Feynman point
The Feynman point is a sequence of six times the number 9 from the 762nd decimal place of the decimal representation of pi .
history
It is named after the physicist Richard Feynman , who allegedly once said in a lecture that he wanted to memorize the passages of pi up to this point so that he could jokingly say "nine nine nine nine nine and so on" when he recited , which would imply that pi was a rational number . It is unclear, however, whether this idea really came from Feynman: It does not appear in any of his books or biographies, and the story is unknown to Feynman's biographer James Gleick . The earliest known mention of the idea of reciting pi up to the six nines is found in Douglas Hofstadter's book Metamagicum (1985), in which Hofstadter writes:
“As a crazy student I once learned 380 digits of π by heart. In my unsatisfied ambition I wanted to reach that point - in the decimal expansion the 762nd place - where it goes on to 999999, so that I could have recited the π constant aloud until I got to those six nines and then with a mischievous one To be able to stop 'and so on'. "
More statistics
The (so far unproven) conjecture was made that Pi is a normal number . The probability that a given six-digit sequence of numbers will appear so early in the decimal notation for a given randomly chosen normal number is usually only 0.08% (or, more precisely, 0.0762%). However, if the sequence is allowed to overlap (such as 123123 or 999999), the probability is lower. The chance of six consecutive nines at this early point in time is about 10% less, or 0.0686%. The probability that any number will be repeated six times in the first 762 digits is ten times as great, 0.686%.
One might, however, ask the question, why are we talking about repeating six digits ? One would also have the probability of finding the same number three times in the first three digits, or four times the same in the first ten digits, five times the same in the first 100 digits, and so on. Each of these possibilities has a probability of about 1%. So if you look at the repetitions up to length 12, there is about a 10 percent chance of finding something as extraordinary as the Feynman point. From this point of view, the fact that we actually find a repetition of some digits at this point is not surprising.
The next sequence of six identical consecutive digits again consists of nines and starts at position 193.034. The next sequence of six following identical digits begins with the number 8 at position 222.299 and the number 0 is repeated six times from position 1.699.927. A sequence of nine sixes (666666666) appears at position 45,681,781. and a string of 9 nines is at position 590,331,982, the next at 640,787,382.
The Feynman point is also the first occurrence of four or five consecutive identical digits in pi. The next position at which four identical digits follow each other is the digit 7 at position 1.589.
The positions in Pi where the number 9 appears for the first time 1 to 9 times in a row are 5, 44, 762, 762, 762, 762, 1,722,776, 36,356,642 and 564,665,206; Follow A048940 in OEIS .
See also
Web links
- Eric W. Weisstein : Feynman Point . In: MathWorld (English).
Individual evidence
- ^ A b D. Wells: The Penguin Dictionary of Curious and Interesting Numbers . Penguin Books, Middlesex, England 1986, ISBN 0-14-026149-4 , p. 51.
- ↑ a b c J. Arndt, C. Haenel: Pi - Unleashed . Springer, Berlin 2001, ISBN 3-540-66572-2 , p. 3.
- ^ David Brooks: Wikipedia turns 15 on Friday (citation needed) . In: Concord Monitor , January 12, 2016. Retrieved February 10, 2016.
- ↑ Rudy Rucker : Douglass Hofstadter's Pi in the Sky . In: The Washington Post , May 5, 1985. Retrieved January 4, 2016.
- ^ Douglas Hofstadter: Metamagicum . Deutscher Taschenbuch Verlag, Munich 1994, ISBN 3-608-93089-2 , p. 133 .
- ↑ a b Pi Search.
- ^ One billion digits of pi. Calculated with editpad lite 7. On: stuff.mit.edu.