Definable number
Definable real numbers are those that can be uniquely specified by a description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge.
Different notions of description give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers.
One way of specifying a real number uses geometric techniques. A real number r is a constructible number if there is a method to construct a line segment of length r using a compass and straightedge, beginning with a fixed line segment of length 1.
Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.
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