Is There An Alternative To Using 0 As A Placeholder?

Question

I'm no math wiz here, but I have a question that I can't wrap my head around. In fact, I don't even know how I would even go about asking the question properly. Is there an alternative to using 0 as a placeholder?

When writing out numbers, 10 doesn't have it's own symbol. So rather we use 1 and 0(the placeholder) to represent 10. Is there a better method than using 0 or keep from going into double digits with the number 10?

Answer 1

If I'm understanding your question correctly:

It's important to remember that the number of digits a number has is only dependent on what base I'm writing the number in. We typically write things in base-$10$. That is, each "slot" can go from $0$ to $9$ before we add another "slot".

However, what if I write $10_{10}$ in base-$2$? I have $1010_2$ (count up only allowing each "slot" to get to $1$ before a new "slot" is added. You'll see where $1010_2$ comes from). In this way of counting, for any even number I'm going to be using $0$ as a "placeholder".

So, my point: you could make $11_{10}$ the fist double digit double if you wanted. Just come up with another symbol to represent $10_{10}$ and then begin writing everything in base-$11$. Whenever we use a "placeholder" or go into more digits is completely dependent on the base, so you can have the first occurrence of $0$ as a "placeholder" occur whenever you want. I hope I helped, because at least I think this is what you were asking.

Answer 2

There isn't a better way, but there are ways. For example, we can use bijective base 10.

In bijective base 10, the digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and numbers are interpreted as

$$5A26 = 5*10^3 + 10*10^2 + 2*10^1 + 6*10^0 = 6026$$

This system is capable of uniquely representing any positive integer. However, we don't gain anything by using it, and it's more awkward to work with. It doesn't generalize as nicely to representing arbitrary real numbers, and even the natural representation of zero (the empty string) is incredibly awkward to use in an equation.

There's nothing to be gained by avoiding zero. Sure, there's no symbol for 10, but if we introduce one, then there's no symbol for 11, and 1 ends up serving a similarly placeholdy role.