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A Celebration of Mathematics and Pi

The Special Properties of Pi-Day 2020

by Keith Greiner

“Pi Day”, March 14, at3:00 pm (3/14/15), is a good time to take a moment to reflect on the importance of mathematics in our lives, and our ability to understand the world. That is especially true for March 14, 2020 at 3:00 pm, as I’ll describe below, where some simple math reveals a special property of about the 2020 pie day. Mathematics gives us all the ability to represent and understand things that otherwise are too complex or too abstract to process. Formulas that usethe constant, “pi” are paramount among those that reveal things that otherwise could never be understood. This essay is about mathematics, and pi, and pi-day: March 14, at 3:00 pm. The date and time are relevant because, as the numbers 3 14 15 [for 3:00 pm, the 15^th hour], the numbers represent the first five digits of pi; the ratio the radius to the circumference of a circle.

Written as the Greek character of the same name, π is an amazing number. Along with the radius, it is essential to finding the diameter, given the radius, or vis. versa. It is necessary everywhere, and is used to find characteristics of both physical objects (ie: the volume of a container) and invisible things like the electricity in this computer. Whoever built the water tower and the water pipes in your community had to use pi to understand the total volume of water contained in those objects. The constant received its name in 1706 when William Jones proposed the Greek name and letter after the Greek word for circumference (περιφέρεια). Leonhard Euler popularized it, beginning in 1737. In modern parlance, he gave pi a “brand”. Now, here is the surprise for 2020. The number of years between 1706 and 2020 is 314: the same as 10 * 3.14. Those 314 years are only a small portion of the thousands of years the number has been contemplated, studied, estimated, and calculated again and again. So, like many things in modern mathematics, pi is deeply embedded in the history of humanity.

Lets look at a few numbers. The number 4 is rational because it can be written as a ratio of 4/1. Similarly the number 0.333333 is rational because it can be written as the ratio 1/3. Pi, however is an irrational number because a simple ratio does not accurately describe it. When it is calculated, it never repeats, and never ends. That is irrational. Now another meaning of the word “irrational” means “not logical” and in many ways the irrational, not a ratio, number is also irrational, not logical, when compared to other numbers. As in irrational, under both definitions, there has been and likely will always be a never-ending puzzle of how one might calculate the never-ending value of pi. Hence the never-ending quest to calculate ever more digits. On March 14, 2019, Google announced that one of its employees calculated more than 31.415 trillion digits of pi, in that effort, using a program called y-cruncher.

Now it will probably be a while before you can run a program on you laptop that will calculate pi to all 31 trillion digits in any reasonable amount of time. So in the mean time, there are a lot of other things about pi that we can explore. Here are a few:

1. How do the hundreds of formulae for pi translate into computer programs, and what are the differences in output from those programs? Some of those calculations are provided in my essay at https://sites.google.com/site/calculatepiincpp/.

2. Given that many of the programs produce slightly different sequences of pi digits, is there one series of “correct” numbers, and if so, are all the others incorrect?

3. Given that today’s computer systems are limited to a 16-digit floating-point numerical system, how does one break through that barrier to explore every digit of truly big numbers?

4. How might the designers of computers be persuaded to incorporate floating point systems that would better allow calculations capable of 31 trillion irrational number digits?

5. Are the numbers of pi random, or is there some kind of obscure pattern among them?

6. Does Benford’s Law apply to pi? See this link for information about Benford's Law.

7. What statistical test would one apply to test the randomness of the list of digits?

To address item 5 on the list, take a look at the table shown below. It is a table of the distribution of 999,999 digits of pi from a publicly available list. If the numbers were truly random, there should be an equal count of numbers for each digit. When a random process is used to create numbers, some variation is allowed, and as more numbers are drawn, the distribution becomes equal. So a truly random distribution would have 99,999.9 numbers for each digit. Given the following distribution, what is your sense of randomness for the almost-million digits of pi?