Pi: Food For After Thoughts
I know...Pi-Day has already passed. But, we can start getting ready for next year.
First some trivia. Have you seen the movie Pi? Did you know that the number written in the opening sequence of the film is not Pi beyond the first 8 decimals? Now, you can impress (or depress) your friends!
Mathematicians have proven that Pi is irrational and transcendental. That is, Pi cannot be written as the ratio of two natural numbers, nor can it be the root-solution to a polynomial equation with rational coefiicients.
But, two big questions emerge:
If you want to learn more about Pi being random or normal (and their connections), see Paul Press's article Are the Digits of Pi Random? (Berkeley Lab Research News, 2001)
and Stan Wagon's article Is Pi Normal? (Mathematical Intelligencer, 1985). Both of these topics are important relative to attempts to use Pi as a random-number generator.
- Is Pi random? That is, in our constant search to produce extended digits of Pi (now beyond 5 trillion places!), will a pattern ever be found? For example, the number 0.101001000100001000001.... is irrational, and probably transcendental as well. However, a definite pattern is evident.
- Is Pi normal? That is, in its base-ten representation, do all ten digits occur "equally often"? For example, the number 0.123456789011223344556677889900111222... is an irrational normal number, and probably transcendental as well.
Now, for some fun with the digits of Pi. Mitacs, a Canadian not-for-profit research organization, invites you to play with its Pi Explorer engine. You can enter either number sequences or a letter sequence, and the program will tell you where those inputs will first occur in Pi's decimal sequence. For example, my zip code "98229" and the word "MATH" occur in the 21,709st decimal place and 584,891st decimal place, respectively.
Finally, for a delightful excurcion, browse Samuel Arbesman's article
A Random Walk with Pi (WIRED, June 2012). He uses the digits of Pi to perform "random" walks. As shown below, an interesting picture of Pi emerges...that should be on the front of t-shirts! NOTE: Color designates time, with blue being digit-walks earlier in the sequence and other colros further along in the spectrum being digit-walks later in Pi's sequence.