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, p. 136).

In particular, let be the Field of Rational Numbers. Now construct an extension field of constructible numbers by the adjunction of , where is in , but is not, consisting of all numbers of the form , where . Next, construct an extension field of by the adjunction of , defined as the numbers , where , and is a number in for which does not lie in . Continue the process times. Then constructible numbers are precisely those which can be reached by such a sequence of extension fields , where is a measure of the ``complexity'' of the construction (Courant and Robbins 1996).

**References**

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.

© 1996-9

1999-05-26