups.edu/download/aata-20150812.pdf Retrieved October 24 2018)

In geometry, algebra can be used in the manipulation of geometric properties; the interplay between geometry and algebra allows for studies of geometric structures such as constructible numbers and singularities. Reducing properties of geometric structures into algebraic structures has created subjects such as algebraic geometry, geometric algebra, and algebraic topology.

Today, the study of algebra includes many branches of mathematics, as can be seen in the Mathematics Subject Classification(citation: ) where none of the first level areas (two digit entries) is called algebra. Algebra instead includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used in 14-Algebraic geometry and 11-Number theory via algebraic number theory.


 * Although it may not be perfect, I certainly think it is better than the current section, and I would like this compared to that. Thank you, IntegralPython (talk) 16:20, 25 October 2018 (UTC)
 * The manual of style and the common sense recommend starting with as few technicalities as possible, and adding technicalities progressively. This is respected in the current version, but not in the proposed one. In particular, the first paragraph involves a synthetic view of mathematics, which outside the knowledge of most readers.
 * The structure of the current section is
 * Paragaph 1: introduction of the basics of algebra that almost everybody knows, with emphasize on its difference with arithmetic (which is less commonly understood)
 * Paragaph 2: the main historical problems of algebra, and how they lead to the concept of algebraic structures, As, for many people, including many historians of mathematics, "algebra" is synonymous with "theory of equations", the relationship between these two subjects must be clarified.
 * Paragaph 3: Relationship between algebra and other branches of mathematics
 * Paragaph 4: Parts of moderns mathematics that belong too algebra.
 * The discussion on possible edits of the article must distinguish between the modification of the structure and the modification of the content of each paragraph. My personal opinion is that the structure of the proposed version is less clear than the current one, and this deserves a discussion.
 * The content of the paragraphs of the proposed version has several issues. My main concern is the theory of "entities" that seems WP:OR and does not appear in any algebra textbook the I know of.
 * Also, the sentence "These questions lead to ideas of form, structure and symmetry", which appears also in the current version is completely wrong, as, firstly this is geometry, not algebra the led to the ideas of form and symmetry, and, secondly, nothing leads to the idea of structure: structures are elaborated as abstractions allowing a better understanding of some problems. Generally, understanding the deep structure of a problem is done by the greatest mathematicians. In algebra one may cite, for example, Galois, Gauss, Hilbert, Grothendieck, who all have deeply changed algebra by understanding hidden structures. I'll remove this sentence in the article. D.Lazard (talk) 16:00, 26 October 2018 (UTC)

Elementary and Abstract Algebra in the First Paragraph
The first paragraph of the page contains the sentence "The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra." I propose a couple of changes to make it more formal. TechnicallyTrue376 (talk) 23:50, 30 October 2018 (UTC)
 * Change "more basic parts" to "basic areas." The word "parts" reads as informal and "more" is unnecessary as it's not being compared to anything yet.
 * Change "more abstract parts" to "more advanced areas." "Parts" has the same issue as before, and it is not helpful to use the word "abstract" when defining abstract algebra.


 * Done, partially. Cheers. ---Wikaviani  (talk) (contribs)  00:16, 31 October 2018 (UTC)
 * Undone, because it is disputed whether elementary algebra and abstract algebra are scientific areas. IMO, these are only course titles or course subjects. D.Lazard (talk) 08:56, 31 October 2018 (UTC)
 * Nobody said that elementary algebra and abstract algebra are scientific areas. The edit aimed to remove the word "part" and replace it with "areas", per the rationale given by . If there is no consensus about that, i'll not insist anymore. Cheers. ---Wikaviani  (talk) (contribs)  15:15, 31 October 2018 (UTC)

Algebra is about combining "objects"
D.Lazard, I do not think "object" is a jargon here as it was used in (Herstein 1964) (i.e., "An algebraic system can be described as a set of objects together with some operations for combining them", see references). --Habil zare (talk) 05:14, 3 May 2019 (UTC)
 * This is a general article for an audience which should not be restricted to people with skills in mathematics. So, if you use a word with a meaning that is not its common meaning, you must define or link it. This is the case here, as, for everybody, an object is a physical object (a chair for example) not an abstract entity. Considering an abstract entity as an object is indeed mathematical jargon. If you look at object (disambiguation) you will see three mathematical meanings of "object", two of them are clearly far from your intended meaning, and the other, mathematical object, includes your meaning, but is much wider, as it includes, for example algebraic systems in the sense of Herstein. I do not see any way of using "object" here in a way that cannot be misinterpreted. So, the best is to not use this word.
 * Also, Herstein does not defines algebra, but algebraic structures, and does not talk of objects in general but of "a set of objects"; that is, it defines implicitly his objects as the elements of a given set. So your quotation is not helpful here.
 * Finally, there are many books on algebra, but few give an definition "algebra". Moreover, the scope of algebra has evolved over the time. So a single reference is not sufficient here. The present formulation results from a WP:consensus among editors, and any change needs a new consensus. This alone would be sufficient for reverting your edit. D.Lazard (talk) 08:04, 3 May 2019 (UTC)
 * What does the word "manipulate" mean in "rules for manipulating these symbols"? We do not change or edit symbols in algebra. We use them to write sound and valid formulas, and algebra has rules for changing (manipulating) the formulas in a way that they remain valid. However, I think this is too general, and applies to other fields of mathematics. The closest field to match this definition is mathematical logic not algebra. Symbols and formulas are used in all fields of mathematics for example in analysis, if $$x$$ is a real number and $$f$$ is a function, then $$f(x)$$ denotes the image of $$x$$. The special thing about algebra is that when we combine the symbols in an algebraic formula, we usually do not think of anything else other than the "combination" itself. That is, when we write $$ab$$ for $$a$$ and $$b$$ as members of a group, we do not pay much attention to what $$a$$ and $$b$$ are (i.e., numbers, matrices, functions, etc.) but in algebra, we are interested to see how they combine and what are the properties of this combination with regards to other objects and combinations we are studying. "Is the function $$f$$ differentiable at $$x$$?" is not an algebraic question because it is about the properties of the object (i.e., $$f$$) thus this is a question in analysis. In contrast, $$f+g=g+f$$ is an algebraic sentence even when we know $$f$$ and $$g$$ are real valued functions because here we are talking about their combination not the identity or properties of each object. --Habil zare (talk) 13:17, 3 May 2019 (UTC)
 * Wiktionary definition of "manipulate: "To move, arrange or operate something using the hands". This is exactly what is done in algebra: "arranging" symbols into formulas and transforming them, using rules. The group axioms is an example of such rules. You wrote "The special thing about algebra is that when we combine the symbols in an algebraic formula, we usually do not think of anything else other than the combination itself". This is exactly what the article says by saying that in algebra one manipulates symbols (and not the objects represented by these symbols, which may have further properties).
 * Nevertheless this page is devoted for discussing how improving the article, not for expressing personal opinions about the subject of the article. So, if you have an edit suggestion that can be the object of a consensus, propose it, and if, possible, give a source for it. You introduction of the word "object" is such a suggestion, but, as far as I know, there is no chance for getting a consensus about it. If you have strong arguments against a formulation, give it, but I don't like it and other personal opinions are not good arguments. Personally, I do not see anything in your last post that can be used to improve the article. D.Lazard (talk) 14:32, 3 May 2019 (UTC)
 * The phrase "rules for manipulating these symbols" is not appropriate to describe algebra in this article because: 1) it is not directly mentioned in, but inferred from, the reference after it, 2) it is not "specific to algebra", e.g., in the formula $$d(x,y)=d(y,x)$$ for a metric $$d$$, we move the symbols $$x$$ and $$y$$ around but metric spaces are studied in analysis not algebra, and 3) it is far from "completely describing algebra" because many of the propositions and discussions in algebra do not need symbols. Example 1: A subgroup of any Abelian group is Abelian. Example 2:Any finite alternative division ring is necessarily a finite field (Artin–Zorn theorem).--Habil zare (talk) 17:27, 3 May 2019 (UTC)

Proposition: Latex conversion
Proposition. Latex should be preferred as the default for mathematical typography on this page because:

1. More popular peer math pages prefer this style, such as Linear Algebra, which uses purely Latex for its mathematical typography.

2. Mathematical typography should be consistent at least within-page even if not between pages.

3. Usage of multiple templates or raw unicode leads to a LOT of rendering variability and browser interaction, whereas Latex focuses efforts of mathematical typography and rendering onto preferred community libraries.

4. Maintenance becomes easier with uniformity. Across different math pages one may find raw unicode, a no-wrap template, a variables template, or a generic Math template. — Preceding unsigned comment added by SirMeowMeow (talk • contribs) 11:46, 22 January 2021 (UTC)
 * Replied at Talk:Module (mathematics) D.Lazard (talk) 12:59, 22 January 2021 (UTC)

Semi-protected edit request on 20 May 2021
Dear Sir/Ma'am I am Ishanya Kirit Jain and I request you to permit me to do editing in your article. Thanking you. Regards, Ishanya Kirit Jain Ishanya Kirit Jain (talk) 11:18, 20 May 2021 (UTC)
 * Red information icon with gradient background.svg Not done: this is not the right page to request additional user rights. You may reopen this request with the specific changes to be made and someone will add them for you, or if you have an account, you can wait until you are autoconfirmed and edit the page yourself.  Mel ma nn   12:16, 20 May 2021 (UTC)

Algebra formulae
From a formula to an answer Lungelo Ndlovu (talk) 21:20, 10 April 2021 (UTC)

working out 105.230.203.194 (talk) 11:48, 5 September 2021 (UTC)

Argebraic expression
work 105.230.203.194 (talk) 11:47, 5 September 2021 (UTC)

easy 105.230.203.194 (talk) 11:48, 5 September 2021 (UTC)

Biased Towards Mathematicians
Mostly section 4. Where's the criticism of algebra, I've seen a lot people with it. If this is for a general audience, general audience opinions from reliable sources should be included. Everyone: OMG, there's no reliable sources because they're all biased and no one ever published anything. Everyone: It's not notable. WP:NOTABILITY Me: * sigh * — Preceding unsigned comment added by Fun81 (talk • contribs) 12:36, 25 May 2021 (UTC)

there are reliable SOURCES 105.230.203.194 (talk) 11:51, 5 September 2021 (UTC)

Broad areas in the first sentence
An editor as suppressed number theory in the first sentence, with the edit summary. I reverted this for the following reasons. D.Lazard (talk) 14:08, 27 October 2021 (UTC)
 * Number theory predates algebra for centuries.
 * : this is definitively wrong, geometry of numbers and analytic number theory, among others, are part of number theory that are not applied algebra, and are far to be secondary.
 * The mention of MSC is dubious, as number theory is a first level entry, while algebra is not. The mention of Bourbaki's classification is dubious either as, AFIK, Bourbaki was against splitting mathematics in separate subjects, and never provide such a classification.
 * The first sentence refers mainly to the state of mathematics from the 17th to the 19th century, or, almost equivalently, to elementary mathematics. As explained later in the article many new areas have been included since the beginning of the 20th century. As this article is intended for non-mathematicians, it is natural to mention here only the areas that are known by almost everybody, and to omit more specialized ones.


 * I agree that analysis also appears in Number theory, but I nevertheless do not see it as one of the top-level areas of math, and I still think it is misleading to picture it as an area distinct from algebra. However the classification of the areas of math is a matter of taste that we cannot decide without sources like MSC.
 * The article on MSC doesn't mention number theory as a first level entry. The first-level areas are : general/foundations (including mathematical logic), discrete math/algebra, analysis, geometry, and applied math.
 * The german article on areas of mathematics (https://de.wikipedia.org/wiki/Teilgebiete_der_Mathematik) which explicitly refers to Bourbaki's classification doesn't mention number theory as a top-level area either. --L&#39;âne onyme (talk) 17:30, 27 October 2021 (UTC)


 * I didn't read the MSC correctly. You were right, numbertheory is a first-level entry. However that makes it only one of many (64) areas of math. I still get the inkling that algebra and analysis are much broader areas than number theory. --L&#39;âne onyme (talk) 18:31, 27 October 2021 (UTC)
 * The field of mathematics called number theory (theory of arithmeticae) has been recognized by people since Gauss's Disquisitiones Arithmeticae. The Disquisitiones covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. The name "number theory" probably comes from Adrien-Marie Legendre's Essai sur la Théorie des Nombres. By that time, Euler had already taken an analytical approach to number theory. But anyway, number theory is a field of mathematics that studies the properties of integers. However, things like "number theory (properties of integers) is not pure algebra" are not uncommon in mathematics, and examples such as Chow's theorem for closed complex projective space, may be of interest to you.--SilverMatsu (talk) 02:17, 28 October 2021 (UTC)

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 20 August 2021 and 11 December 2021. Further details are available on the course page. Student editor(s): Gfurga2.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 13:49, 16 January 2022 (UTC)