3
$\begingroup$

I've drawn two random points on a two dimensional plane. If the distance between these two points is one unit, can a line of π units be drawn using a compass and straight edge?

Or, is there a proof that shows this is impossible?

  • 1
    This is called "squaring the circle" and is one of three impossible problems that have been around from the times of the Ancient Greeks. The best you can do is construct an approximation to $\pi$: have a look at http://en.wikipedia.org/wiki/Squaring_the_circle.2012-03-17

1 Answers 1

5

No (to your first question). A constructible number must be algebraic. However $\pi$ is not an algebraic number ($\pi$ is a transcendental number). Also of interest with regards to this is the famous "squaring the circle" problem.

  • 0
    @DavidMitra: Thanks, I didn't know that. (For future reference, [Wikipedia has$a$construction](http://en.wikipedia.org/wiki/Square_root#Geometric_construction_of_the_square_root).)2012-03-17