Talk:Algebra/GA1

GA Review

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Reviewer: Bilorv (talk · contribs) 19:41, 9 March 2024 (UTC)

I'm excited to take this one. An enormous topic, obviously, and so I'll spend quite a bit of time thinking hard about scope and breadth (criterion 3). That this article should exist at all is non-obvious and controversial—as the definition and history sections say, there are different interpretations of what "algebra" is. Elementary/symbolic algebra is used in every field of maths and by "algebra" laypeople often refer to notation rather than concept. "Abstract" algebra is a better-defined field of maths but one of the broadest, and one that overlaps with unexpected fields. Nonetheless, I think there is enough of a connection between elementary and abstract algebra that there is an encyclopedia article in it, not just a dictionary definition or a disambiguation page to elementary algebra and abstract algebra. — Bilorv (talk) 19:41, 9 March 2024 (UTC)

Hello Bilorv and thanks for doing this review. These were also some of the concerns I was struggling with when I got started with this article and I arrived at a similar conclusion. Phlsph7 (talk) 07:58, 10 March 2024 (UTC)

Okay, apologies for the verbosity that follows but I'd rather be unambiguous than concise. Feel free to push back on points or make counterproposals. — Bilorv (talk) 23:33, 9 March 2024 (UTC)

Scope comments

• Maybe in "Other branches of mathematics", I feel more could be made of the frequent transformation between geometric problems and algebraic problems. An illustrative example: we can prove that a line intersects a circle either 0, 1 or 2 times by defining some axes and showing that the algebraic equations $y=mx+c$ and $(x-a)^{2}+(y-b)^{2}=r^{2}$ yield a quadratic whose discriminant determines the number of solutions. Algebraic symbol-pushing is a completely different conceptual approach to geometric reasoning. "Geometry" might be its own subsection.

  Good idea, I added a similar example. I considered having separate subsections for the applications of algebra to different mathematical fields but my impression was that this would lead too much into details and is better covered in separate articles, like Algebraic geometry. Phlsph7 (talk) 12:45, 11 March 2024 (UTC)

• Perhaps in "Elementary algebra" or in "History", I feel there should be some treatment of notational conventions. For instance, we typically name an isolated variable x; an equation of a line is typically $y=mx+c$ rather than $A=bC+\theta$ (on a b-A axis graph); arbitrary numbers of variables typically use subscripts like $x_{i}$; polynomials often order terms by decreasing order and coefficients precede variables; vectors might be lowercase bold and matrices uppercase bold. At least noting the existence of conventions with an example would be good.

  I found a spot in the subsection "Elementary algebra" to explain the basics. I left out the part about vectors and matrices since they are not named in any of our examples. Phlsph7 (talk) 13:35, 11 March 2024 (UTC)

• In elementary algebra, some authors make a distinction between "equation" and "identity" and the latter can be an object of study in its own right (like proving that $\cos 3\theta {\text{cosec}}\theta +{\frac {\cos \theta }{\sin 3\theta }}\equiv 2\tan 2\theta$, though I don't know where trigonometry falls with respect to algebra). Identities might deserve some sort of treatment (a possible place where it would naturally follow: the end of the paragraph containing "their truth value usually depends on the values of the variables"). Formulas also, and it seems it is invoked in only one of the three definitions the articles gives of "elementary algebra" (where another word may fit better).

  I added an explanation about the difference between identity equations and conditional equations, which, I assume, is what you meant. I'm not sure if this information is essential but it probably does not hurt either. I didn't get your last comment about formulas. Phlsph7 (talk) 14:02, 11 March 2024 (UTC)

  Yes, that addresses identities/conditional equations. On "formula": the article mentions the word twice ("examines how formulas may be transformed" and "making it possible to express equations as mathematical formulas") but doesn't define it. It might be better just to avoid using this word. To some authors "formula" has a specific definition (e.g. 'an equation in two or more variables') that may not agree with other specific definitions. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  Ah, I see. I manage to reformulated those passages. Phlsph7 (talk) 08:57, 17 March 2024 (UTC)

• Is abstract algebra only about binary operations? Am I right in saying a normed vector space has two binary operations ($+,\times$) and a 1-ary operation (the norm)?

  According to the reliable sources, the answer is yes to both questions, which kind of creates a paradox. I added a short footnote to clarify this point. Phlsph7 (talk) 16:45, 11 March 2024 (UTC)

• Perhaps in "Definition and etymology", can we talk about how mathematical ideas might be categorised today by whether they are "algebra" or not? e.g. an exam syllabus; Dewey Decimal 512; Mathematics Subject Classification doesn't separate "algebra" as its own thing.

  In principle yes, but I'm not sure that there is an easy answer to this. For the Mathematics Subject Classification, the answer would be quite messy and probably not particularly useful to the reader. Does the Dewey Decimal Classification provide a precise definition of each of its branches or does it rely on the people who use it to classify items correctly? Phlsph7 (talk) 16:54, 11 March 2024 (UTC)

  From my experience the classification just gives the name of an area, like "Algebra", and it is down to each library to sort books into those areas as it wishes. In practice, however, there are databases with book-classification pairs that many libraries use. I think the Dewey number and maybe Library of Congress QA 150-272.5 are significant to mention as they imply that most libraries sort books by whether or not they are about "Algebra". — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  I added a footnote to cover the different classification systems. Phlsph7 (talk) 09:29, 17 March 2024 (UTC)

• In "Elementary algebra", substitution of variables by algebraic expressions is treated as a method of solving, but variables can be substituted for numbers and substitution can be something to study in its own right e.g. substituting natural numbers into $an+b$ generates an arithmetic progression.

  I mentioned substition for numbers. I don't think that the arithmetic progression is central enough to elementary algebra to deserve a separate discussion. Phlsph7 (talk) 17:00, 11 March 2024 (UTC)

• "Elementary algebra has applications in many branches of mathematics, the sciences, business, and everyday life" and "Because of its presence throughout mathematics, the influence of algebra extends to many sciences and related fields" – To spell out every 'application' of 'algebra' would be a neverending task, but I feel that "In various fields" could be renamed "Applications" with an introductory section or a subsection beginning with these ideas. It might say that algebraic notation and terminology is used in science (e.g. chemical equation with its own idea of "balancing"; to express physical laws like $F=ma$), in computer science (with its own idea of variables), and anywhere it is desirable to model systems that change over time (business/economics/geography) or have systems of calculations (finance).

  Done. This topic is a little difficult to properly cover since most sources only talk about specific applications rather than the general overall influence. Phlsph7 (talk) 17:57, 11 March 2024 (UTC)

• I think isomorphism needs some treatment in "Universal algebra", the key being: it is a type of homomorphism that preserves all structures, so in some sense two isomorphic objects are "the same", and we say things like "there are 3 groups of order 8 (up to isomorphism)". Perhaps also automorphism might be worth describing (as it feels slightly different—rather than saying "two structures are the same" it says "one structure has symmetry" or "these elements of the structure have the same role").

  I added an explanation of isomorphisms. You are probably right to say that this should be discussed. I hope it doesn't add too much to the difficulty of this section. Phlsph7 (talk) 18:38, 11 March 2024 (UTC)

• In "Education", can there be some treatment of the psychology of algebra or the reason it is deferred until secondary education? For instance, it might be that algebra is too abstract until a certain maturity or prerequisite knowledge/skills—I don't know if there's some model like concrete-pictorial-abstract that could explain this. Or: is it only notation that is deferred to secondary education (do students discover commutativity when learning times tables by rote, or is it even taught to them explicitly?)?

  I expanded the introductory explanation of that point. The details of how it is taught probably differ a lot form country to country, both concerning the specific curriculum and at which grade primary education ends and secondary education starts. Generally speaking, I would assume that primary students already train some forms of algebraic thinking when solving certain word problems but the topic is not explained to them in a systematic manner. Phlsph7 (talk) 09:20, 12 March 2024 (UTC)

Prose comments

Lead

I have a lot to say about the first paragraph as it defines the scope of this article and its existence as a coherent topic that contains subtopics like elementary algebra and abstract algebra.

• I'm not sure how formal it is to use phrases like "algebra is interested in", "algebra studies" etc. as if it is sentient. Alternatives would be that algebra "encompasses", "includes", "comprises" etc. This is a pattern from top to bottom e.g. "It [education] aims to familiarize students" could be "Educators aim to familiarize students".

  I understand your concern but it is often the most convenient way to express something and should be easily understandable to readers. This language is also used in reliable sources. For example, from [1]: An important part of algebra, linear algebra, studies linear spaces,...; and from [2]: Algebra ... seeks to solve equations ... abstract algebra is interested in such question as.... I wouldn't recommend a general change to this practice but I'm open to fixing individual cases where this type of language does not fit well. Phlsph7 (talk) 09:32, 12 March 2024 (UTC)

• First sentence: I think self-reference ("Algebra ... studies algebraic systems") is to be avoided and I personally don't like the focus on "systems" (less clear than something like algebraic structure) and "equations" (inequalities are also of interest, as are algorithms like "simplifying an expression" that go beyond the equivalence relation properties of equations). Would it be fair to say instead:

  Algebra is the branch of mathematics that encompasses abstract mathematical objects and the manipulation of such objects.

  Or: the first sentence doesn't necessarily have to be a definition e.g. "Algebra is a mathematical concept that includes ..."

  The main difficulty I see is to find a definition that covers the different branches of algebra. I think your first suggestions does a good job at that. One difficulty may be that it is too wide since algebra does not study all abstract mathematical objects. For example, proofs can be understood as mathematical objects but they are not directly studied by algebra.

  According to MOS:LEADSENTENCE, If its subject is definable, then the first sentence should give a concise definition. So starting with a definition would definitely be preferable. What do you think about talking of algebraic structures instead of algebraic systems. Algebaric structures can be defined without reference to algebra. The sentence could be: "Algebra is the branch of mathematics that studies algebraic structures and the manipulation of equations within those structures." If the focus on equations is too narrow, we could replace it with "statements" or "algebraic expressions". Phlsph7 (talk) 09:59, 12 March 2024 (UTC)

  I'd be happy enough to see Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. I see what you mean about my first suggestion being too broad. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

  Done. Phlsph7 (talk) 09:32, 17 March 2024 (UTC)

• I don't like "regular numbers" at all (especially as "regular" is often used for maths jargon). How about: "It is a generalization of arithmetic that introduces variables and algebraic operations other than ..."?

  Done. Phlsph7 (talk) 10:02, 12 March 2024 (UTC)

• Is it possible to say a bit more on the abstract algebra side about how numbers themselves are generalized? In algebraic structures, we treat all sorts of things as though they are numbers (by taking a set of them and applying operations to them): the congruence class 3 (mod 7); 4x6 matrices; symmetries of a hexagon; or just formal variables. Is that abstraction part of what defines algebra?

  We could provide some examples but I usually try to keep the lead as concise as possible so it may not be the right place for this. Phlsph7 (talk) 12:26, 12 March 2024 (UTC)

• I think [[equation solving|isolate variables]] is an Easter egg/overly specific link that could just be dropped e.g. b is isolated in the inequality $b\leq -3$ or the equation $b=cx+d$ (but is changing the subject of a formula the same as "solving"?).

  Done. Phlsph7 (talk) 12:27, 12 March 2024 (UTC)

• "investigates more abstract patterns that characterize algebraic structures" – is it worth saying (or even accurate to say!) universal algebra is about studying classes of algebraic structures?

  Done. Phlsph7 (talk) 12:30, 12 March 2024 (UTC)

• To stick to the recommended four paragraphs of lead, can "Algebra is relevant to ..." be merged with the previous paragraph?

  Done. Phlsph7 (talk) 12:30, 12 March 2024 (UTC)

Up to "Abstract algebra"

• Not all variables are maths formatted, like "x-y-pairs" without italics, and the minus sign is its own thing, which needs fixing in phrases like "either 2 or -2".

  Done. I hope I got everything. Phlsph7 (talk) 12:49, 12 March 2024 (UTC)

• In note (a), "an algebraic operation is mapping" should be "is a mapping", but would "function" be better than "mapping"?

  Done. Phlsph7 (talk) 12:49, 12 March 2024 (UTC)

• For consistency I think "polynomial" should be defined in a footnote.

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "A set is a collection of elements" – I think "unordered" and "distinct elements" needs to be said somewhere in the definition of "set".

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• It's not clear from the prose that The Compendious Book... is alternately titled Al-Jabr.

  I reformulated it to clarify this point. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "it is possible to express a general law that applies to any possible combinations of numbers" – "commutativity" should be mentioned and linked as the example used here.

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "Elementary algebra is interested in algebraic expressions, which are formed" – As well as the "interested in" issue from above, I feel this would be more concise and direct as "Algebraic expressions are formed ..."

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• Is $\neq$ an important enough inequation to be mentioned?

  Done. Phlsph7 (talk) 13:25, 12 March 2024 (UTC)

• "To achieve this, it relies on different techniques used to transform and manipulate statements" – This would be simpler as "Techniques to transform and manipulate statements are used to achieve this"

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "A key principle guiding this process is that whatever is done to one side of an equation also needs to be done to the other side of the equation." – Again I'm not sure whether this needs limiting to equations as this is also broadly what is done with inequations too (though multiplying and dividing by non-positives can break things). "whatever is done" might also more formally be "whatever (algebraic) operation is applied" (you can't "do" things like "append a '3' digit to the end of each number" and keep balance).

  I implemented the second part. For the sake of simplicity, I think it's better to just stick with "equation" here. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• On isolating variables and $x-7=4$: a key idea is the inverse of an operation or the inverse of an order of steps (to isolate x in $3x^{2}+2=c$, note that x has had $^{2},\div 5,+2$ done to it so we perform $-2,\times 5,{\sqrt {}}$).

  That's a good point. It's just that we haven't defined the inverse of an operation so it might be better to just have the simple intuitive example here and leave the details of the different approaches to the article Equation solving. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "The polynomial as a whole is zero if one of its factors is zero" – This is an "if and only if" (otherwise we wouldn't have solved completely, just found a subset of solutions).

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "Other techniques rely on commutative, distributive, and associative properties" – These terms and links might best be deferred to "Abstract algebra". I'm not sure what these techniques might be other than those already mentioned plus expanding (which might be worth mentioning).

  Done. Phlsph7 (talk) 13:47, 12 March 2024 (UTC)

• "For example, the system of equations ..."– These might be better as bullet points or (a), (b), (c) than numbers, to avoid confusion. Also, is it not more common to show matrix multiplication without a symbol, and the dot might be confused with the dot product.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• Rather than a footnote, I think multiplying an equation by a constant is an important enough technique to say (and source) in prose.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "Addition is its binary operation and takes two numbers as input to produce one number in the form of the sum as output" – I think this wordiness will create more confusion than it solves. I would prefer "... has the natural numbers as the underlying set and addition as its binary operation". You could add to the "black box" image caption: "... as output, like addition and multiplication do".

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "The underlying set can contain mathematical objects other than numbers and the operations are not restricted to regular arithmetic operations" – I think showing rather than explaining would be more helpful. A symmetry group might make a good example.

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• The ring of integers is a ring of the form $\langle \mathbb {Z} ,+,\times \rangle$. – I don't think "of the form" fits because there is only one ring of integers and what's given is just notation for it. Maybe "a ring denoted by"?

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

• "The rational numbers, the real numbers, and the complex numbers each form a field" – since operations are emphasised here, "... form a field with the operations addition and multiplication"?

  Done. Phlsph7 (talk) 17:53, 12 March 2024 (UTC)

From "Universal algebra"

• "It is a generalization of identity in the sense ..." – I think this sentence is redundant to the previous sentence (and the meaning of the prefix "quasi-").

  You are right. My reason for adding it anyways was that it is a difficult topic of which the reader may not have heard before. So having some redundancy could be a good thing. Let me know if you think otherwise then I'll remove it. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "A homomorphism is a function that... Its special feature is that..." – Can the wordiness be improved? Most of the key information is conveyed in the following sentences. I would just replace these two sentences with "A homomorphism is a function from one set to another that preserves some of the algebraic structure".

  I reformulated this passage to make it more concise. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "in the form of a theory of equations" – I think this phrase needs explaining, at the very least in a footnote.

  I reformulated the sentence. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• In the caption, should "al-Khwarizmi's ..." be lowercase even at the start of a sentence?

  No. I fixed it. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "the use of zero and negative numbers" – is a link to 0 justified (as a way to learn more about its history)?

  Done. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• vector algebra is a disambiguation link and also has unclear meaning (is this the invention of the Euclidean vector and associated notation and basic operations?)

  I disambiguated it to Vector space. The basic idea here is that operations on vectors were conceived in terms of algebraic structures. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "The basic idea of the even more general" – I would drop "basic", and "first conceived" in the same sentence should just be "conceived".

  Done. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "Closely related developments" – were these all 20th-century developments?

  Most of them but I think not all. The formulation "Closely related developments" is intentionally vague to not imply too much. Phlsph7 (talk) 18:35, 12 March 2024 (UTC)

• "Another key aspect is to apply structures to model how different types of objects interact" – I'm not sure I understand the meaning of this sentence.

  I simplified this sentence. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• the connective "if...then" – I think the spacing of MOS:ELLIPSIS should apply here.

  Done. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "students need to learn how to transform them according to certain laws until the unknown quantity can be determined" – The "until" clause isn't always the aim (e.g. "write $y={\frac {2}{5}}x+3$ in the form $ax+by+c=0$").

  I reformulated the sentence to not imply that this is the only goal. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "A common example to introduce students" – Rather than an "example" I feel like this should have some technical name like "mental model" or "pictorial approach".

  Done. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "The mass of some weights" – Maybe this is just me but the mixture of "mass" and "weight" is confusing. A weighing scale moves according to weight, right, so I'd say "the weight of some objects"?

  True, that can be confusing. I kept the term mass since the examples usually focus on the mass of objects rather than the forces acting on those objects. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

• "For example, students may be presented with a situation in which Naomi" – I think the solution should be given, in prose or a footnote: $2x+x=12$, $x=4$ and Naomi has $2x=8$ apples.

  Done. I slightly modified the example so that x represents Naomi's apples. Phlsph7 (talk) 09:00, 13 March 2024 (UTC)

Referencing and other

Most of the references are introductory books on math/algebra, clearly by expert authors and reliable publishers. They go beyond Wikipedia's requirements of mere verifiability into accessibility to layperson readers or interested learners. Math encyclopediae and books for laypeople like the Very Short Introduction series are suitable here as we only want an overview of an enormously broad topic. Britannica is a source to assess case-by-case and for the (elementary) facts it verifies I think it's a good reference. I've read a bit more into MathWorld and the Stanford Encyclopedia of Philosophy and am happy they are reliable. Other sources are clearly academic and appropriate.

I've been spotchecking ad hoc as I go so I'm only going to do a few systematically (chosen by random number generator): #22, #41, #44, #87, #95. No issues found. Also obviously no copyright issues. Very impressive.

The images are all free and strike a good balance of illustrating concepts and providing historical information. I'm not 100% sold that File:Venn A subset B.svg illustrates much of use (to me the important idea is that operations can't take you out of the subset) but I understand its relevance; however, I think it should be moved a paragraph down so it precedes the subalgebra paragraph as the use of A and B and existing placement could make a reader think it is about the homomorphism example. (Remember that most readers are on mobile and see an image directly where it is placed in wikitext.) — Bilorv (talk) 21:35, 16 March 2024 (UTC)

Done, I moved the image. Phlsph7 (talk) 09:40, 17 March 2024 (UTC)

Overall

Thanks for everything you've done so far and I'm looking forwards to discussing and reviewing this further! From what I've briefly spotchecked so far, it looks like the references are all very reliable, accessible and direct but I'll have to check verifiability at the next stage. This is formally on hold but that might be for longer than seven days (especially given the scope of the topic and that some of my comments might require significant amounts of research), as long as progress is being made. — Bilorv (talk) 23:33, 9 March 2024 (UTC)

@Bilorv: Thanks for the in-depth review and all the helpful and actionable comments. I tried to address all the points and look forward to hearing your responses. Phlsph7 (talk) 09:12, 13 March 2024 (UTC)

I've been broadly following the changes as you've made them but a proper response may have to wait until the end of the week. Things are looking good so I'll move onto spotchecks and a second runthrough for any minor tweaks that need to be made. — Bilorv (talk) 16:50, 13 March 2024 (UTC)

@Phlsph7: I'm happy that almost all of these have been addressed, with a reply on the points about formulas, the Dewey Decimal Classification and the first sentence. None of these are dealbreakers if you prefer the status quo. I've also added a subsection commenting on references and one point on images. — Bilorv (talk) 21:35, 16 March 2024 (UTC)

@Bilorv: I've made the corresponding adjustments. I appreciate all the time and effort you have put into this review! Phlsph7 (talk) 09:42, 17 March 2024 (UTC)

Pass for GA. Thanks for all your work in this review, too. It's an extraordinary achievement to get this vital article to such a high standard. — Bilorv (talk) 09:56, 17 March 2024 (UTC)