1/9998 Produces Binary Output

The pattern will break down once you get past 8192, which is 2^13. That means that the pattern continues for an impressive 52 significant figures (well, it actually breaks down on the 52nd digit, which will be a 3 instead of a 2). The reason it works is that 9998 = 10^4 - 2. You can expand as   1 / (10^n - 2) = 1/10^n * 1/(1 - 2/10^n) = 1/10^n * (1 2/10^n 2^2 /10^2n 2^3 /10^3n ...) which gives the observed pattern. It breaks down when 2^k has more than n digits, which happens approximately when   2^k > 10^n => k > n log(10) / log(2) which comes out to 4 * log(10)/log(2) = 13.28 when n = 4. --- Another pattern can be generated from the power series expansion   x / (1 - x)^2 = x 2x^2 3x^3 4x^4 ... setting x = 1/10^n gives the infinite series   1/10^n 2/10^2n 3/10^3n ... which leads to the neat fact that   1 / 998001 = 0.000 001 002 003 004 005 006 007... --- Another example is the fraction   1000 / 997002999 = 0.000 001 003 006 010 015 021 ... which goes through the triangle numbers[0] in its expansion, or   1 / 998999 = 0.000 001 001 002 003 005 008 013 021 ... which goes through the Fibonacci numbers[1]. --- Getting the squares is harder, but you can do it with   1001000 / 997002999 = 0.001 004 009 016 025 036 049 ... [0] http://en.wikipedia.org/wiki/Triangle_number [1] http://en.wikipedia.org/wiki/Fibonacci_number reply

VeryVito 4 hours ago | link If you'd like to continue the pattern beyond 52 digits, just keep adding 9s to the original fraction... 1/9999999999998 = 1.0000000000002 0000000000004 0000000000008 0000000000016 0000000000032 0000000000064 0000000000128 0000000000256 0000000000512 0000000001024 0000000002048 0000000004096 0000000008192 0000000016384 0000000032768 0000000065536 0000000131072 00000002621440... × 10^-13 reply

phamilton 1 hour ago | link For the fibonacci, add a 9 on both sides of the denomator 1/998999 1/99989999 1/9999899999 To get more 0 spacing and avoid overflow

Notes:

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Concepts:
Taylor series (0.989340): dbpedia | freebase | yago
Power series (0.717437): dbpedia | freebase
Series (0.701319): dbpedia | freebase
Laurent series (0.677218): dbpedia | freebase
Mathematical analysis (0.657370): dbpedia | freebase
Radius of convergence (0.582307): dbpedia | freebase
Mathematical series (0.574263): dbpedia
Infinity (0.479767): dbpedia | freebase