March 10th, 2009 - Post # 3.14159265358979323846264338327950288419716939937510... A Piece of Pi

dot. dot. dot. Or as Kurt Vonnegut would say, "And so on."
A handy phrase to have around. A recent post had me waxing philosophically (re: babbling) about numbers and British idioms, which of course, reminds me of a funny story.
Once upon a time I travelled to Sweden to install some photographic equipment. (and this reminds me of another story, making notes here...)
I was staying at a hotel in the city of Malmo. Most of the younger generation in Sweden speak English pretty well, especially those in positions dealing with a lot of travellers, and my hotel receptionist was no exception. But every once in a while she'd stumble on a phrase or verb form or whatever, as you'd expect any second-language (or for her, probably fourth or fifth) speaker would.
It was fun and reassuring to have someone to speak your home tongue with (especially when it's the only language you speak), and extremely helpful when you are in a foreign country and don't speak the language, understand signs, or order food properly. 
I did take German in Jr. High and High School however, and Swedish is a Germanic language, but then again, so is Lowlands Scots. So that only goes so far, you can recognize the roots of some words and puzzle a few things out, but just as easily get yourself into all sorts of trouble and embarrassment thinking you know what something means when it doesn't.
One day I asked the receptionist where a certain shop was, and if it was hard to get there from the hotel. She said, "No, no, it's right down the road. It's as you Americans say, 'Piece of Peanuts'!"
Well, it's actually the Americans that use the idiom, "Peanuts" for describing something insignificant or easy, and the Brits often say, "Piece of cake" in the same way, sometimes in sort of a back-handed fashion, for something insurmountably difficult. Since they meant basically the same thing, my receptionist neatly combined the terms.
Americans also have been known to say, "Piece 'o pie" in the same fashion, to describe a bagatelle or triviality.
My pun on "Piece of Pi" though, is really quite a stretch if you know anything about "pi" the value. Though "Pi" the number may sound like it would be a simple thing, it's anything but trivial. And, since March 14th is "World Pi Day," I thought you might need a little prepping.

The word Pi itself they say, was first used in 1706 by William Jones, but since he didn't have a very famous sounding name, Swiss mathematician Leonhart Euler (and no, he didn't invent the ruler) popularized it by using it in Latin in 1748. 
Pi was derived (in a grammatical sense, not mathematical) from Hebrew. Literally, it means, "little mouth." Heh, as in round. It's an abbreviation of the Greek peripherei meaning "periphery" and represented by the Greek letter "π." 
Cool stuff, eh, wat...? But wait, there's more. And more, and more, and more. And so on.
While the value of Pi (π) has been currently computed to more than a trillion (about 1012) digits, elementary applications; such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. 
For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the Earth with a precision of a millimeter, (that is, if the Earth were actually round, as if) and one truncated to 39 decimal places is "sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom." (!) Is about all I can muster to say to that.

Beyond all that trivia, Pi is an amazing value. Behold:

Pi is:
  • An "irrational number." Meaning a "real" number (and don't even get me started on "imaginary numbers") that cannot be expressed as a ratio between two integers*.
    *An integer, in case you've forgotten, is a member of the set of positive whole numbers {1, 2, 3, . . . }, negative whole numbers {-1, -2, -3, . . . }, and zero {0}. You know, your basic numbers 'n stuff.

  • A "transcendental number." Which just plain sounds cool and means it's "a real number that is not the solution of any single-variable polynomial equation whose coefficients are all integers." Oh, momma.  Let's just skip down to the rest of the bullet points:
  • All transcendental numbers are irrational numbers. Okay.

  • But the converse is not true; there are some irrational numbers that are not transcendental. (Grind, crunch. Okay. I think.)

  • The case of Pi has historical significance. The fact that Pi is transcendental means that it is impossible to draw to perfection, using a compass and straightedge and following the ancient Greek rules for geometric constructions, a square with the same area as a given circle. This ancient puzzle, known as squaring the circle, was, for centuries, one of the most baffling challenges in geometry. Schemes have been devised that provide amazingly close approximations to squaring the circle. But in theoretical mathematics (unlike physics and engineering - which I now know why I like A LOT better), approximations are never good enough; a solution, scheme, or method is either valid, or else it is not. (or snot, as my dad would say)

  • It can be difficult, and perhaps impossible (hah! blasphemer!), to determine whether or not a certain irrational number is transcendental. Some numbers defy classification (algebraic, irrational, or transcendental) to this day. (don't I know it!)
    Two examples are the product of Pi and e (we'll call this quantity "P pie") and the sum of Pi and e* (We'll call this "S pie." For some illogical reason). 
    *BTW, "e" is the base of the natural logarithm (just trust me on this one, and we won't even go there)
    It has been proved that Pi and e are both transcendental. It has also been shown that at least one of the two quantities P pie and S pie are transcendental. But as of this writing, no one has rigorously proven that P pie is transcendental, and no one has rigorously proved that S pie is transcendental. Gah. Bashes head to desk.

  • Pi's decimal expansion never ends and does not repeat. Well, as far as we know... (To me, THAT is cool, he said geekily.)

  • We do know that the Egyptians and the Babylonians knew about the existence of the constant ratio Pi, although they didn't know its value nearly as well as we do today.
    (Not bad though, considering it was 2000 BC) 
    They had figured out that it was a little bigger than 3; the Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians had a somewhat worse approximation of 4*(8/9)^2 (about 3.160484). Why Egypt made it, and Babylon crumbled, is probably the subject of another blog post. But not on my blog.

  • This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Stupid computers.
Now the thing that this begot in my thinking was, "which came first, numbers or words...?"
I mean when you think about it, numbers may have had more significance than words in the development of language and the psyche. Like when an animal becomes self-aware, now it is "one." What was it before? When Johnny Weissmuller uttered the famous words, "Me Tarzan, you Jane," what did Jane think? Geez, I thought I was one, now you are telling me you are one AND I am one? What does that make both of us? And um, I understood there would be no math...
So I don't know, maybe language sprang up to describe numbers. Too bad I hated math in school. I hated it so much I would write in any number in math homework with pages of long division. Maybe it was my "visual-learning" rebelling against this nauseating "conceptual-learning." These numbers are so boring to look at! Ahhhh! And they never end!!!! Ahhhh! And so on.

Which brings me to my last piece of Pi:
Memorizing digits, or piphilology.

Recent decades have seen a surge in the record number of digits of Pi to be memorized.
Long before computers had calculated π, memorizing a record number of digits became an obsession for some (very sick) people.
In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places. This, however, has yet to be verified by Guinness World Records.
The Guinness-recognized record for remembered digits of π to be recited is (drum roll please...) 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China. (Rim-shot, high-hat)
It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error. I wonder if he said at the press conference, "That was the best day of my life."

There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit.
Here is an example of a piem:
"How I need a drink, alcoholic in nature after the heavy lectures involving quantum mechanics." Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. 
The "Cadaeic Cadenza" contains the first 3834 digits of π in this manner. (And I thought Facebook was a vapid waste of time...)
Piems are related to the entire field of humorous (I didn't write that) yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology.
In other languages there are similar methods of memorization. Or lack thereof.
However, this method proves inefficient for large memorizations of Pi. (I think in my case it would prove inefficient for even small quantities.) Other methods include remembering patterns in the numbers. (Wikipedia leaves off... somewhat cryptically.)
Holy Smokin' Moses!  I have to go stick my head into a snowbank now. The reactor core is going critical....

"Life is good for only two things, discovering mathematics and teaching mathematics"--Siméon Poisson

"Ahhhh!" -T.
(Also known as πB)