It's one of the most wonderful, confounding, nerve-racking and sublime mathematical conundrum of all times. In the spin of quantum mechanics, in the accretion of galactic clouds, in a little girl's twirl, when a pebble is dropped on a quiet lake; wherever and whenever there is rotation, it lurks, right at the center. This then, is Pi, the ratio of a cirlce's circumference to its diameter, an infinite series, a transcendental number, one of the uncountable many that is familiar and distant, like a far away galaxy.
Euclid proved every circle has a Pi in it, Archimedes dabbled in it, Ramanujan in his characteristic style snatched out an astonishing formula for it from nowhere, Gauss approximated it, Newton obsessively calculated it, Euler set it into the greatest equation of mathematics (see). Great mathematicians have on many occasions measured themselves against Pi's Himalayan peak of inscrutability. It has relentlessly driven people insane and has attracted insane people relentlessly.
At last count, Professor Yasumasa Kanada at the University of Tokyo has calculated Pi's value to 1.2 trillion digits (that's 550 Gigabytes of storage space, or 2.4 billion pages if you write 500 digits on a page). There are no patterns in the decimal digits of Pi; the brightest and most intuitive of mathematicians could not discern any pattern in the digits over the past few thousand years. Pi's digits occur randomly. If so, we would expect each of the ten digits 0 to 9, to occur one tenth of the time in Pi's decimal part. If we consider the first 800 billion decimal digits, then, 0 must occur 80 billion times, so should 1, 2, and all digits upto 9. Well, behold the mind-boggling statistical truth that shows why our universe is not just queer but queerer than we can imagine.
Digit - Frequency
0 - 79,999,604,459
1 - 79,999,983,991
2 - 80,000,456,638
3 - 79,999,778,661
4 - 80,000,238,690
5 - 79,999,773,551
6 - 79,999,935,320
7 - 79,999,775,965
8 - 80,000,650,170
9 - 79,999,802,555
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Statistically, that's no less than a perfect distribution for mortals like us. Astounding. More astounding is the fact that we still don't have a clue whether Pi is random all the way to the end of time, or if there is a pattern in it somewhere.
Can we know what Pi is? Why is it woven into the fabric of our universe? A layman's bafflement (mine), a mathematician's holy grail.
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Wow. It is pretty amazing that pi is that well distributed. It's also pretty amazing that we've calculated it out 1.2 trillion digits.
I like Pi too. Pi can be used to give you an even break.
Pi is the ideal travel companion for the pessimist. You can take it to any number of dismal places.
It might have be Borges who said that Pi also contains the lost plays of Sophocles, a score of the Art of Fugue as Bach fully intended to finish it,a Greek translation of the Q document with glosses in the original Welsh, the complete genetic code of the LUCA and many other things including this comment. You might think that finding them would be hard but a much bigger problem is distinguishing the real thing from the infinite number of false versions.
Define "pattern."
Mathematicians know that Pi is transcendental. Not only is it not rational (a ratio of integers) but it has been proven to be non-algebraic (not a root of any polynomial with integer coefficients). In particular that means that no sets of repeating digits will ever be found (because there are none).
There are lots of numbers that have the same uniform distribution of their digits in decimal representation - infinite in fact! For example, the square root of 2 will also have a similar property (if you want to write that out 800 billion digits to verify). So will e - the base of the natural logarithm.
paul, my definition of pattern (non-mathematical) is a string of digits that allow abstraction. I am not sure if I understand 'In particular that means that no sets of repeating digits will ever be found (because there are none).'