Is there any mathematical reason for this “digit-repetition-show”?

The number 308642 has a crazy decimal representation : 555.5555777777773333333511111102222222719999970133335210666544640008

Is there any mathematical reason for so many repetitions of the digits ?

A long block containing only a single digit would be easier to understand. This could mean that there are extremely good rational approximations. But here we have many long one-digit-blocks , some consecutive, some interrupted by a few digits. I did not calculate the probability of such a “digit-repitition-show”, but I think it is extremely small.

Does anyone have an explanation ?

Answer

The architect’s answer, while explaining the absolutely crucial fact that 3086425000/9=555.555, didn’t quite make it clear why we get several runs of repeating decimals. I try to shed additional light to that using a different tool.

I want to emphasize the role of the binomial series. In particular the Taylor expansion
1+x=1+x2x28+x3165x4128+7x525621x61024+
If we plug in x=2/(5000)2=8108, we get
M:=1+8108=1+410881016+3210241601032+.
Therefore
308462=50009M=50009+2000091084000091016+16000091024+=59103+29104491012+1691020+.
This explains both the runs, their starting points, as well as the origin and location of those extra digits not part of any run. For example, the run of 5+2=7s begins when the first two terms of the above series are “active”. When the third term joins in, we need to subtract a 4 and a run of 3s ensues et cetera.

Attribution
Source : Link , Question Author : Peter , Answer Author : Jyrki Lahtonen

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