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 √308642≈5000/9=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+x2−x28+x316−5x4128+7x5256−21x61024+⋯

If we plug in x=2/(5000)2=8⋅10−8, we get

M:=√1+8⋅10−8=1+4⋅10−8−8⋅10−16+32⋅10−24−160⋅10−32+⋯.

Therefore

√308462=50009M=50009+200009⋅10−8−400009⋅10−16+1600009⋅10−24+⋯=59⋅103+29⋅10−4−49⋅10−12+169⋅10−20+⋯.

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*