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

The number has a crazy decimal representation :

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 ?

The architect’s answer, while explaining the absolutely crucial fact that $$√308642≈5000/9=555.555…,\sqrt{308642}\approx 5000/9=555.555\ldots,$$ 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.
$$√1+x=1+x2−x28+x316−5x4128+7x5256−21x61024+⋯ \sqrt{1+x}=1+\frac x2-\frac{x^2}8+\frac{x^3}{16}-\frac{5x^4}{128}+\frac{7x^5}{256}-\frac{21x^6}{1024}+\cdots$$
If we plug in $$x=2/(5000)2=8⋅10−8x=2/(5000)^2=8\cdot10^{-8}$$, we get
$$M:=√1+8⋅10−8=1+4⋅10−8−8⋅10−16+32⋅10−24−160⋅10−32+⋯. M:=\sqrt{1+8\cdot10^{-8}}=1+4\cdot10^{-8}-8\cdot10^{-16}+32\cdot10^{-24}-160\cdot10^{-32}+\cdots.$$
√308462=50009M=50009+200009⋅10−8−400009⋅10−16+1600009⋅10−24+⋯=59⋅103+29⋅10−4−49⋅10−12+169⋅10−20+⋯. \begin{aligned} \sqrt{308462}&=\frac{5000}9M=\frac{5000}9+\frac{20000}9\cdot10^{-8}-\frac{40000}9\cdot10^{-16}+\frac{160000}9\cdot10^{-24}+\cdots\\ &=\frac{5}9\cdot10^3+\frac29\cdot10^{-4}-\frac49\cdot10^{-12}+\frac{16}9\cdot10^{-20}+\cdots. \end{aligned}
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 $$5+2=75+2=7$$s begins when the first two terms of the above series are “active”. When the third term joins in, we need to subtract a $$44$$ and a run of $$33$$s ensues et cetera.