Everyone knows that —the ratio of any circle’s diameter to its circumference—is irrational, that is, cannot be written as a fraction . This also means that ‘s decimal expansion goes on forever and never repeats …but have you ever seen a proof of this fact, or did you just take it on faith?
The irrationality of was first proved (according to modern standards of rigor) in 1768 by Lambert, but his proof was rather complicated. A more elementary proof, using only basic calculus, was given in 1947 by Ivan Niven. You can read his original paper here, but it’s rather terse! Just as I did for Calkin and Wilf’s paper, Recounting the Rationals, I plan to write a series of posts explaining Niven’s proof in a bit more detail, with some accompanying intuition. I’ll assume a basic knowledge of calculus; if you don’t know calculus, just hang tight for a few posts!
Here’s the basic outline of the proof. We begin by supposing that is rational: in particular, suppose for some integers and . We’ll then use these values of and to define a special function , about which we will show the following:
is an integer, AND
But this is absurd! There are no integers greater than zero and less than one. The inescapable conclusion will be that our initial assumption—that —was false.
In my next post, we’ll define the special function and begin exploring some of its properties.