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