Six nines in pi

The first 1000 digits of π contain ample double consecutive digits (marked yellow), and a few triples (marked green). The presence of the sextuple (marked red) in such a small sample is an intriguing anomaly. For comparison, 2π (τ) contains a run of 7 nines (marked purple) near the same point.

A sequence of six 9s occurs in the decimal representation of π, starting at the 762nd decimal place.[1] It has become famous because of the mathematical coincidence and because of the idea that one could memorize the digits of π up to that point, recite them and end with "nine nine nine nine nine nine and so on", which seems to suggest that π is rational. The earliest known mention of this idea occurs in Douglas Hofstadter's 1985 book Metamagical Themas, where Hofstadter states[2][3]

I myself once learned 380 digits of π, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9's, and then impishly say, "and so on!"
— Douglas Hofstadter, Metamagical Themas

The sequence of six nines is sometimes called "Feynman point" after physicist Richard Feynman, who has also been claimed to have stated this same idea in a lecture. [4] It is not clear when, or even if, Feynman made such a statement, however; it is not mentioned in published biographies or in his autobiographies and unknown to his biographer, James Gleick.[5]

Related statistics

π is conjectured to be, but not known to be, a normal number. For a randomly chosen normal number, the probability of a specific sequence of six digits occurring this early in the decimal representation is usually only about 0.08%.[4] However, if the sequence can overlap itself (such as 123123 or 999999) then the probability is less. The probability of six 9s in a row this early is about 10% less, or 0.0686%. But the probability of a repetition of any digit six times starting in the first 762 digits is ten times greater, or 0.686%.

One could ask the question, though, "Why talk about a repetition of six digits?" We could have had a repetition of a digit three times in the first three digits, or four times starting in the first ten digits, or five times in the first 100 digits, and so on. Each of these has about a 1% chance. So if we look at repeats up to length 12, there is about a 10% chance of finding something as surprising as the six nines. From this point of view, the fact that we really do find a repeat of several digits after 762 digits is not really very surprising.

The next sequence of six consecutive identical digits is again composed of 9s, starting at position 193,034.[4] The next distinct sequence of six consecutive identical digits starts with the digit 8 at position 222,299, and the digit 0 repeats six consecutive times starting at position 1,699,927. A string of nine 6s (666666666) occurs at position 45,681,781[6] and a string of nine 9s occurs at position 590,331,982 and the next one at 640,787,382.[7]

The early string of six 9s 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.[6]

The positions of the first occurrences of 9, alone and in strings of 2, 3, 4, 5, 6, 7, 8, and 9 consecutive 9s, are 5; 44; 762; 762; 762; 762; 1,722,776; 36,356,642; and 564,665,206; respectively (sequence A048940 in the OEIS).[1]

Decimal expansion

The first 1001 digits of π (1000 decimal digits), including the consecutive six 9s underlined and coloured red, are as follows:[8]

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 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

See also

References

  1. 1 2 Wells, D. (1986), The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, p. 51, ISBN 0-14-026149-4.
  2. ↑ Hofstadter, Douglas (1985). Metamagical Themas. Basic Books. ISBN 0-465-04566-9.
  3. ↑ Rucker, Rudy (May 5, 1985). "Douglass Hofstadter's Pi in the Sky". The Washington Post. Retrieved 4 January 2016.
  4. 1 2 3 Arndt, J. & Haenel, C. (2001), Pi — Unleashed, Berlin: Springer, p. 3, ISBN 3-540-66572-2.
  5. ↑ David Brooks (12 January 2016). "Wikipedia turns 15 on Friday (citation needed)". Concord Monitor. Retrieved 10 February 2016.
  6. 1 2 Pi Search
  7. ↑ calculated with editpad lite 7
  8. ↑ The Digits of Pi — First ten thousand

External links

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