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I am not a mathematician.

I have some number $A_b$ which has a digit in a position $p$. That number can be written in another base as $V_d$ and it will have a range of digits $q_{start}$ – $q_{end}$ which hold (potentially added with other values) data from that digit.

I need to find the range $q_{start}$ – $q_{end}$.

And I can do a part of that for non-negative positions $p$ (to the left of the point) we know that the smallest non-zero digit in any base is $1$, thus $q_{start}$ is simply $\log_q\left(b^{p}\right)$ (round down), and $q_{end}$ is somewhat trivial once we bound it to $1$ in the next position, so it becomes $\log_q\left(b^{p+1}\right)$ (round up?).

However, I get completely lost once we approach digits to the right of the point. I know that in some situations $q_{end}$ will be infinite. I understand that it cannot happen when $d$ is a multiple of $b$. But somehow I am stuck.

mwt
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v010dya
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    Even for non-negative positions $p$ it's not that easy. In general, changing $p$-th digit of a number in base $b$ can affect even the most significant digit of a number in base $d$. For example two numbers ${473523889}{11}$ and ${47352388A}{11}$ in base $11$ differ only by the least significant digits. This change, however, makes a huge difference in base $10$ where these two numbers are written as, respectively: ${999999999}{10}$ and ${1000000000}{10}$. – mwt Aug 01 '20 at 15:00
  • @mwt Thanks, but as i have said i differentiate about the value of that digit and the values that it is added with due to other digits. But you're correct, i should have made it more clear. – v010dya Aug 01 '20 at 18:46

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