Constructible Numbers

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GenePeer
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Constructible Numbers

Post by GenePeer » Fri Apr 16, 2010 1:17 am

I've been doing some research on solving cubic equations with compass and straight-edge. Found it's impossible but it can be done using neusis i.e. straightedge with marks on it. I learnt about constructible numbers. The idea is a real number n is constructible if a segment of length n can be constructed by a finite number of steps. the solution of a cubic equation requires constructing an angle trisection which is generally not part of constructible numbers.
What i want to know is the new limit that neusis brings. does pi become constructible when neusis is allowed like the cbrt(2)?
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uws8505
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Re: Constructible Numbers

Post by uws8505 » Fri Apr 16, 2010 6:24 pm

Pi cannot be in the expanded group because it is a transcendental number, not an algebraic number (which means it cannot be a solution of a polynomial equation with rational coefficients).

Maybe marked edge can allow all algebraic numbers?
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GenePeer
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Re: Constructible Numbers

Post by GenePeer » Sat Apr 17, 2010 1:33 am

not all algebraic numbers either. on my second day of research i came up with quite a lot. With the normal straight-edge(unmarked) and compass, the following "arithmetic equivalent" constructions could be made: addition, subtraction, multiplication, division, and square root. That was it. and constructible numbers were numbers that could be calculated by a finite number of these operations on the number one i.e. sqrt(3) = sqrt(1+1+1), 12 = (1+1+1+1)x(1+1+1)... or to make it simpler constructible numbers are numbers that could be calculated by a finite number of these operations on rational numbers. these were minor and useless examples but i hope you get the point.

Now from what i've gathered, neusis (using a marked ruler) only adds one more construction to the list, cube root. so we can't say it will create all algebraic numbers because polynomials of degree >4 will still need functions more complex than cube root and neusis construction can't help.

When i checked the cubic formula, i realised that these operations are used: addition, subtraction, division, multiplication, square root AND cube root. this is why normal construction will not be able to solve cubic equations. but since neusis can find cube roots as well it will be able to solve cubic equation and quartic equations too(they also have the same operations as cubic equations but just longer).

PS: I'm in my last year of high-school doing the IB(International Baccalaureate) program. It requires us to write an extended essay on a subject & topic of our choice. I'm writing it on this :D
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Re: Constructible Numbers

Post by euler » Tue Apr 20, 2010 9:23 pm

You may already have plenty of references for your project, but if you haven't seen a proof already then I wrote a short article some time ago which proves why "any geometrical construction involving straight-edge and compass that can be shown to be equivalent to a cubic equation containing rational coefficients and irrational roots is impossible."
http://mathschallenge.net/index.php?sec ... structions

If you click "FAQ" (on the navigation bar) then you will be find proofs for the other impossible constructions which make use of this result. Also there are articles which show how you can perform the arithmetic operations, find square roots, and solve quadratic equations using compass and straight-edge. In the article which proves that trisecting an angle cannot be done with compass and straight-edge in the note I explain how Archimedes was able to trisect an angle with a slight adaptation.
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