Compass and Straightedge = rationals and square roots
Origami = rationals, square roots and cube roots (I think)
How far can we get if we use other tools, like rulers, protractors, pieces of string, etc? Can we cover all the algebraic numbers?
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1If we cannot obtain all algebraic numbers then to prove this we need a formalization of possibilities of tools which we use. – Alex Ravsky Dec 21 '14 at 20:56
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"Other tools" is a fairly vague term. What does a ruler mean, for example? A theoretical ruler would let you mark out any real number length, which would make all algebraics a snap. – Thomas Andrews Dec 21 '14 at 20:56
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Note, the "constructible" question involves proposing theoretical versions of real world objects. It doesn't actually make sense to talk about real-world constructibility, because all constructions there are merely approximations, and we can certainly approximate any algebraic number as close as we want, up to the limits of quantum mechanics. – Thomas Andrews Dec 21 '14 at 21:00
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2One thing we know, from Galois, is that just being able to construct $n$th roots, for all $n$, is not enough to construct all algebraic numbers. – Thomas Andrews Dec 21 '14 at 21:03
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Some closely related questions are still open, e.g. see my remarks here on Heine's radical integers. – Bill Dubuque Dec 21 '14 at 21:40