I was wondering idly this morning about real numbers $r$ with the property that the decimal representations of $r$ and $1/r$ both contain the same nonzero digits (not necessarily the same number of times or in the same order). Some obvious example are:
- The only nonzero digit in the decimal representation of $1/3$ is $3$.
- The only nonzero digit in the decimal representation of $10^n$ and $1/10^n$ is $1$.
Of course this property depends on the number base one uses to represent the real number. In binary, every number has this property (because there is only one nonzero digit available!). In any base $b$, the numbers $b^n$ all have this property. Somewhat more subtly, if the base $b$ has the form $n^2+1$, then the representation of $1/n$ will be $0.nnnnnn \dots$ (which generalizes the example of $1/3$ in base $10$).
But all of these examples involve whole number $r$ and seem in some sense trivial. Are there any nontrivial examples? Is it possible to prove whether there exist irrational numbers with this property?
Here's a nice class of non-trivial examples that will work in any base. In particular, suppose that $r$ satisfies $$ r = 1/r + n $$ for some integer $n$. Then $r$ and $1/r$ will have all the same digits after the exact same digits after the decimal place. Usually, both will contain all $10$ digits somewhere in there.
For example, with $n = 1$, we have the solution $$ r = \frac{1 + \sqrt{5}}2 $$ (aka the golden ratio), and we find that $$ r = 1.61803\dots\\ 1/r = 0.61803\dots $$