Irrational But Well Rounded!

3.141592653589793238462643383279502884197169399375105820974944592307816406286 ….digits after digits arrayed effortlessly in his mind as this student recited the first two hundred and fifty five digits of π to the precision of a computing Machine as the rest of the school watched with a gasp! Whether the rest of them can memorize first five digits or hundred, they get the idea- this mysterious number can go on for ever, never ending, never repeating. This particular student won the π- digit memorization contest that our school proudly conducts every year as part of the π -day celebrations!

Each year, we celebrate π – day a little differently. In the past, we ate apple pie, doodled with circles, played bingo with problems that involves pi or just decorated the room representing the non terminating feature of this number. This year, it falls on a Saturday-and since we are racing to make up school days we missed due to snow, I was not planning to do anything special in my class. But students were quick to remind me today that Saturday is March 14th and wanted to celebrate it! They welcome any distractions wholeheartedly!

In a rare coincidence, it is also Albert Einstein’s birthday. π – Day is hence a tribute to the great scientist! As a high school teacher, I am also aware of a group of students anxiously waiting for this auspicious moment when MIT releases their irrational but well thought out college admission decisions every year. There is a pang of anxiety surrounding this date with certain section of the senior class here in Boston and around the country.

Coming back to the irrational nature of this number, the very existence of it as the ratio of the most symmetric geometric figure's circumference to its diameter has intrigued human mind from early on. How did this number come to exist? Who used it first? What is the context? Answers can be found in this great book: ” pi- the biography of world's most mysterious number”. Rereading this book after few years is my past time this weekend

Written for the general audience without much tedious mathematical processes, this book is an attempt to understand π and its beautiful aspects. The book is divided into six parts- what is π, history of π, calculating value of π, π enthusiasts, π Curiosities, applications of π and π paradox. I was fascinated by how the number evolved into today's form.

The first chapter explains in detail one of the greatest challenges the ancient mathematicians faced – trying to measure a circular figure in terms of a straight line.  The problem they experienced was that the circular arcs and straight line could not find a common measure; there were always something left over when trying to compare these types of measurements. The various approximation of the value of in ancient times was the attempt to find out what is left out from a circular measure to the straight line measure.

One of such early attempts was from Egyptians who tried to make a square with the same area as that of a circle. In Rhind Papyrus, a mathematical handbook from early Egyptians, there is reference to this construction which has fascinated the mathematicians for generations until finally they came up with a proof that it is an impossible construction. If we begin with a circle of radius d/2 and a square with side length 8d/9, we can see that Egyptians were reasonably close to the approximated value of. Babylonians approximated the value of to 3.125 in an attempt to find the ratio of the perimeter of a regular hexagon with the circumference of its inscribed circle.  It was Archimedes who is credited with calculating first most accurately. The symbol π, which is the sixteenth letter in Greek alphabets, was used in 1706 for the first time to represent the ratio of the circumference to the diameter of the circle by an English Mathematician.

The book also deliberates π’s appearance in unexpected places, like in Euler’s identity where it join its distant cousin e to form one of the most beautiful mathematical equations (e^(iπ)+ 1 = 0) or in harmonic series or where, at times it almost appears in places like that of square root of 10 or cube root of 30 or square root of 2+ square root of 3 (the values of these expressions being close to that of π). They are fun to read and experience the wonder where things that are seemingly unrelated fit as pieces of jigsaw puzzle.

Now, there are supercomputers that can approximate the value of π to millions and millions of decimal places. While just reciting or memorizing the digits in this number to some random decimal places don’t mean much in the long run, it is a chance for young children to come face to face with what a non terminating non repeating decimal number might look like and to appreciate the beauty of mathematics in their own ways! Just like Egyptians and Babylonians thousands of years ago, this mysterious number, its disguise in a well round figure with an irrational behavior, will continue to fascinate the mathematicians to come!