TITLE: What would base $1$ be?
QUESTION [100 upvotes]: Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be?

Let's say we define Base $1$ to use: $\{0\}$.
Because $10_2$ is equal to $010_2$, would all numbers be equal?
The way I have thought Base 1 might be represented is tally marks, $0_{10}$ would be represented by nothing. So, $5$ in Base 1 would be represented by $00000$? Or we could define Base 1 to use: $\{$|$\}$ and $5$ would be |||||?

REPLY [4 votes]: $\displaystyle \sum_{i=0}^k a_i b^i$ with $1 \le a_i \le b$ can represent any positive integer uniquely for fixed $b$ and has the tally mark interpretation when $b=1$.
This is called bijective numeration. When $b$ is ten you lose the $0$ numeral but gain a numeral for ten ($A$ say).  Many decimal numbers look the same when written in bijective base-ten, such as $432$, but others do not, such as $402$ which becomes $3A2$.