Let $b\geq 2$ be an integer. A real $r \in [0,1]$ is said to be *normal with respect to $b$* if every finite string made from the elements $\{0,\ldots,b-1\}$ appears in the $b$-ary expansion of $r$.


Are there integers $b, b'\geq 2$ as well as a real number $r\in[0,1]$ such that $r$ is normal with respect to $b$, but not with respect to $b'$?



Does normalcy in one base imply normalcy in any other base?
A [1960 paper by Wolfgang Schmidt](https://msp.org/pjm/1960/10-2/p22.xhtml) states the following:



> 
> We write $r \sim s$, if there exist integers $n$, $m$ with $r^n = s^m$. Otherwise, we put $r \not\sim s$.
> 
> 
> In this paper we solve the following problem. Under what conditions on $r$, $s$ is every number $\xi$ which is normal to base $r$ also normal
> to base $s$? The answer is given by
> 
> 
> ### THEOREM 1.
> 
> 
> A) Assume $r \sim s$. Then any number normal to base $r$ is normal to base $s$.  
> 
> B) If $r \not\sim s$, then the set of numbers $ξ$ which are normal to base
> $r$ but not even simply normal to base $s$ has the power of the continuum.
> 
> 
> 


