From: israel@math.ubc.ca (Robert Israel) Newsgroups: sci.math Subject: Re: Squaring the circle ?! Date: 29 Dec 1998 00:34:17 GMT Keywords: What does it means to say one cannot square the circle? In article <768ruj\$68g\$1@nnrp1.dejanews.com>, ijalab@hotmail.com writes: |> I didnt quite understand the problem of squaring a circle, and how you go |> about proving it. I do know, however that 'pi' is transcendental , that |> 'pi' cannot be the root of an algebraic equation. So what ? How does it |> follow that you cant 'square a circle ' ? The problem of "squaring the circle" is the following: given a circle, to construct with straightedge and compass a square with the same area as the circle. By analytic geometry, geometric problems can be related to algebraic problems. That is, the coordinates of all points produced in a straightedge-and-compass construction are related by algebraic equations. We can assume that the original circle has centre at the origin and radius 1. Then all further points produced in a straightedge-and-compass construction starting with this circle will have coordinates that are algebraic numbers. But in order to square the circle, you would have to be able to construct a point with coordinates [sqrt(pi),0]. Since, as you noted, pi is transcendental, so is sqrt(pi), and therefore this is impossible. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2