Computing Pi

Nowadays, many people take the concept for granted, that they ‘know the meaning’ of certain Mathematical functions and constants, but that if they ever need a numerical equivalent, they can just tap on an actual calculator, or on a computer program that acts as a calculator, to obtain the correct result.

I am one such person, and an example of a Mathematical constant would be (π).

But, in the 1970s it was considered to be a major breakthrough, that Scientists were able to compute (π) to a million decimal places, using a computer.

And so the question sometimes bounces around my head, of what the simplest method might be, to compute it, even though I possess software which can do so, non-transparently to me. This is my concept, of how to do so:

http://dirkmittler.homeip.net/ComputingPi.pdf

http://dirkmittler.homeip.net/Pi_to_5000.pdf

(Update 08/28/2018 : )

The second link above points to a document, the textual contents of which were created simply, using the program ‘Yacas’. I instructed this program to print (π) to 5000 decimal places. Yet, if the reader was ever to count how many decimal places have been printed, he or she would find there are significantly more than 5000. The reason this happens is the fact that when ‘Yacas’ is so instructed, the first 5000 decimal places will be accurate, but will be followed by an uncertain number of decimal places, which are assumed to be inaccurate. This behavior can be thought related, to the fact that numerical precision, is not the same thing as numerical accuracy.

In fact, the answer pointed to in the second link above is accurate to 5000 decimal places, but printed with precision exceeding that number of decimal places.

Dirk

 

How Accuracy-Of-Prediction may not be Adequate Proof.

What modern people may be tempted to think, is that if a hypothesis exists, which predicts real-world events with great precision, this hypothesis must also be a true statement, of how the real world operates. And I’d like to point out a Historic example, where this was not the case. That example, was the Ptolemaic system of Astronomy.

This system of Astronomy taught, since thousands of years, that the stars in the nighttime sky are attached to rotating shells, which are somehow pinned to the Earth’s axis, as well as being hinged to each other. Thereby, certain stars now known as fixed stars, would exist as belonging to one such shell, while other stars which are now known as planets of our own Solar System, exist as belonging to an additional shell, which is rotating around the first shell, I described, along an additional axis.

What modern people may not see, is that this system was able to predict with extreme precision, where each planet or star was going to appear, at any point in time, which in turn was seen as adequate proof, that the system was ‘true’.

In fact, when Nikolaus Copernicus suggested that the planets, including the Earth, in fact orbit our Sun, his model had a weakness: The Copernican hypothesis suggested circular orbits. And what this did, was to produce inaccuracies, with respect to the observable positions of each planet, over which the Ptolemaic system was far more accurate.

This weakness was only corrected by Johannes Kepler, who discovered that the orbits of the planets are elliptical. Making these orbits elliptical, made them consistent both with Newtonian Physics and Newtonian Gravity, as well as with the observable positions of planets in the nighttime sky.

And so a precedence exists, where a system of ideas was presented, which was extremely accurate in its numerical predictions, but which was ultimately false. I strongly suspect that today’s notions of Quantum Mechanics are a repeat of that.

Dirk