Squaring the circle: Encyclopedia II - Squaring the circle - Transcendence of π

Squaring the circle - Transcendence of π

A solution of the problem of squaring the circle by straightedge and compass demands construction of the number , and the impossibility of this undertaking follows from the fact that π (pi) is a transcendental number—that is, it is non-algebraic and therefore a non-constructible number. The transcendence of π was proved by Ferdinand von Lindemann in 1882. If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π, which is impossible.

It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations.

Bending the rules by allowing an infinite number of ruler-and-compass constructions or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can in Gauss-Bolyai-Lobachevsky space.

Adapted from the Wikipedia article "Transcendence of π", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki