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Definition. A real number r ∈ R is called constructible if there is a finite sequence of compass-and-straightedge constructions that, when performed in order, will always create a point P with at least one coördinate equal to r.

Definition. A real number r ∈ R is called constructible if there is a finite sequence of compass-and-straightedge constructions that, when performed in order, will always create a point P with at least one coördinate equal to r.

One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.) In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems: Doubling the cube.

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.

The Constructible Number Theorem: Every number α that you can con- struct has the following properties: (i) α is an algebraic number. (ii) The degree of the characteristic polynomial of α is a power of 2. Before we prove this, we note a couple of significant corollaries: Corollary 3.2.

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.

One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.) In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems: Doubling the cube.

To determine if a number is constructible, express it in terms of addition, subtraction, multiplication, division, and square roots. If you can, it's constructible. If you can prove it can't be done, it's not constructible. If you can't do either, then you don't know.