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Constructible Number


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.

All rational numbers are constructible, and all constructible numbers are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic equation with rational coefficients has no rational root, then none of its roots is constructible (Courant and Robbins 1996, p. 136).

In particular, let F_0 be the field of rationals. Now construct an extension field F_1 of constructible numbers by the adjunction of sqrt(k_0), where k_0 is in F_0, but sqrt(k_0) is not, consisting of all numbers of the form a_0+b_0sqrt(k_0), where a_0,b_0 in F_0. Next, construct an extension field F_2 of F_1 by the adjunction of sqrt(k_1), defined as the numbers a_1+b_1sqrt(k_1), where a_1,b_1 in F_1, and k_1 is a number in F_1 for which sqrt(k_1) does not lie in F_1. Continue the process n times. Then constructible numbers are precisely those which can be reached by such a sequence of extension fields F_n, where n is a measure of the "complexity" of the construction (Courant and Robbins 1996).


See also

Algebraic Number, Compass, Constructible Polygon, Euclidean Number, Euclidean Tools, Rational Number, Straightedge

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References

Bold, B. "Achievement of the Ancient Greeks" and "An Analytic Criterion for Contractibility." Chs. 1-2 in Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 1-17, 1982.Courant, R. and Robbins, H. "Constructible Numbers and Number Fields." §3.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 127-134, 1996.

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Constructible Number

Cite this as:

Weisstein, Eric W. "Constructible Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConstructibleNumber.html

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