<<

Life of

Perhaps the most well-known irrational , or even most well-known constant, pi, has had quite a history from its very inception to the current date. People have been striving to calculate this number for about 4000 years, starting with the Egyptians and Babylonians. So what is this mysterious number? Well, in its simplest form it can be described as the between a ’s diameter and circumference. While this may not sound like a tough number to crack down and calculate, it has proven to be the very opposite. To this day, mathematicians and supercomputers have been trying to calculate pi, be it by , formulas, or even patterns in the non-, all to no avail.

While pi has been approximated and is most often used as 3.14, the challenge of finding an exact number or pattern inside of the ludicrously long decimal remains in tact.

Around 1650 BC, an Egyptian scribe named Ahmes wrote on piece of papyrus that if you “Cut off 1/9 of a diameter and construct a upon the remainder; this has the same as the circle”.1

When put into an algebraic context, this essentially implies that pi = 4*(8/9)*2 = 3.16049, which was quite accurate for how far back the claim dated. The Chinese on the other end however, continued using what they thought was an accurate value of pi = 3, even hundreds of years after Ahmes’ claim.

While the value of pi = 3 is probably the most used and well-known of the time, there was a much lesser-known approximation found in the Hebrew language. This is very interesting, because in Hebrew, each letter equates to a certain number, and the word’s value is calculated by getting the sum of its letters. So in the same place that the approximation of pi = 3 was made popular, 1

Kings 7:23 of the Old Testament, the word “line” is written as Kuf Vov Heh although Heh is not pronounced and does not need to be there. Including that extra letter, the word has a value of 111 and without it the value would be 106. The fascinating part comes from the ratio of pi to 3 being very close to the ratio of 111 to 106, or pi/3 = 111/106 roughly, bringing pi to equate into 3.1415094.2 1 David Blatner’s “The Joy of Pi”, 8 2 Boaz Tsaban and David Garber’s “On the Rabbinical Approximation of pi”, 78 Unfortunately, this approximation was more of a hidden gem in the rough compared to the more popularized value of pi = 3.

Now, think of an equilateral triangle, where the interior angles are all 60 degrees each. This means that their exterior angles are 120 degrees, and three exterior angles makes this 360 degrees.

When looking at a pentagon, five exterior angles at 72 degrees each makes 360 degrees, and a hexagon has exterior angles of 60 degrees with six of those making 360 degrees again. The point of all of this is to show that as the number of sides of a polygon increase, each exterior angle decreases, and their sum always equates to 360 degrees; a circle. So if you could take an n-polygon a near infinite number of sides, always being circumscribed by the same circle, you would get a polygon closely resembling the shape of that circle. This method was crucial to methods that some mathematicians were using to try and find pi, such as of Syracuse and Liu Hui of China. Another Chinese mathematician,

Zhu Chongzhi had taken Liu Hui’s algorithm and applying it to a 12288-polygon came up with pi measured accurately to 7 digits, and 22/7 being a rough approximation of pi. This value proved to be the most accurate approximation for the next 900 years. 3

Those 900 years were going by and very little progress was being made on finding what pi actually was, until the end of the 16th century when French lawyer and amateur-mathematician

Francleois Viete came into the picture. With yet another application of Archimedes’ method, Viete finished with 393,216 side, with a final result of 3.1415926535 < pi < 3.1415926537, which is quite accurate. Perhaps more known for the fact that he was the first man in history to define pi using an infinite product.4 While not entirely useful, this innovation was what opened the minds of many in the 3 Lecture 15, 96-97 4 Petr Beckmann’s “The History of Pi”, 92 future to go about in a similar way, leading to many more accurate findings in the future. 1647 brought in William Oughtred, who introduced the “x” symbol for multiplication among abbreviations for and cosine functions, and “/” to denote the ratio. This was not entirely accepted until 1737 when

Leonhard Euler began using it in conjunction with the symbol pi, in which it frantically became the norm.5 Then, by the year 1750, the number pi had been computed to over 100 digits, been expressed by infinite , and has been given its symbol that is even used in present day. Irrationality was coming to fruition, with the first arguable proof of pi’s irrationality coming from Swiss mathematician Johann

Lambert in 1761 which most argued was not rigorous enough to be accepted. This inspired Adrien

Marie Legendre to come up with a more solid proof in 1794, that was accepted by everyone, this is the same person that went on to prove that the of two is also irrational.6

Once again, the next hundred years brought nothing substantial in the context of pi aside from just finding more and more digits. 1882 did bring forth a proof of the transcendence of pi by , laying rest to the uncertainty about since pi is not a solution of any algebraic .7 This was rather significant because there was finally an answer in stone that the circle could not be squared. When the 20th century came about, computers were starting to get powerful enough to the point of taking over the burden of calculation, allowing people to exceed previous records to get what were once seen as incomprehensible results. D. F. Ferguson discovered the error in

William Shanks’ calculation from the 528th digit onward in 1945, presenting his results two years later after an entire year of calculations which resulted in 808 digits of pi.8 Not even two years later, the

1000 digit mark was hit, and in 1949 the speed at which calculations could be done was improved significantly by the Electronic Numerical Integrator and Computer, punching out 2037 digits in just 70 hours.

5 Florian Cajoris’ “A ”, 158 6 Berggren’s “Pi: A Source Book”, 297 7 Berggren’s “Pi: A Source Book”, 407 8 Berggren’s “Pi: A Source Book”, 406 At this point, the number of digits found was getting larger and larger with the time gap between the discoveries getting smaller due to the progress of computers. Today, the current record is set at 13.3 trillion using a program called the y-cruncher by Alexander Yee with a supercomputer.

While the number of digits beyond 39 decimal places is largely irrelevant, people still continue to push the power of these supercomputers to find these digits of pi hoping for an end. Mathematicians will continue looking for patterns or of as some subset of the decimal of pi, but the mystery remains. They want to be able to distinguish the digits of pi from other numbers, much how you can tell someone’s writing from their style, and as persistent as they are this still has yet to be seen.

Pi has had a bit of a journey from its beginning, and the end does not seem to be near so maybe more significant discoveries will come from pi throughout what remains. As of now, though, those of relevance to pi seem to have been found, and now these mathematicians are just trying to reach the end of a long and mysterious number.