Constructible number

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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 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.

The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the smallest field extension of the rational numbers which is closed under square root and complex conjugation. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack.

Contents

Geometric definitions

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

Since the order of E, F, G, and H in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, Z is constructible from E, F, G and H if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by E, F, G, and H, in the above sense.

Now, let A and A′ be any two distinct fixed points in the plane. A point Z is constructible if either

Put simply, Z is constructible if it is either A or A′, or if it is obtainable from a finite sequence of points starting with A and A′, where each new point is constructible from previous points in the sequence.

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