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. Constructibe 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").