Correlation between $\sqrt{1/10}$ and length of powers of integers.

Question

I'm searching information about this simple problem involving square roots and length of powers.
It's very simple but it seems interesting, at least for me.
I'm not a mathematician.

Description

By observing the sequence of all positive integers where the length of their square increments:

  1  4  10   32   100     317
  1 16 100 1024 10000  100489

We can see that it happens alternatively at powers of 10 and at some particular numbers.

This is the algorithm in Ruby which I think it's self explanatory: to_s converts a number to string so that we can take its length, we print x if its power increases in length.

length = 0
temp   = 0
1.step{|x|
 if length < temp = (x**2).to_s.size
  length = temp 
  print"#{x}\n"
 end
}

Try it online!

If we discard powers of 10 we have those particular numbers:

4 32 317 3163 31623 316228 3162278 ..

It turns out that they are the sequence of digits of √(1/10)

31622776601683793319988935444327185337195551393252168268575048527925944386392382213442481083793002951873472841528400551485488560304538800146905195967001539033449216571792599406591501534741133394841240853169295770904715764610443692578790620378086099418283717115484063285529991185968245642033269616046913143361289497918902665295436126761787813500613881862785804636831349524780311437693346719738195131856784032312417954022183080458728446146002535775797028286440290244079778960345439891633492226526120677926516760..

The algorithm was improved to this giving a lot of digits with less resources.
It takes advantage of the observation that the next number differs only on the last two digits. More precisely we can observe that after x we have a number in the range [ (x-1)10 .. x10 ).
But this is just a shortcut, let's stick to the original algorithm involving lengths.

One fellow noticed it was the digits of √(1/10) and explained me.. "The length of integers increases at each power of 10. The length of squares of integers increases at (the ceiling of) each power of √10. Half of these powers are also powers of 10, while the other half are powers of 10 multiplied by √(1/10)" which has same digits of √10 or √1.
That makes sense obviously but at the same time it seems like a different point of view. I can't find any proof, articles, anything about this.

So my question is: Does there exist any information about this simple concept?

I'm not a mathematician. As a side note it seems like we can compute square roots by observing the length of powers, in fact , we can also get the digits of $\sqrt 2$ by inspecting the lengths of $(x^2)/2 $ or $(x^2)*5$ and similarly √3 for example.

Answer 1

If $n$ is the number where the square change from having $k$ digits to $k+1$ digits then $(n-1)^2<10^k\le n^2$. So $n-1 < 10^{k/2}\le n$. If $k$ is even we have $n = 10^{k/2}$. If $k = 2m+1$ then $n-1 < 10^{m+1}\sqrt{1/10}< n$ which explains why the first digits of $n$ coincide with $\sqrt{1/10}$.