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The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of π. It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip "nine nine nine nine nine nine and so on", suggesting, in a tongue-in-cheek manner, that π is rational.[1][2]

Related statistics

π is conjectured, but not known, to be a normal number. For a randomly chosen normal number, the probability of any chosen number sequence of six digits (including 6 of a number, 658020, or the like) occurring this early in the decimal representation is only 0.08%.[1]

The next sequence of six consecutive identical digits is again composed of 9s, starting at position 193,034.[1] The next distinct sequence of six consecutive identical digits starts with the digit 8 at position 222,299. Of the remaining digits, 0 is the last to repeat 6 times consecutively, starting at position 1,699,927.[3]

The Feynman point is also the first occurrence of four and five consecutive identical digits. The next appearance of four consecutive identical digits is of the digit 7 at position 1,589.[4]

The positions of the first occurrences of 9, alone and in strings of 2, 3, ..., 9 consecutive 9s, are 5; 44; 762; 762; 762; 762; 1,722,776; 36,356,642; and 564,665,206; respectively (sequence A048940 in OEIS).[2]

The number 2π (sometimes referred to by the Greek letter τ (tau)) has a corresponding sequence of seven consecutive 9s beginning at the 761st decimal. This can be proved or explained by multiplying the 761st to 768th digits of pi by 2 since τ = 2π, or 2 x 49999998, which is equal to 99999996, therefore tau has 7 nines in a row starting from the 761st digit. By contrast, the first appearance of seven consecutive copies of any digit in π is 3333333 at position 710,100.
Full decimal expansion

The digits of π up to three digits beyond the Feynman point are as follows:[5]
3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837
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Piphilology
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Ramanujan's constant
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References

^ a b c Arndt, J. & Haenel, C. (2001), Pi — Unleashed, Berlin: Springer, p. 3, ISBN 3-540-66572-2.
^ a b Wells, D. (1986), The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, p. 51, ISBN 0-14-026149-4.
^ Pi Search
^ See, for example, the online Pi Search.
^ The Digits of Pi — First ten thousand