Why is the common positional notation unintuitive

Question

Let's say I want to invent my own positional numeral system. I start with several symbols representing the smallest amounts: A, B, C, D, E, F. Since I don't want to have infinitely many symbols and since I'm making a positional system, the symbol representing the next amount will be AA.

So I created this notation:

  A,   B,   C,   D,   E,   F,
 AA,  AB,  AC,  AD,  AE,  AF,
 BA,  BB,  BC,  BD,  BE,  BF,
...
 FA,  FB,  FC,  FD,  FE,  FF,
AAA, AAB, AAC, AAD, AAE, AAF,
...

Of course all numbers in the table are meant to be unique and naturally ordered.

My question is: Why this intuitively created positional notation doesn't correspond to our common base-10 system? Does my positional system have any design flaws? Or is it just a historical thing?

When the chosen symbols are 1-9 then the numbers 10, 20, 100, 130, ... are missing. When the chosen symbols are 0-9 then there weird numbers like 01, 001, 0001 appear.

Answer 1

The usual positional system has a symbol for $0$, which causes that there are several notations for the same number, e.g. $6$ and $06$. A system without this feature is called a bijective numeral system, since the correspondance between symbols and numbers is... well, bijective.

Thus, if we have $k$ symbols $\{1,\dots k\}$, the string $a_n \dots a_0$ represents the integer $\sum_{j=0}^n a_j k^j$. Note that the zero must be represented by an empty string, i.e. it has no representation.

Apart from the lack of a symbol for zero, arithmetic operations behave much in the same way as in the usual system, except that carries occur one unit higher, i.e. when exceeding $k$, rather than when reaching $k$.

For instance, the OP suggests a base-6 bijective numeral system, where the integer $6$ can be represented as a single digit $F$, rather than the $10$ it would be in usual base-6 positional system with zero.

As for the historical choice of the standard system, the Wikipedia article has some additional comments: https://en.wikipedia.org/wiki/Bijective_numeration#Historical_notes

Also relevant is the final remark in the paper https://doi.org/10.2307%2F3029479 which I reproduce here literally:

The foregoing manipulations indicate that the 0-less system has substantially the elasticity of the conventional decimal system, and as a consequence challenges the assertion that modern science, in industry, or commerce would be inconceivable without the zero-symbol, even though its discovery happened to be an historical condition to their development. While facility in ordinary arithmetic manipulations may not be dependent on the symbol, it must nevertheless be realized that the development of pure mathematics would have been retarded without it, since the study of classes necessitates the identification of the 0-class. This paper, therefore, is not to be interpreted as an argument that the values of the zero-symbol to mathematics are wholly accidental, but as a discussion of its alleged essential character in an easily manipulated system of numbers.

Answer 2

In positional notation you need a symbol for $0$, which is why $1$ through $9$ don't work.

The leading zeroes in things like $001$ are correct and harmless. They're usually omitted, as you know, but writing them doesn't hurt.

So your new system really is the standard system - base $6$ in your example.

As @Miguel points out, if $A$ is $0$ then the second row repeats the first.