What would base 0 be? How would/could it work?

Solution 1:

Base 0 does not make any mathematical sense.

Look at binary (base 2). There are two digits, 0 and 1. Thus, every other number you need to roll over the 1 back to a zero, and add 1 to the next column.

Now, look at base 1. Now, every number requires rolling over to the next row. This is essentially a tally system, where each '1' (in base ten) gets it's own column.

Now, if you think about base 0, that would mean every increase by '1' in any non-zero base represents an infinite amount of columns that need to be created to support the overflow. Thus, every number in base 0 would essentially be infinite, or even worse, every number would be the same number.

Solution 2:

In base $10$, we use ten symbols.

In base $2$, we use two symbols.

In base $1$, we use one symbol (tally marks).

In base $0$, we'd use zero symbols. We can't express anything with zero symbols.

Solution 3:

Base 0 unfortunately does not make any sense, for the very reason you specify.

Most digits in the number would be worth exactly zero, and the digit in the "ones position" would not even have a defined place value.

Solution 4:

When you express a number in base $b$ you find it as a sum of various powers of $b$. For example to express $65$ in base 3 we first note $65= 27+27 + 9 +1 +1$ so $65=2\cdot 3^3 + 1\cdot 3^2 + 2\cdot 3^0$, hence $65=(212)_3$. Unfortunately all powers of zero are zero, and so sums of powers zero cannot be anything other than $0$, so zero is powerless to be the base for number system.