Is the decimal notation the "right" notation for arithmetic? I am considering here the pre-decimal notations such as Roman numerals, Egyptian numerals etc. It seems reasonable that these must all be equivalent. And it seems that decimal notation (i.e. place-value notation) replaces all these earlier notation. So without getting into Peano's notations, if we were looking for a notation for arithmetic, could the decimal notation be considered the "right" solution? Thanks!
 A: Positional notation is an optimally efficient encoding of numbers into strings of characters drawn from a fixed set. Any other identification of numbers with strings will require strings at least as long as those in positional notation. 
In this sense, positional encoding is 


*

*fundamentally superior to all the historical encodings that came before it (Roman numerals, Egyptian numerals, etc), and

*better or equally as good (in terms of string length needed) as any possible notation people can ever think of.


The reason is that a notation system is an association (bijection) of numbers with strings, and positional encoding systematically enumerates all possible strings of digits of length $k$, leaving none out, before moving on to strings of length $k+1$.
For example, suppose our character set consists of the symbols "0" and "1" (Ie., we are working in binary), and we wish to represent all numbers from 0 to 7, represented by circles. We have the positional encoding,
\begin{align}
\text{none} & \longleftrightarrow 0 \\
\circ & \longleftrightarrow 1 \\
\circ \circ & \longleftrightarrow 10 \\
\circ \circ \circ & \longleftrightarrow 11 \\
\circ \circ \circ \circ & \longleftrightarrow 100 \\
\circ \circ \circ \circ \circ & \longleftrightarrow 101 \\
\circ \circ \circ \circ \circ \circ & \longleftrightarrow 110 \\
\circ \circ \circ \circ \circ \circ \circ & \longleftrightarrow 111
\end{align}
On the right are all possible strings of length 3 (when prepended by an appropriate amount of zeros). It would be impossible to create an encoding scheme for this using only strings of length 2, because there are only 4 strings of length 2, whereas we need to represent 8 numbers. One could create other optimal encodings by permuting the order of the strings, but that's it.
A: 
If we were looking for a notation for arithmetic, could the decimal notation be considered the “right” solution?

Given the context, I will assume that by “decimal notation” you actually mean positional notation, since the other numeral systems that you've mentioned are also decimal or base $10$, since they consist of symbols representing various powers and multiples of $10$. In which case, my answer would be: Even if not “the right solution”, it's definitely the best one we've got so far, at least when compared to the others you've mentioned $\big($i.e., think about doing division and multiplication using Roman numerals, for instance$\big)$.
