Wikipedia:Reference desk/Archives/Mathematics/2016 October 18

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October 18

Triple bar: real or hooey?

Our article on, say, triple product contains lots of instances of $\equiv$, and I don't get the point. The cross product of two vectors is straightforward, if annoying, to work out as pure algebra, and if I worked it out as an equation, I'd use a plain vanilla equals sign in that equation. I looked up the triple bar article and Identity (mathematics) and I don't really see anything, outside of specialized contexts, where it has any general significance I can understand. To me the sign is confusing because there have been times when I've sat in a classroom and seen the sign used in the same way as "<-" in a computer program (or "=" in languages that use "==" for equals). So I have to look and see, are they saying these things work out to be equal, or are they defining them to be equal?

Is there a way to defend this usage, to say that yeah, anyone looking at this knows this triple-bar thingy has to be there instead of an = sign, and everyone knows what that signifies? Wnt (talk) 18:01, 18 October 2016 (UTC)

Huh, that looks like simply a mistake in the article. Equal signs all around seem to be warranted. But now, I feel like I may be missing something, as well... Tigraan^{Click here to contact me} 18:09, 18 October 2016 (UTC)

I don't think you all are missing anything. The article used equal signs until April 2016, when someone went about tweaking the equations with no explanation or edit summaries. Per MOS:MATH#NOWE, a simple equal sign is preferred over $\equiv$ or ":=". That an equation serves as a definition should, in my opinion, be explicitly indicated in the prose rather than rely on a possibly obscure symbol. If no one, objects, let's revert to equal signs. --Mark viking (talk) 19:30, 18 October 2016 (UTC)

I'm pretty sure triple product is intending to use $\equiv$ to mean roughly "equal by virtue of explicit prior definition or assumption.", as opposed to e.g. "equal because we did the same thing to both sides". And not necessarily doing anything with any consistency. But I agree with Mark that WP MOS is pretty clear on this, and we should probably revert to all '='. SemanticMantis (talk) 20:16, 18 October 2016 (UTC)

My interpretation was the triple bar was meant as identity as opposed to equality. The distinction is subtle and it's much better to spell out in words what is meant rather than using an obscure symbol. For example if "$\sin ^{2}\theta +\cos ^{2}\theta \equiv 1$" is meant to mean "$\sin ^{2}\theta +\cos ^{2}\theta =1$ for all $\theta$" then just write it out that way instead of confusing half of your readers. In the triple product article the "for all a, b, c" is implied by context so the distinction isn't needed. A bigger question is whether $\equiv$ as identity should be in any WP article other than to mention it as being used by "some authors". --RDBury (talk) 21:57, 18 October 2016 (UTC)

As someone who used to use the triple bar back when I published research, I disagree with the above discussion and the preference stated in the MoS. A double bar means equals, maybe for some value of a variable or maybe for all values of parameters and variables. A triple bar means more specifically the latter, and so should be preferred on grounds of clarity in my opinion. And $f(a,b)\equiv g(a,b)$ is more succinct than, and hence preferable to, $f(a,b)=g(a,b)$ for all a and b. Loraof (talk) 23:39, 18 October 2016 (UTC)

This distinction continues to evade me. Do we write a = a + 0 or do we write a $\equiv$ a + 0 or a = a + 0 "for all a"? I mean, the cross product is just a shorthand for some algebra, $\mathbf {u} \times \mathbf {v} =(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} +(u_{3}v_{1}-u_{1}v_{3})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k}$. At least it's a double bar in cross product where I stole this math code from. Or is that a triple bar when we say it is equal ... or identical... to something else? Wnt (talk) 23:52, 18 October 2016 (UTC)

Essentially, the triple bar is used for equations that are always true, whereas the double bar is used for those that are not always true (and you can solve for the values that make it true). So you would write sin θ = 1 with the double bar (it's not always true, but you can easily find solutions, like θ = π/2); but sin² θ + cos² θ ≡ 1 needs the triple bar because it doesn't matter what θ is, the equality always holds. Double sharp (talk) 02:58, 19 October 2016 (UTC)

So E=mc^2 needs to be "E$\equiv$mc²? I feel like when working out math problems we use quite a bit of "identity", and I'm loath to give up my ascii equals sign for all of it. As much as it is used in triple product, by this standard there are many other sections and instances (like under "Proof") where the triple bar still needs to be put in to replace the double. Wnt (talk) 11:45, 19 October 2016 (UTC)

There probably are situations where it is helpful for the reader if the text tries to distinguish identities (equations that are true for all values) from definitions and from constraints (equations we want to be true and where we want to find the specific values that make them true). However, I don't think the triple bar is widespread enough to be used without explaination, and I fear it might confuse the reader rather than help him. (For what it's worth, in situations where it is important to distinguish identities from constraints I have seen an equals sign with an exclamation point above used to denote constraints, like $\sin \theta {\stackrel {!}{=}}0$. Similarly, sometimes a question mark above an equals sign is used to denote identities we want to show to be true. I have only seen this usage on blackboards, for what it's worth.) Tea2min (talk) 12:16, 19 October 2016 (UTC)

The double-bar/triple-bar distinction is only useful when it is important to distinguish constraints from identities. When that is not the issue at hand, it is often not used, because it is then not serving any useful purpose. Oh, and I would also confirm what Tea2min has said about ? and ! modifying equals signs on blackboards. Double sharp (talk) 13:14, 19 October 2016 (UTC)

If the idea is to use the triple bar for every expression that evaluates to true no matter what values the variables around it have (i.e. as a shorthand for "forall..."), it looks pretty ridiculous to me. I could be convinced it is helpful as a shorthand for "analytic equality", i.e. an equality that is deduced purely from logical axioms and the form of the variables of the LHS and RHS (type but not value in a programming language). But it would take a lot of convincing, a better definition, and anyways, it is the MOS that needs convincing (good luck with that). Tigraan^{Click here to contact me} 13:22, 19 October 2016 (UTC)

The article references that I can access use just an ordinary equal sign. Does anyone object if we put the article back to that format (with a mention in the text that the identity is true for all values if anyone would possibly read it differently)? Dbfirs 15:52, 19 October 2016 (UTC)

I think article should match MOS. While I understand the complaints against the content of MOS, I think it is still best to follow something consistently. As it stands, we have three (or more) notions what that symbol means. Any given (good) math text that uses the symbol can simply define it at first usage, but that is not how we are supposed to approach math articles on WP. SemanticMantis (talk) 18:14, 19 October 2016 (UTC)

Methinks we should move that whole conversation to the article TP, and give page watchers a few days to react (or do the change now, but be prepared for a possible on-sight revert; we cannot really claim a meaningful consensus from here). Tigraan^{Click here to contact me} 16:41, 19 October 2016 (UTC)

I suspect that it would be only Dangvugiang who would disagree, but we don't want to start an edit war. I've copied this content to the talk page of the article in the hope that we get further opinion. If Dangvugiang doesn't respond, would it be permissible to revert his change to "equiv" back to plain "equals"? Dbfirs 07:29, 21 October 2016 (UTC)

If you scan that user's contributions, 100% of them are to article space, and there is not a single edit summary (not counting the auto-generated ones for section headings and undo's). So I wouldn't hold my breath waiting for him/her to engage on the question. Leave a note on the talk page (and on the user's talk page too; won't help probably but it's worth a try), and if there's no response, after a decent interval (which in my opinion can be pretty short; I don't have a lot of patience for this sort of editor), change it back. --Trovatore (talk) 07:45, 21 October 2016 (UTC)

Thanks. Yes, I see what you mean. I've left a courtesy note on his talk page so that all opinions can be taken into account. Dbfirs 08:27, 21 October 2016 (UTC)

I've seen this often enough on Wikipedia that I don't think we should make this about one article or one editor. There's a broader issue here and I've seen people here weigh in with points toward both sides. My preference remains for simple equals without philosophy but I'm not declaring a jihad over it. Wnt (talk) 11:09, 21 October 2016 (UTC)

I agree that the issue is much wider, but the question was raised here about a relatively recent change to the article mentioned above. Our article at Identity (mathematics) uses a plain "equals", not "equiv". Dbfirs 12:05, 21 October 2016 (UTC)

equation of circle

how to find equation of circle which touches two parallel lines 3x-4y=7and 3x-4y+43 and center lie on the line 2x-3y+13=0.SonishaShrestha (talk) 23:31, 18 October 2016 (UTC)

Heh, that will get ya thinking. The first thing I'd look at is the perpendiculars. You know which way to go from the line to the center of the circle -- exactly perpendicular to it. Per the first equation, y = 3/4 x + something. So the three lines you specify have slope 3/4, 3/4, and 2/3. Oh, they're parallel lines! Well, that makes it easy - you know the circle is on a line in the middle, i.e. average 7 and 43. Now you just have to see where that line intersects the other line, solve the two equations simultaneously, and you're done.

I'll stop there because the Refdesk has a sort of spoiler policy about homework, but it wouldn't take much to get someone to finish up if you really are confused, but by now you should be done. Wnt (talk) 23:57, 18 October 2016 (UTC)

If they weren't parallel lines, I think you'd take the negative inverse of the slope to get a perpendicular, i.e. -4/3. Then you can say that the circle is centered on the third line and the radius extends outward in the two different directions by equal amounts ... ? ... PROFIT! Maybe I have to think that over again. Meanwhile, the triangle article explains an inscribed circle is on each of the bisectors of the angles, but this seems to invoke more trigonometry (arctangent or something) than I'd prefer. Wnt (talk) 00:07, 19 October 2016 (UTC)

Actually, it is rather easy to prove that a circle tangent to two non-parallel lines has its center on the angle bisector of the angle formed by the lines (without trigonometry). This simplifies the algebraic brute-forcing of the problem a lot. (As a general rule, geometry problem can always be solved by calculus means, but it is much more dirty than geometric solutions.) Tigraan^{Click here to contact me} 10:50, 19 October 2016 (UTC)

Using a free tool like Geogebra might help your geometric intuition, find thesolution via an (interactive) construction, which you can use to verify solution you've found by algebraic computation.--Kmhkmh (talk) 00:09, 19 October 2016 (UTC)

Regarding the case with three non-parallel lines: I made a crappy graphic showing various equal angles and right angles and lines and such, then realized it was all irrelevant. The way to find the circle in the arbitrary case is to take the two bounding lines and solve simultaneously to get the intersection point. Then use the second formula in distance from a point to a line to get the x and y values for the nearest point on the third line (which the circle is centered on). Then use the first formula in distance from a point to a line, i.e. the distance from the point to the line, with either of the first two lines, to find the radius of the circle. Not a single sine or cosine or god forbid arctangent in sight. Wnt (talk) 12:02, 19 October 2016 (UTC)

• By "3x-4y+43" do you mean 3x-4y=43 or 3x-4y+43=0? It's fairly obvious that the center lies on the line 3x-4y=a, where a is 25 in one case and -18 in the other. So you have two simultaneous linear equations determining the center. —Tamfang (talk) 21:19, 19 October 2016 (UTC)

The + and = are on the same button, so I didn't think that was a hard one to guess. :) Wnt (talk) 21:23, 19 October 2016 (UTC)