Talk:Constructible number

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 Field:  Number theory

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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)