Can every number in $\mathbb{N}$ be written in base $-10$?

Question

So I was playing around with base $-10$ and noticed that some numbers can be written in more than one way e.g. 47 can be written as both $167_{-10}$ and $1967_{-10}$. Since $167_{-10} = 1\cdot (-10)^2 + 6\cdot(-10)^1 + 7\cdot(-10)^0 = 47$. But can every natural number be written in this base?

Answer 1

Yes, and you can write negative numbers as well. Say you want to expand $n$, then you can use the division algorithm to write:

$$n=(-10)q+r$$

$0\le r\le9$ is the rightmost digit, and recursively expanding $q$ the same way gives the remaining digits. Note that the recursion terminates, since $\left|q\right|<n$ for $\left|n\right|\ge10$. This generalizes to other interesting bases as well, such as, with a bit of work, base $2i$.