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").