Pi Approximation Day or Casual Pi Day

Hello dear readers and a Happy Pi Approximation Day (or as some also call it, Casual Pi Day) to all! :-D Now just like Yellow Pig Day I had no prior knowledge of this math holiday. Before Yellow Pig Day the only math holiday that I knew about was Pi Day. Imagine my surprise then upon finding out about another holiday related to Pi Day while perusing the internet for special events in July. 

Excitement and wonder aside there is some explaining to do regarding Pi Approximation Day. First of all, why is it held July 22nd of all days? The answer is clear if you look at its date number 7/22 and take the inverse of that (or just look at it European style). That will give you 22/7 which is a pretty good approximation to pi (better actually than the 3.14 provided by the digits of the well established Pi Day. If you do the division you will see that 22/7 comes out to about 3.142857142857143. Besides this slightly closer approximation to pi I also find the notion of a pi approximation day more appealing than just a pi day since the best we can do with a finite number of digits is approximate the constant. Besides that, I also find the holiday appealing from a computer science perspective. It's natural to wonder how people have tried to approximate pi over the years and how people have been able to generate all of the immense digits of pi that are known today. 

The literature on numerical methods for approximating pi is immense. A quick browse through left me getting excited over topic after topic after topic, and unsure what would be appropriate for this post. After discarding the notion of trying to write a comprehensive post on such methods, I decided to expand on a historical note on the topic (people have been trying to approximate pi since antiquity), which brings me to a name I'm sure many of you have heard: Archimedes. In relation to pi, he managed to produce surprisingly accurate bounds on the constant, approximating it to be greater than 223/71 and less than 22/7. His methods of proof involved a bit of geometry, specifically regular polygons with many sides inscribed within and around circles. 

In much more recent years (relative to Archimedes' time) a very elegant mathematical proof surfaced of the upper bound: pi < 22/7. As Lucas notes, (see reference below), the integral that implies the result was first officially published by Dalzell in 1971, and may have been known a little earlier in the mid 60s by Kurt Mahler. This very special integral is shown below in my paint drawing (sorry for the rather unelegant drawing of the otherwise elegant integral; I don't believe that blogger has special symbols for mathematical formulae and I don't have any special math writing software). 

With a little bit of Calculus 2 knowledge the integral will not be hard to evaluate (though maybe a little time consuming to expand in the numerator). To evaluate you expand the numerator and divide by the denominator. After long division there will be one term at the end that has a (1 + x^2) in the denominator. The thing to remember with that is the antiderivative which is the arctangent function. Evaluating the expression at the upper and lower bounds and taking limits as necessary will give the answer of 22/7 - pi. The answer will be positive because the integrand is positive and the upper integration limit of 1 is greater than the lower limit of 0. Together you get the awesome magical inequality of 0 < 22/7 - pi which implies that pi is less than 22/7. Pretty cool. 

Some readers may wonder where such a magical integral could have come from. Did it just pop out of the sky or is there some deeper mathematical pattern underlying all of this? It turns out to be the latter. The above integral can be generalized to arbitrary powers m and n for x and (1-x). This generalization was recently introduced by Backhouse and can provide many more approximations of the constant pi. The reason this is so is because it will evaluate to ax + b*pi + clog2. The coefficients a, b and c are all rational, a and b have opposite sign, and if it is the case that 2m is congruent to n modulo 4, then c will equal zero (Backhouse proved this) and we once again have a result like ax + b*pi > 0. As m and n increase the results to pi will become finer and finer. Even more so, it turns out that this is not the only such integral that provides approximations of pi. A skimming over the Lucas reference provided below will show another similar integral as well as a table comparing the approximations provided by the two integrals with increasing values of m and n combined.

And that's that my dear readers. It would be great to continue exploring approximation methods to pi, but if you are in more of a casual pi day kind of mood then I recommend snacking on some tasty tarts (especially pecan: my favorite!, they kind of look like/approximate pies too) lay back, listen to some Herb Alpert, and take it easy. :-) 

References: 
1.  N. Backhouse, Note 79.36, Pancake functions and approximations to , Math. Gazette 79 (1995),
371–374.

2. S.K. Lucas, Integral proofs that 355/113 > pi, Gazzette Aust. Math. Soc.
32 (2005), 263–266.