Pi Day Once Again

 [Apologies, readers; this post belongs in the—mostly—nonmusical blog I could be Totally Wrong, but.]

Well, it's π day in the USA, and though we wish it's an international feast, it really isn't!

I, and I'm sure many other mathematicians, sneer at this celebration, but I'm thinking: who am I to spoil the fun of so many mathematician wannabes?  Let them eat π, to paraphrase Marie Antoinette!

A few bits of trivia about the fabled mathematical constant:

1.  Though it's commonly thought of as 3.14, one of the cardinal properties of the number is that it could not possibly be represented by a decimal number that stops.  Cannot be done.  However, you can represent it as accurately as you want, but it will never be exact.  It can't be written as any fraction, either.

2.  HOWEVER: Archimedes had discovered an excellent approximation to the number Pi, namely 22/7.  If you've got a circle of radius 10 inches, and if you want to know what its circumference is, we know that, in the abstract, it will be 10 inches × Pi × 2.  This will be perfectly accurate.  But since we cannot represent Pi exactly, we can only find this circumference approximately.  (This means not exactly, but closely—in fact as closely as desired.)  If you want an estimate to as close as 1/1000th of an inch, we need to use about 6 decimal places of the value of Pi.  (It's  been 10 years since I've done this sort of thing, so I might be off by a couple of decimal places!)  So basically, what approximate value of Pi you must use depends on how close you want your calculation to be.  You can easily Google Pi, and compare it with 22/7, and you'll find that they agree to more than 5 decimal places. 

3.  But guess what.  It was known by Eastern mathematicians (Indians, Persians, Egyptians, Chinese, etcetera) that 355/113 was even a better approximation to Pi!  The miracle of these two approximations to Pi is how close they come using such small numbers!  The next fraction that comes even closer, is a fraction of two enormous integers. Google sends us to a website that gives 100798/32085 ~ 3.14159264,correct to 8 decimal places. But see how huge the numbers in the fraction are?