Talk:Constructible number

In the table at the end, in the row about trisecting an angle, the"counterexample" does not match the "associated set of numbers", i.e. it is not obviously a counterexample. Could one of the authors correct this?

e^(2πi) / 7 is not constructible, because 7 is not a Fermat prime

This can lead to confusion, because its not a necesarely condition that n should be a fermat prime in order to a regular n-gon to be constructible.

A regular 15-gon is constructible, cause cos(2pi/15) is constructible, but 15 is not a fermat prime

You are correct. But 15 is the product of the two Fermat primes 3 and 5. 7 is neither a Fermat prime nor the product of 2^n and one or more Fermat primes (the only known ones being 3, 5, 17, 257 and 65,537). I've edited the main article accordingly. --Glenn L (talk) 05:50, 13 April 2010 (UTC)

Unclear definition

What is a "fixed" coordinate system? Our article on coordinate systems lists many systems. Could we fix the system to bipolar coordinates? Even if we have the standard Cartesian system, how does that impact the rules for what is allowed in construction with Straightjacket & Compassion? And next, what does it mean for a point to be constructible from the axes? It is all very unclear.

Why not simply define a constructible point as a point that can be constructed with S&C, starting from two given distinct points. Then we can define constructible complex numbers as those whose points in the complex plane are constructible when the given starter points are (0,0) and (1,0).

Comments?  --Lambiam 13:03, 31 January 2008 (UTC)

The intro isnt that nice i think. those 2 parts should be merged into one :

old: "A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), the point can be constructed with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge.[1] It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible."

isnt "constructible from the usual x- and y-coordinate axes." in the above text a fuzzy term? Note: The above version somehow differentiates between point and number. is that differentiation improtant/necessary? in my opinion both are tied so close together one can/should be aware of it and then treat both as somehow the same object.

New suggestion to improve article: "A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), the point can be constructed with unruled straightedge and compass. It can be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge.[1] It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible." — Preceding unsigned comment added by 2A02:8108:1A00:3000:94FE:109A:4DE0:2BB2 (talk) 00:19, 27 May 2017 (UTC)

This introduction has some real problems. It seems to me to be a mish-mash of synthetic and coordinate geometry. From the synthetic point of view, one doesn't talk about constructible points, but rather constructible lengths since there is no "origin" (a coordinate concept) to relate a point's position to. On the other hand, if you assume coordinate axes then you already have all real number lengths and every point is construcible. I notice that there are no references for this approach and I am willing to assume that it is all OR and needs to be removed. --Bill Cherowitzo (talk) 04:06, 27 May 2017 (UTC)

See "Clean-up" section below. --Bill Cherowitzo (talk) 18:55, 28 May 2017 (UTC)

Fourth Root of 2

It's possible that you can construct line ABC which AC=2^(1/4)*AB. 220.255.2.63 (talk) 03:48, 30 October 2011 (UTC)

Other Tools other Numbers

The number 2^(1/3) can be constructed if the ruler is used in a certain way:

http://demonstrations.wolfram.com/ConstructingTheCubeRootOfTwo/

Jan Burse (talk) 22:35, 3 October 2016 (UTC)

meaning of notation in geometric part not clear

In the geometric part it says:

"The geometric definition of a constructible point is as follows. First, for any two distinct points P and Q in the plane, let L(P, Q ) denote the unique line through P and Q, and let C (P, Q ) denote the unique circle with center P, passing through Q. (Note that the order of P and Q matters for the circle.) By convention, L(P, P ) = C (P, P ) = {P }. Then a point Z is constructible from E, F, G and H if either"

I dont get this part:

"...By convention, L(P, P ) = C (P, P ) = {P }. Then a point Z is constructible from E, F, G and H if either"

what does L(P, P ) = C (P, P ) = {P } mean? (Ive never seen such notation. Is L(P, P ) a line connecting P to itself?) Where do the points Z,E,F,G and H suddenly come from? — Preceding unsigned comment added by 2A02:8108:1A00:3000:94FE:109A:4DE0:2BB2 (talk) 00:32, 27 May 2017 (UTC)

This is poorly phrased and is probably WP:OR. I'll try to come up with something to replace it with in a few days. --Bill Cherowitzo (talk) 04:10, 27 May 2017 (UTC)

See clean-up section below.--Bill Cherowitzo (talk) 18:55, 28 May 2017 (UTC)

thanks --2A02:8108:1A00:3000:D8B3:7D52:4006:188 (talk) 01:46, 16 June 2017 (UTC)

Clean-up

While attempting a clean-up of this page and the introduction of some references I decided to remove the statements about the complex numbers. The extension to the complexes is trivial once one has a coordinate plane and there does not seem to be any advantage to viewing things this way. Also, I could find no references that supported this viewpoint. If I am wrong, or being too harsh, and someone can provide references, I'll be happy to reinsert that material. --Bill Cherowitzo (talk) 18:51, 27 May 2017 (UTC)

I have found a reference for the complex numbers and it seems, the source for most of the problematic parts of this article. I'll add the reference but will reduce the mention of complexes to an aside. Even the source I have doesn't do anything in the complex realm except describe the field extensions in those terms. He needs to restrict to the reals to be compatible with the rest of the literature, so, in keeping with the larger literature, I will stay with the reals and just mention the extension. --Bill Cherowitzo (talk) 18:55, 28 May 2017 (UTC)

I thinking keeping the bulk of the complex article free of complex numbers might be good idea, there reason why they nevertheless often occur in sources is probably due to some textbooks base their treatment/description of geometry on the complex number plane.--Kmhkmh (talk) 11:38, 1 June 2017 (UTC)

I've actually found a few more references using the complexes. They are all coming from the algebraic side of things, so I am assuming that some results might be easier to prove in algebraically closed fields. I will look at this more closely as I move on to the section on field extensions. I still see no obvious advantage in describing things using the complexes from the geometric side of this topic, but perhaps my viewpoint will change as I continue to flesh out this article. --Bill Cherowitzo (talk) 15:42, 1 June 2017 (UTC)

More illustrations

I've added illustrations on commons (and an assocaited category) for multiplication, inverse and root construction (one identical to the recently added one and one based on the pythagoras/cathetus theorem).--Kmhkmh (talk) 11:18, 1 June 2017 (UTC)

P.S. I added the geometric mean theorem to the existing illustration, that is based on it. It might be also a good idea to mention it in the article's normal text as well as the intercept theorem that can be used for mutliplication and the inverse.--Kmhkmh (talk) 11:22, 1 June 2017 (UTC)

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Kavyansh.Singh (talk) 13:47, 19 November 2021 (UTC)

(

• Comment or view
• Article history

)

• ... that a point in the plane has a straightedge and compass construction if and only if its coordinates have closed-form formulas using only arithmetic and square roots? Source: Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, New York, doi:10.1007/978-1-4612-0629-3, ISBN 0-387-98276-0, MR 1483895, Corollary 2.16, p. 41, footnote [1] of the nominated article. For an explanation of the "E" in Martin's statement of this corollary, see earlier pp. 36-37.
  • ALT2: ... that a number is "constructible" if it can be graphed with only a straight-edge and compass?
  • ALT3: ... that any number that can be created with only simple functions can also be graphed with a straight-edge and compass?

• Reviewed: Teraura

Improved to Good Article status by David Eppstein (talk). Self-nominated at 22:59, 8 November 2021 (UTC).

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

QPQ: Done.

Overall: —valereee (talk) 18:51, 11 November 2021 (UTC)

• Starting review —valereee (talk) 18:51, 11 November 2021 (UTC)

• David Eppstein, hm on the hook interest. Is there anything else about this that might be more accessible to the average reader? Because I've got a master's in a statistical field and I don't even know what these words mean. —valereee (talk) 19:00, 11 November 2021 (UTC)
  • ALT1 ... that an equivalence between algebraic and geometric definitions of constructible numbers helps prove the impossibility of compass-and-straightedge cube-doubling, trisection, and circle-squaring? —David Eppstein (talk) 19:18, 11 November 2021 (UTC)

LOL yeah that's what I thought. :) Okay, I'll go ahead and approve it with a request to the promoter that if to them this seems like it's not 'of general interest' maybe we discuss further? I do understand that this is a topic for which it may be challenging to find a hook that's of general interest, and that DYK shouldn't avoid entire topics because of it. Approving both hook suggestions. —valereee (talk) 19:22, 11 November 2021 (UTC)

Thanks! I would only add that, if math-aversion is so severe that it causes readers not to be able to understand popular literary tropes in the works of Aristophanes, Dante, Gilbert & Sullivan, O. Henry, and Thomas Mann, I can't help those readers by hiding all of the mathematics in a hook about a mathematics article — even if successful in getting those readers to click on the article, that would only cause those readers to have a bad experience once they clicked. —David Eppstein (talk) 20:09, 11 November 2021 (UTC)

• @Valereee and David Eppstein: This is super interesting to me, and I'm going to spend a bunch of time reading about this. I'm less sure about whether this'll be interesting to a broad audience. What I am sure of is that even if this hook were interesting, it wouldn't be hooky to a broad audience—this might make sense to half of its readers if they thought about it, but realistically, a large percentage of readers aren't going to get that far, especially next to all the uncomplicated hooks nearby. I like to pretend that instead of trying to lure in adults, we have the full, undivided attention of a group of worldly ten year olds—but not for very long. theleekycauldron (talk • contribs) (they/them) 08:59, 18 November 2021 (UTC)

• Theleekycauldron, we do include hooks that are generally much more interesting to people who have specialized knowledge. Nearly all of our classical music hooks are probably only interesting to people interested in classical music, and we include one in most sets. If this is a hook that is exceptionally interesting to those who have specialized math knowledge, then I think that's an okay compromise. — Preceding unsigned comment added by valereee (talk • contribs)

• For running so many of them, it's interesting that none of them made it to the october 2021 stats page, which roughly 29% of hooks cleared in October. We shouldn't be optimizing for view counts, obviously, but I digress. At least classical music hooks don't have a high barrier to entry, even if they aren't super broadly interesting. If we just want to cover the basics, I added an ALT2 above to appeal to the lowest common denominator (it's funny that the expression is lowest common denominator but the mathematical concept is greatest common denominator), and an ALT3 that kind of walks a middle ground. I'm going to say that ALTs 0 and 1 are too jargon-y to be sufficiently interesting to a broad audience, and if you don't like ALTs 2 or 3, we'll have to find something else. Or maybe I'm wrong. theleekycauldron (talk • contribs) (they/them) 23:35, 18 November 2021 (UTC)

• Theleekycauldron if you are unfamiliar with the jargon please do not string together likely words in the hope that they might be meaningful. The result of doing so is worse than the correct use of technical terminology: it creates all of the appearance and impenetrability of actual jargon to the uninitiated, adds that "fingernails screeching on blackboard" sensation to those familiar with the terminology, and conveys none of the meaningfulness. Your hooks ALT2 and ALT3 are bad and should not be used. I don't know what you think you mean when you talk about a number being "graphed", or for that matter "created". "Simple functions" is too vague to be useful. And your hooks miss most of the point of the article, and despite doing that fail to be more interesting. As for ALT1: Maybe you missed my earlier comment about Dante, O. Henry, etc, and its link. These authors of what is now classical literature clearly expected that every literate person would understand that circle-squaring is an impossible task in geometry, even if they wouldn't understand exactly what it means. The hook points to why it is impossible. I think the audience of people who have been forced to study some geometry in school is much much broader than the audience of people who care about who conducted some early-20th-century opera performance. —David Eppstein (talk) 01:35, 19 November 2021 (UTC)

• Fair cop, I made a mistake with the phrasing. I wasn't familiar with circle-squaring expression beforehand, although it seems to be common. Maybe it gets rephrased/shortened to ALT1a: ... that an equivalence between algebraic and geometric definitions of constructible numbers helps prove the impossibility of squaring the circle and doubling the cube with a compass and straightedge?

In fact, doubling the cube doesn't have to be in the hook either. Squaring the circle is going to be the one most people are going to pick up on, and it should be rephrased so that what you're riffing on is easily recognizable. theleekycauldron (talk • contribs) (they/them) 02:30, 19 November 2021 (UTC) this was my ten thousandth edit? really? not exactly a high note

also, I don't know what "" means, I'm not your reviewer, unless it just means "get off my back". theleekycauldron (talk • contribs) (they/them) 02:31, 19 November 2021 (UTC)

I wasn't trying to use it to complain about you or valereee at all, so if that's what you got from it, then I apologize for the confusion. I intended it to mean that this discussion appears to have superseded the previous "good to go" icon, and the bot or people who recognize which hooks are good to go and which need another round of review should be given a signal that this one still needs another round of review. If valereee is happy to do that, I'm happy too. As for whether the angle-trisector cranks are going to have hurt feelings by being snubbed in a tighter hook, I suppose it's not really a big problem. And if we're going for concision over precision, we can also drop the compass and straightedge part. So how about:

ALT5 ... that an equivalence between algebraic and geometric definitions of constructible numbers helps prove the impossibility of squaring the circle?

—David Eppstein (talk) 07:25, 19 November 2021 (UTC)

• nice :) i'd sign off on it myself, but that tends to be a big delay for hooks because i'm handling promotions at the moment, and non-geographic hooks are in short supply as well—so i'll ping valereee back here to do the formal honours. theleekycauldron (talk • contribs) (they/them) 10:44, 19 November 2021 (UTC)

Good to go with ALT5. Oh, I like that one much better! That one feels much more inviting! —valereee (talk) 12:54, 19 November 2021 (UTC)

Promoting ALT5 to Prep 6, and this one is interesting! – Kavyansh.Singh (talk) 13:47, 19 November 2021 (UTC)