Monday, March 14, 2016

Pi Day: Musings on a Constant

Happy pi day, Physics fans! This 3/14 is a particularly special one, as 3/14/16 is as close as we're going to get to the actual value of pi using our current date format (at least for the next hundred years!) While some here at PhysicsCentral are adherents of the Tau philosophy, there's no reason we can't celebrate again on 6/28! In my book, any excuse to eat pie or pontificate a bit on the nature of transcendental numbers is a good one.


Image Credit: Wiki user MGTom, CC BY-SA

Assuming that the digits of pi are pseudo-randomly distributed after the decimal place, it's theoretically possible to construct any finite series of digits from a subset of pi. If the probability of any given digit being your "target digit" is 1/10, then the probability of any given string of digits being your "target string" is (1/10) * (1/10) * (1/10)... with another power of .1 for each digit in your target string. This means that, in order to find a random 3-number string of your choosing, you should have to go about 1000 digits into pi. But that's not that much; we already know pi to some few trillion decimal places.

Pi is a transcendental number, which is a fancy word meaning that it's infinite, and it never starts over. So if it's infinitely long, and there's no discernable pattern, then every number and every possible combination of numbers is in there somewhere.

If you use a numerical-to-alphabetical encoding, assigning whatever values you please to whatever letters you want, somewhere in pi is the answer to any question you can ask, along with all the works of Shakespeare written in English, then again in piglatin. It's probably buried waaaaay way down there in the 10^10^10^10th digits, but they're there, just hanging out. It's actually mathematically guaranteed, sort of the Pythagorean equivalent of the "ten thousand monkeys on typewriters for ten thousand years" argument, only the random monkey-gibberish is already there, encoded into the basic framework of our universe's geometry as a result of our base-10 number system.

Go ahead and try it—use your browser's "find" function to search, and you'll likely be able to find any nine-digit string you like in the first billion digits of pi! Search your own phone number, and you'll likely find a few instances—with ten billion digits, you could likely find it with the area code included!

In addition to being everybody's favorite math-related holiday, today also happens to mark the first day of the American Physical Society's annual March meting, so you can expect the PC crew to be pretty busy these next few days; we're providing physics outreach resources to the ~10,000 physicists who have invaded Baltimore for the week!

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