Talk:Equation

syed zain ul abdeen

syed zain ul abdeen — Preceding unsigned comment added by Syedzainulabdeen (talk • contribs) 11:01, 7 November 2014 (UTC)

Equation vs equality

I reverted three edits by Zedshort in the lead section. Here are my reasons in more detail.

(1)

I deleted the following text (and restored an earlier sentence):

"There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. An conditional equation is true for only particular values of the variables"

The initial 3 sentences of the lead establish the terminology that an equation is something to be solved, i.e. (usually) not true for all values of the variables, but often only for a few (called the solutions). The deleted text is inconsistent with that; what it calls an "identity equation" is called an "identity", or an "identity equality" in the article. The text would fit perfectly into the equality article, after changing "equation" to "equality". Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

First source: College Algebra, fifth ed. 1964, author Lyman M Kells, publisher Pretice-Hall Inc, page 80.: "An identity is an equation satisfied for all values of the symbols for which its members are defined. ... A conditional equation is satisfied for all values of the symbols for which its members are defined" (bolding theirs). Second source; College Algebra 2nd ed. Brooks/Cole Publishing Co. 1996, page 69 Chapter 2: Equations and Inequalities: (w -4)(w + 4)= w² - 16 and 4x + 7 = 19 the letters w and x are variables. In the first of these equations, the equation is true no matter what value the variable w stands for. This equation is the "differnce of squares" formula from Chaper 1; it is true for all w, so we say it is an identity." Zedshort (talk) 19:09, 4 August 2016 (UTC)

I suggest to stick with that terminology, even if some references use another one. All references should agree that it makes sense to distinguish between

1. something that is asserted to be valid for all values of their variables (like "x+y = y+x for all x,y", called identity equality in wikipedia),
2. something that is asserted to be valid for a given value range of their variables (like "sqrt(x)²=x for x>=0", called conditional equality), and
3. something that is to be made true by finding appropriate variable values (like "for which x is 14x+15=71 ?", called equation in wikipedia).

I suggest that the wikipedia terminology is kept, and deviating terminology is mentioned. Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

I could not disagree more, simply because a crew of wiki-people agree to disagree with published references is not sufficient reason to mislead people. You are losing insightfull information about the subject that is vital for people who are new to the subject or want a review. Zedshort (talk) 19:09, 4 August 2016 (UTC)

(2)

I deleted the following text:

An equation is analogous to a scale into which weights are placed. When equal weights of something (grain for example) are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. Likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality.

My reason was the same as above: the scale analogy fits well with "equality", but doesn't fit with "equation". It could be adapted to fit with the latter by asking something like "how many grains have to be put on the left scale to obtain balance if ...?", i.e. by giving a Word problem (mathematics education) that amounts to solving an equation. Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

When did equation and equality come to be separated into totally different things? I am puzzled.Zedshort (talk) 19:09, 4 August 2016 (UTC)

When an equation is solved, it is important to keep track of the solution set, while the deleted text suggests that "balance" should be kept. I repeat my counterexample from my edit summary: squaring an equation usually changes the solution set (and therefor has to be handled with care), but doesn't change "balance" (whatever that word means in absense of particular values for the variables). Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

The balance is an analogy. As has been said many times before "Do not carry the analogy too far." You could successfully push the analogy a bit further, but obviously it will fail at some point. The point being made with the analogy is simply to say "To keep the balance in balance you must remove equal weights from each side; likewise, to keep an equation as an equation you must perform the same operations to each side, to do otherwise makes it into an inequality. Going on about the solution set is getting carried away. You would be surprised by how many people get an "ah ha" moment when they are introduced to that simple analogy. Zedshort (talk) 20:14, 4 August 2016 (UTC)

(3)

I deleted the following text:

Each side of an equation is called a member of the equation. Each member will contain one or more terms. The equation, $Ax^{2}+Bx+C=y$ has two members: $Ax^{2}+Bx+C$ and $y$. The left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, and C.

It introduces unusual (to say the least) terminology; I suggest to change "members" to "left-hand side and right-hand side", which is more common. I know of two meanings of "term", explained at term (logic) and Addition#Notation and terminology, but none of them makes sense in the deleted sentence "Each member will contain one or more terms". If the former notion is meant, it should be "Both the lhs and the rhs is a term"; if the latter notion is meant, it should be "Many, but not all, equations contain one or more terms on their lhs and rhs". Finally it suggests that the distinction between vairables and parameters (a notion that wasn't even mentioned before) is obvious, probably from the naming. On the contrary, I think that the equation $Ax^{2}+Bx+C=y$ can be solved wrt. an arbitrary nonempty subset of {x,y,A,B,C}, thus giving rise to 31 mathematical tasks, some of which are more, and some less common. Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

Third source, College Algebra, 3rd ed, 1966, M. Richardson, Prentice-Hall, page 15, "A statment of an equality like a = b is called an equation. The expressions on either side of the equals sign are called the left member and right member of the equation, respectively." Notice that there are only two sides of an equation, left and right, and "member" is used in both cases. Also, same source, page 84, "An equation which becomes a true statment for all allowable values of the variables is called an identical equation or simply an identity. ... An equation which becomes false for some allowable value(s) of the variable(s) is called a conditional equation or simply an equation." (bolding author's) The fact that the author has contracted it to "equation" does not negate the fact that an "identity" is also an equation. Also, from that source, page 34: "A series of numbers separated by + and - signs may be called an algebraic sum. One of the individal numbers separated by the + and - signs, taken together with the preceeding sign, constitutes a term of this algebraic sum. Thus in the expression 2 + 3 - 4 + a - b the terms are +2, +3, -4, +a, -b." Zedshort (talk) 20:21, 4 August 2016 (UTC)

More general, while it may be a good idea to have an example already in the lead, I think it should be kept simple (and understandable to a wide audience). So I'd prefer to use the historical equation "14x+15=71" from the picture, for which the solution x=4 is easily understood. Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

How about both? Zedshort (talk) 19:09, 4 August 2016 (UTC)

BTW @Zedshort: you should take notice of the WP:BRD policy, although I won't insist on it here. - Jochen Burghardt (talk) 17:27, 4 August 2016 (UTC)

Last, but not least, we should be constantly asking ourselves, "For whom am I writing?" If you are a mathematician and are writing to please yourself, you have the wrong idea. These articles should be directed at an audience that are lower down on the scale. That is not to suggest we should reduce it to the fourth year of education level, but try to make the writing explicit, complete, and ideally interesting enough that the reader will feel compelled to continue reading and to possibly follow the links and learn a bit more. One of the best ways to learn how to write is to learn how not to write. To that end I suggest you experience the Simple English Wikipedia, where you will find pedantic, and grotesquely overwritten articles that are over the head of their intended audience: (elementry school level and people who are learning English as a second language. Those articles typically parse at the 12th year of formal education.Zedshort (talk) 19:59, 4 August 2016 (UTC)

I support Jochen's reverts, they were the correct thing to do. Zedshort's proffered sources are mostly a half century old and quite out of date. The only one of them that is only twenty years old actually supported Jochen's position on terminology. Much has changed in the last half century with respect to the teaching of mathematics. Language has been refined and honed because older terminology was not working well and led to confusion among learners. To try to dredge up forgotten terminology and imprecise language does not help our readers understand these simple concepts, especially if they have to turn to the modern literature, which is what they should be doing in the first place as Wikipedia is "not a textbook". --Bill Cherowitzo (talk) 22:15, 4 August 2016 (UTC)wp:

Oh wow, I did not know that facts became old and stale, how insightful. What exactly is a "proffered" source? Is a "proffered" source a bad source? What terminology did I use that is "out of date"? Note that the application of the word "modern" is just a password for contemporary and does not mean, necessaryly, that it is genuinely improved. Please be explicit in your response. Finally, your argument that Wikipedia is not a textbook is just strawman. Zedshort (talk) 03:42, 5 August 2016 (UTC)

Bill Cherowitzo was referring to terminology as outdated, not facts. I agree the article was clearer before your edits. M∧Ŝc²ħεИ_τlk 09:56, 5 August 2016 (UTC)

Of what "facts" do you speak of? I see nothing but terminology and explanations which can and do become old and stale. The word "proffer" means to offer for approval and has no negative connotations. Calling the sides of an equation "members" is clearly out of date, such terminology is just not used by anyone in this century. Your source's definition of "term" does not hold up beyond the simplest of expressions. For instance, is "b" a term in the expression sin(a + b) - 3? While you may disagree with it, Wikipedia not being a textbook is policy (WP:NOTTEXTBOOK). While I applaud your attempt to simplify the text, your sources (or your interpretation of them) do not seem to be up to the task, which needs to be done without introducing errors or confusion. --Bill Cherowitzo (talk) 06:08, 5 August 2016 (UTC)

Take a breath and get some O2, it will help you a lot. Zedshort (talk) 06:49, 5 August 2016 (UTC)

You have been warned about personal attacks before. Be careful. - DVdm (talk) 08:32, 5 August 2016 (UTC)

Zedshort has inserted his comments in the middle of Jochen Burghardt post. This is a kind of vandalism, as it makes difficult to distinguish Jochen's contribution from Zedshort's one, and this destroys the logical structure of Jochen's argumentation. Therefore, I have added Jochen's signature before each Zedshort's comment, and I have increased the indentation of Zedshort's comments. This does no restore completely the logical structure of Jochen's post, but I have not found any better solution (reverting Zedshort's edit is not possible, as others have already answered).

By the way, I agree with Jochen edits. D.Lazard (talk) 09:08, 5 August 2016 (UTC)

I agree with your revert. By the way, see also this revert to Linear_equation. - DVdm (talk) 10:16, 5 August 2016 (UTC)

My main issue is that the distinction I sketched above in (1), Nr.1.-3. should not be blurred. So, ignoring naming issues for the moment, can we agree that distinction makes sense?

If yes, I'd suggest to agree next on a name for each of them, to be used as "main name" in wikipedia articles, while deviating names are to be mentioned in side remarks. Moreover, I think we'd need a name for the general concept that is common to Nr.1.-3; I'll call it "Nr.0." for brevity in the following discussion.

My suggestions for Nr.1., 2., and 3. are "identity equality", "conditional equality", and "equation", respectively (as I already wrote in (1)), and to use "equality" for Nr.0.

@Zedshort: It seems that your 1st source Kells.1964 agrees with my naming suggestion for 1. and 2., but prefers "equation" for 0.; your 2nd source Brooks.1996 also uses "identity" for 1. and "equality" for (I think) 0.; your 3rd source Richardson.1966 uses "identical equation" or "identity" for 1., "conditional equation" for (I think) 2., and again "equation" for 0. From this, I guess your point is to use "equation" for 0., while you may agree with me on Nr.1.-3. - did I get that right?

In this case (provided there is consensus about the above distinction), we'd need to distinguish 3. from 0. by their use (3.="equation as problem task", 0.="just an equation, with no particular use given", 1.="asserted equation, or identity equation"). My point is that this would complicate the article text (we'd need to replace "equation" by "equation to be solved", and then "equality" by "equation"), and would even require renaming of the article (as it is about Nr.3, i.e. solving, except for a brief subsection "Identities" and your recent lead insertions). Ease of presentation is the main reason why I like(d) the terminology of the article (as suggested above for keeping).

To complicate things, we need to delimitate in any case from other articles, viz. equality (mathematics) (a mess) and identity (mathematics) (giving undue emphasis to trigonometrical and exponential laws) about Nr.1., and theory of equations (should better be integrated here as a history section) about Nr.3.

I split off some minor points below into subsubsection, signing each one separately, and hoping that this makes it easier to reply in an appropriate place. - Jochen Burghardt (talk) 10:48, 5 August 2016 (UTC)

I agree with Jochen that we need to clarify the terminology. In fact, it is me who has introduced the clarification,which has been restored by Jochen. However the terminology must not result from our discussion (this would be WP:OR, but it must reflect the current practice of professional mathematicians. I agree that their terminology may differ from that of teachers, but, generally, teachers terminology corresponds to older mathematicians terminology, and teachers terminology follows mathematician terminology after some while (for example set theory terminology was adopted by mathematicians in the beginning of 20th century, and introduced in elementary teaching only about 50 years later). This general trend implies that teachers terminology, when it differs from mathematicians terminology, must appears only as a variant in WP.

I agree also with Jochen's terminology for item 1., 2. and 3. However, "identity" seems, presently, to be used only in elementary mathematics and traditional expressions (for example, trigonometrical identities or Bézout's identity). The reason seems to be that identities are, in fact, equalities in some structure (equality in a polynomial ring or equality of functions).

About the general concept (item 0.). We must distinguish between the "equality operator", which is an operator used for constructing expressions, which may be, depending on the context, an identity, a conditional equality or an equation to be solved, and the property of equality, which means that two different expressions represent the same mathematical objet. D.Lazard (talk) 13:58, 5 August 2016 (UTC)

I was surprised when I first saw in Wikipedia that an identity is not an equation, since I had always heard it referred to as a particular kind of equation. Here are two sources that agree with me:

identity: An equation which states that two expressions are equal for all values of any variables that occur, such as x ..., from The Concise Oxford Dictionary of Mathematics, 4th edition, 2013 http://www.oxfordreference.com/view/10.1093/acref/9780199235940.001.0001/acref-9780199235940-e-1396?rskey=DAadHg&result=1425

A definitional equation sets up an identity between two alternate expressions that have the same meaning. For such an equation, the identity equality sign ≡ ... is often employed.... from Chiang, Alpha C., Fundamental Methods of Mathematical Economics, third edition, 1984, p. 9.

I think the article should be clear that there are two different sets of terminology for how expansive "equation" is. Right now the article contradicts itself by taking both positions—the lead says there are two kinds of equation including identities, while the section "Identities" says An identity is a statement resembling an equation. Loraof (talk) 01:27, 14 August 2016 (UTC)

All sources given so far agree in using "equation" for the general concept (meaning "Nr.0" above, i.e. two expressions joined by "=").
(I found three more references from the fields of automated theorem proving and theoretical computer science with the same usage (Caution: Jargon):

• David A. Duffy (1991). Principles of Automated Theorem Proving. New York: Wiley.
  — "equation" is defined in sect.2.2.3.1, p.32 as "universally closed equality atom", and in sect.7.1.1, p.148 as "equality atom" (from the context, universal closure may be tacitly understood, too)
• N. Dershowitz and J.-P. Jouannaud (1990). "Rewrite Systems". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 243–320.
  — an "equation" is defined in sect.2.2, p.251 as "unordered pair {s,t} of terms", written "s=t", with all occurring variables "understood as universally quantified"; in sect.6, p.279 the universal quantification convention is dropped, since equation solving ("unification") is investigated there
• M. Wirsing (1990). "Algebraic Specification". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 675–788.
  — sect.3.1, p.691 defines a "Σ-equation" to have the for "t=_st’" where Σ is a many-sorted signature, s is a sort from Σ, and t,t’ are terms of sort s; quantification has to be expressed explicitly in this approach

)
Thus I meanwhile think Zedshort was right (apologies!).

As a consequence, I suggest:

• the article should give a short explanation of the common meaning ("Nr.0") and a brief overview of the usages ("Nr.1-3"), preferrable in the lead.
• After that, it should elaborate on usage Nr.1 ("identity", currently sections Analogous illustration, Identities), and mention its generalization, Nr.2, ("conditional equation"). Alternatively, that part could be kept short, and refer for details to the articles Equality (mathematics) and Identity (mathematics); I'd also like to integrate the latter (merely a stub) into the former.
• Finally, it should elaborate on usage Nr.3 ("equation to be solved", currently sections Properties, Algebra, Geometry, Number theory, Differential equations, Types of equations; I suggest to integrate the stub Theory of equations here also). Alternatively, that part could be kept short, too, and refer for details to the article Equation solving; in that case, most of the current sections should be moved to there.

- Jochen Burghardt (talk) 16:41, 22 August 2016 (UTC)

Today, I started editing a new suggestion for the article at User:Jochen Burghardt/sandbox2. Up to now, I only changed the lead, using stuff from de:Gleichung and simple:Equation which both support the above suggestions (while the quality article fr:Équation seems to be restricted to equation solving), and incorporating some sources from this talk page. Comments (best placed on this talk page in a new section, I suppose) and constructive edits in the sandbox are welcome. - Jochen Burghardt (talk) 11:19, 13 September 2016 (UTC)

Balance analogy

As for the balance analogy: I'm not against it, but it needs to be phrased carefully in order not to be misleading. I think it is most useful for notion Nr.1 and 2., but dangerous for Nr.3., as my above squaring example should illustrate (another one is multiplying both sides with zero; this no doubt keeps the balance, but widens the solution set to all numbers). Moreover, the analogy is given (and depicted) in an own section "Analogous illustration" (which should be cleaned up, too, in this respect). - Jochen Burghardt (talk) 10:48, 5 August 2016 (UTC)

Term

@Zedshort: your notion of "term" (sourced from Richardson.1966) appears to agree with that of Addition#Notation and terminology. However, there are certainly equations that don't have a sum on either side, so your sentence "Each member will contain one or more terms", while fitting on your particular example, doesn't hold for all equations, as already Wcherowi (thanks!) pointed out. - Jochen Burghardt (talk) 10:48, 5 August 2016 (UTC)

Second example in lead

As for $Ax^{2}+Bx+C=y$ as a second example in the lead: I'd feel it would make the lead too long, and would better fit into section "Polynomial equations". Maybe, if we can illustrate another important point on this example, it's worth to be in the lead, nevertheless. Maybe, we can combine it with the lead paragraph starting "In geometry, ...", giving e.g. the parametric equation of a circle ($x^{2}+y^{2}=R^{2}$, to be moved up from section "Parameters and unknowns" in this case)? This whould have 2 variables and a parameter, too, and in addition a geometric meaning. - Jochen Burghardt (talk) 10:48, 5 August 2016 (UTC)

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Grouping types of equations

I recently grouped the essentially-identical classes of differential equation, integral equation and integro-differential equation together. The change was reverted as "not an improvement". I think that showing the connection between these nominally different types of equations is indeed an improvement, even if my particular wording was not perfect. I have made my changes again. If anyone wants to revert them, please explain here why that is done. LachlanA (talk) 13:57, 28 May 2019 (UTC)

degree zero?

At: Equation#Types of equations, should the list begin with something like: "constant equation for degree zero", since equations like "y+2=5" are typically the first type of equations that students solve? (And to be more in sync with this article: Degree of a polynomial#Names of polynomials by_degree) DKEdwards (talk) 06:45, 23 January 2022 (UTC)

IMO, the answer is no, since "equation of degree zero" and "constant equation" are rarely used, while "polynomial of degree zero" is common. D.Lazard (talk) 10:45, 23 January 2022 (UTC)

What are they called then? Just "linear"? It seems they determine a point in a one dimensional space rather than a "line", unless we implicitly assume a 2nd dimension. Should there at least be a comment explaining the convention? "linear equation for degree one and zero"? DKEdwards (talk) 08:00, 24 January 2022 (UTC)

An equation that determines a point on a line has the form $x=c,$ and is therefore linear. Typical equations of degree zero would be $3=0$ and $0=0.$ They are so trivial that it is not interesting to name them. In other words, for being interesting, an equation must contain unknowns or variables, and this excludes the degree zero. D.Lazard (talk) 08:55, 24 January 2022 (UTC)