Constructible number

A real number r is said to be constructible if a line segment of length r units can be constructed with a compass and unruled straightedge, given a line segment one unit long. Constructible numbers comprise a field.

The nonconstructibility of certain numbers proves the impossibility of certain problems attempted by the philosophers of ancient Greece. For example, the cube root of a constructible number is generally not constructible (hence the impossibility of "duplicating the cube"), the sine of (1/3)arcsin(x) is in general not constructible for arbitrary constructible x (hence the impossibility of "trisecting the angle"), nor is the constant pi (hence the impossibility of "squaring the circle").