Why don't we have a distinctive symbol for 10?

Question

For example: Why don't our digits go: $1,2,3,4,5,6,7,8,9,a$ instead of using only nine symbols and $0$: $1,2,3,4,5,6,7,8,9,10$.

Answer 1

A number $N$ can be represented in our standard base 10 number system as follows

$$ N = \sum_{i=0}^{m} d_i10^i $$ with $$ 0 \leq d_i \leq 9. $$

To represent $N$ in zero-less base 10, we have the same sum, but with $$ d_i \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, a= 10\} $$

Some examples of numbers written in zero-less base 10

$N = 100 = 9a$

$N = 1000 = 99a$

$N = 1025 = a25$

Note that both number systems have ten symbols available to represent a number. The zero-less representation is simply 'shifted forward' by one number. Both number systems are equivalent in expressive power except that the zero-less system cannot represent zero naturally. One could write $1-1$ to mean zero, but that is the tradeoff for not having to use two symbols to represent ten.

Answer 2

Because $0$ is absolutely necessary (arguably, it's the most important number), and adding another digit would make it base 11.

Base $11$, unlike base $12$, which is an awesome, makes for a terrible numeral system because $11$ is a prime number. Number table for base 11:

$$ \begin{array}[pos]{r} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & a \\ 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 1a \\ 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 2a \\ 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 3a \\ 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 4a \\ 50 & 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 5a \\ 60 & 61 & 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 & 6a \\ 70 & 71 & 72 & 73 & 74 & 75 & 76 & 77 & 78 & 79 & 7a \\ 80 & 81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 & 8a \\ 90 & 91 & 92 & 93 & 94 & 95 & 96 & 97 & 98 & 99 & 9a \\ a0 & a1 & a2 & a3 & a4 & a5 & a6 & a7 & a8 & a9 & aa \\ \end{array} $$

11 is a prime number, so all fractions not of the form $x/a^y$ will recur, making arithmetic difficult:

$$ \frac{1}{2} = .\overline{5}, \ \ \ \frac{1}{4} = .\overline{28}, \ \ \ \frac{1}{5} = .\overline{2}, \ \ \ \frac{1}{a} = .\overline{1} $$