Small Numbers (243)

Feynman (1997, p. 116) noticed the curious fact that the decimal expansion

(1)

repeats pairs of the digits 0, 1, 2, 3, ... separated by the digits 4, 5, 6, 7, .... Just after this point, the pattern breaks, since the fraction is given exactly by the repeating decimal

(2)

This pattern is related to the fact that

(3)

and

(4)

REFERENCES:

Feynman, R. P. 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, 1997.