Pi Day 2024 –A Really Big PI-related near integer

Pi Day 2024 –A Really Big PI-related near integer

There is a really good Youtube channel done by a math prof that I follow on Youtube, Richard E. Borcherds. I will have a little more to say about him later, but first I will discuss the Pi-related content in one of his videos.

https://www.youtube.com/watch?v=a9k_QmZbwX8

I won’t attempt to get into any high-level explanation, as that would involve very advanced abstract algebra, which is far above the pay grade of a lowly retired statistician. That said, I did take one university-level abstract algebra course long ago and can confidently say that this type of math is very abstract indeed. Stats degrees are usually heavy on calculus, linear algebra and probability, not so much on abstract algebra. Arguably though, these subjects are all inter-related, though most of us can only learn so much math in one lifetime.

 

Anyway, the screenshot above shows a couple of very big numbers:

• the exact integer: 262537412640768000.

• the near integer: 262537412640768143.99999999925...

As you can see, the second number is quite close to the first number, up until the last 3 digits before the decimal point. They differ by almost 144.

And here is the Pi Day angle: that number is e raised to the power Pi times the square root of 163, and it nearly comes out to an integer, with the first 12 places after the decimal point being 9s. So, that’s a transcendental number taken to the power of another transcendental number multiplied by an irrational number. Given that these are numbers that are non-repeating and non-ending, that seems pretty weird.

The professor notes that a normal pocket calculator will actually show an integer after doing the calculation.

Even better, a Windows-based spreadsheet shows the result to 20 places past the decimal, as an integer.

And here is what the stats language R does. It just rounds the number, long before getting to the decimal. There are some R packages that handle really big numbers, but I couldn’t get any of them to work, so we will have to leave it at that.

Professor Borcherds recommends a particular online calculator, called free42, which emulates a particular high-level HP calculator on the PC. It looks interesting.

Apparently Martin Gardner (who wrote a recreational math column for Scientific American back when it was good) claimed that this was an integer, though Ramanujan had an accurate calculation of this number around a century ago, showing that it was not an integer.

 

The professor goes on to show a few more numbers involving e to the Pi times the square root of some other numbers, that end in a remarkable run of 9s, though not so spectacular as the first one.

You are probably now getting the feeling that something is going on here that is much more than a coincidence. And you would be right.

From this point, the explanation gets into elliptic modular function theory and other highly complicated areas of math. 

 

You can see that this all involves mathematics with complex numbers, which is where the factor of Pi sneaks in (note the presence of e to the 2 Pi in some of those formulas).

The video then segues into a number of areas, touching on some more of Ramanujan’s work, as well as Euler’s. These involves number theory (primes), as well as group theory. Eventually, the aptly named “Monster Group” makes an appearance. This object exists in 198663 dimensions, and has order 10 to the 53.  Here's a SF comic book that makes use of the concept (though with some slight errors).

This all seems rather esoteric, but these matters do come up in “sort-of” practical endeavors such as particular physics (group theory) and cosmology (string theory in 26 dimensions).

It turns out that Richard Borcherds actually was a co-winner of the Fields Medal, based on work related to the monster group. I didn’t know that when I stumbled over his Youtube channel and watched a bunch of videos on complex analysis and group theory. Then I picked up a book that I have about the Monster (just a recreational math book) and there he was! For those who don’t know, the Fields Medal is similar in status to the Nobel prizes, maybe more so, as the Fields are only handed out once every four years.

The professor’s videos are quite compelling and often rather oddly amusing. So, I would definitely recommend his channel if you are interested in math, even if the videos are probably something of a stretch for most of us.

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So, now that you have dutifully read some Pi Day math, you should read a science fiction book, or even better, a whole series. Book 1 of the Witches’ Stones series even includes a reference to pi.:

Kati of Terra

How about trying Kati of Terra, the 3-novel story of a feisty young Earth woman, making her way in that big, bad, beautiful universe out there.

So, go on a romantic adventure, a sort of spaceship and planet-side road trip, while escaping from dangerous slavers, with our gal Kati of Terra and her companion Mikal (and, of course her ever helpful but cantankerous Granda node):

 

http://www.amazon.com/gp/product/B00811WVXO

http://www.amazon.co.uk/gp/product/B00811WVXO

The Witches’ Stones

Or, you might prefer, the trilogy of the Witches’ Stones (they’re psychic aliens, not actual witches), which follows the interactions of a future Earth confederation, an opposing galactic power, and the Witches of Kordea. It features Sarah Mackenzie, another feisty young Earth woman).

 

https://www.amazon.com/dp/B008PNIRP4

https://www.amazon.co.uk/dp/B008PNIRP4

The Magnetic Anomaly: A Science Fiction Story

“A geophysical crew went into the Canadian north. There were some regrettable accidents among a few ex-military who had become geophysical contractors after their service in the forces. A young man and young woman went temporarily mad from the stress of seeing that. They imagined things, terrible things. But both are known to have vivid imaginations; we have childhood records to verify that. It was all very sad. That’s the official story.”

A short story of about 6000 works.

https://www.amazon.com/dp/B0176H22B4

https://www.amazon.co.uk/dp/B0176H22B4

The Zoo Hypothesis or The News of the World: A Science Fiction Story

In the field known as Astrobiology, there is a research program called SETI, The Search for Extraterrestrial Intelligence. At the heart of SETI, there is a mystery known as The Great Silence, or The Fermi Paradox, named after the famous physicist Enrico Fermi. Essentially, he asked “If they exist, where are they?”.

Some quite cogent arguments maintain that if there was extraterrestrial intelligence, they should have visited the Earth by now. This story, a bit tongue in cheek, gives a fictional account of one explanation for The Great Silence, known as The Zoo Hypothesis. Are we a protected species, in a Cosmic Zoo? If so, how did this come about? Read on, for one possible solution to The Fermi Paradox.

The short story is about 6300 words, or about half an hour at typical reading speeds.

https://www.amazon.com/dp/B076RR1PGD

https://www.amazon.co.uk/dp/B076RR1PGD