I was in Spain a couple of weeks ago, more precisely in Salamanca, which as you know has one of the oldest universities in Europe. I always take advantage of my travels to search for old books of mathematics. In this case, in a quite small old library close to the cathedral I found a book of mids XIX century. As I am not proficient in Spanish, with the help of a friend I translated the part talking about number representation.

The author talked about an ancient Moorish way of representing natural numbers in a decimal system, using a special symbol for 10. He uses the letter “D” to do so, in part because it is like a “1” and a “0” put together, in part because “D” is the beginning of “Diez” (ten in Spanish) – and I guess many ancient typesets lacked special symbols.

So I found curious the way of representing natural numbers in this decimal system:

```
1,2,3,…. …, 8, 9, D,
11,12,13… …, 18,19,1D,
21,22,23… …, 28,29,2D,
…
91,92,93… …, 98,99,9D
D1,D2,D3… …, D8,D9,DD
111,112,113…
```

This is a valid representation of all natural quantities, because there is one and only one representation for every quantity. For instance,

`(4D8)`_{D} = (508)_{10}
(DDD)_{D}= (1110)_{10}

The numbers not containing any "D" represent exactly the same quantity in both systems.

This author pointed out that this was the way that numbers were written anciently in a decimal system before the introduction of the “0” for representing quantities. He did not describe any historical facts -he talks about the Moors but maybe he was wrong-, but thinking about the way Egyptians used to represent numbers in a decimal system, they had special symbols for powers of 10, but the symbol for Zero (nfr) was never used as a symbol to write other numbers. So the author’s explanation makes sense.

Then I understood that one thing is having a symbol for Zero for operations and another thing is using that symbol to represent numbers. The question is,

If Zero is not a natural number, why do we introduce it inside the representation of them? If we have the set of natural numbers, and we want to have a decimal representation, in a pure natural representation we should not introduce an element we do not have. So for example, lets accept that we have powers of ten represented as D^{i}. Then

`(703)`_{10} = 7 x D² + 0 (???) x D + 3 * 1

What is that ‘0’ element?

Then, extending this way of representing numbers to other systems, we have for example:

```
Octal: 1,2,3,4,5,6,7,8,11,12,13,14,15,16,17,18,21,22…
Binary: 1,2,11,12,21,22,111,112,121,122,211,212,221,222…
Unary: 1,11,111,1111,1111…
```

The conclusion is: *The unary system is totally valid.*

And as a side advantage, with this system we use less digits. For instance, with the normal binary system we can count up to (111)_{2}= 7, while with this other binary system we can count up to (222)_{2}= 14.

Of course there are advantages of using the Zero for representing natural numbers, but this led us to a confusion, making some people think -as we could see in some answers- that the unary system was not a valid system, or that Zero could be represented as a blank. It is indeed a valid system, and for sure the most ancient -at least it has been proved to be 20000 years old. And the answer to this question is that *Zero in the unary system is the same as in any other system, something introduced not for representing quantities but as an element of more complex mathematics* -as in algebraic structures like groups or rings, or in set theory, as the cardinality of the empty set.