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
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
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 Di. 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:
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.