## Constructible Numbers

Shape and space, angle and circle properties, ...
GenePeer
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### Constructible Numbers

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)? uws8505
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### Re: Constructible Numbers

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?
Math and Programming are complements
GenePeer
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Joined: Sat Apr 03, 2010 1:14 pm
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### Re: Constructible Numbers

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  euler 