A sequence of six consecutive nines occurs in the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
representation of the number
pi (), starting at the 762nd decimal place.
[.] It has become famous because of the
mathematical coincidence
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
For example, there is a near-equality close to the round number 1000 between powers ...
, and because of the idea that one could memorize the digits of up to that point, and then suggest that is
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
. The earliest known mention of this idea occurs in
Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange ...
's 1985 book ''
Metamagical Themas'', where Hofstadter states
This sequence of six nines is colloquially known as the "Feynman point", after physicist
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
, who allegedly stated this same idea in a lecture.
[.] However it is not clear when, or even if, Feynman ever made such a statement. It is not mentioned in his
memoir
A memoir (; , ) is any nonfiction narrative writing based on the author's personal memories. The assertions made in the work are thus understood to be factual. While memoir has historically been defined as a subcategory of biography or autob ...
s and unknown to his biographer
James Gleick.
Related statistics
is
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d, but not known, to be a
normal number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to ...
. For a normal number sampled uniformly at random, the probability of a specific sequence of six digits occurring this early in the decimal representation is about 0.08%.
[
The early string of six 9s is also the first occurrence of four and five consecutive identical digits. The next sequence of six consecutive identical digits is again composed of 9s, starting at position 193,034.][ The next distinct sequence of six consecutive identical digits after that starts with the digit 8 at position 222,299.][
]
Decimal expansion
The first 1,001 digits of (1,000 decimal places), showing consecutive runs of three or more digits including the consecutive six 9's underlined, are as follows:
See also
* 0.999...
* 9 (number)
9 (nine) is the natural number following and preceding .
Evolution of the Hindu–Arabic digit
Circa 300 BC, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bo ...
* Mathematical coincidence
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
For example, there is a near-equality close to the round number 1000 between powers ...
* Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".
Ex ...
* Ramanujan's constant
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, the ring of algebraic integers of \Q\left ...
References
External links
"Feynman Point"
��''MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
'' article
{{Douglas Hofstadter
Pi
Recreational mathematics
Richard Feynman