# The Archimedean Origin of Modern Positional Number Systems
## 3 From Arenarius to Zeroless Decimal Systems
In this way, the first period of the DAS in the XDS becomes  
\[1,2,3,4,5,6,7,8,9,X\]  
For example, the XDS translation of \(1^{\infty}3^{0}6\) is XX3X6. The logic of the XDS enumeration is based on powers of ten: 1, X, XX, XXX,.... Any numeral is the concatenation of multiples of these powers, and their sum provides the number expressed by the numeral. It is interesting to remark that this structure resembles exactly the construction of numerals in many natural languages. However, neither the DAS nor XDS are positional systems in the strict sense of our usual decimal system with zero, but they could be better characterized as polynomial systems (where monads are monomials). Polynomial systems of number representation occur, in primitive forms, in many ancient systems and in the measurement of angles. The mathematician and astronomer [PERSON] (first century, author of the _Almagest_) used circled digits, and in some contexts his circle resembles zero. Going back to DAS numerals, we could avoid putting circles to digits when in a numeral all the monads smaller than the leftmost monad occur. In fact, in that case the level of any monad corresponds to its position. For example, \(1^{\infty}3^{0}6\) is completely expressed by 136. But, if circles are deleted in \(1^{\infty}6^{0}\) and \(1^{\infty}6\), we obtain, in both cases, 16, which does not distinguish between the two different numerals. However, we can avoid circles if the missing monads are indicated. Therefore, zero, which was discovered at the end of the fifth century within the Indo-Arabic mathematical tradition [4], has a natural motivation in [PERSON]'s periodical system, as a new digit 0 expressing the absence of any monad having a number of circles corresponding to its position (distance from the rightmost digit). Nevertheless, the zero digit is not necessary for having a positional system, because zeroless positional systems in the sense of the 0 DS can be defined. One such system is based on the strings that can be constructed over a finite sets of digits [5; 6]. Let us assume the ten digits (without zero) in the order  
\[1,2,3,4,5,6,7,8,9,\text{X}.\]  
For each digit, the strings of two digits are generated according to the following square, where ordering is from left to the right in the columns and from the top to the bottom for the rows:  
\(\text{X}(1,2,3,4,5,6,7,8,9,\text{X})\)  
In general, numerals are generated by orders \(L_{i}\), for \(i=1,2,\dots\)  
\[L_{1}=1,2,3,4,5,6,7,8,9,\text{X}.\]  
\[L_{i+1}=1L_{i},2L_{i},\ \dots\ \text{X}L_{i}\]  
where numerals of \(L_{j+1}\) follow those of \(L_{j}\), and for any digit \(D\), the following equation holds:  
\[DL_{i}=\{D\alpha|\alpha\in L_{i}\} \tag{1}\]  
with \(D\beta>D\alpha\) for any \(\beta>\alpha\) in \(L_{j}\), and \(j>1\). This is the structure of any enumeration system, over strings, based on orders and periods. The ordering associated with this enumeration corresponds to the lexicographic ordering, characterized by the following conditions (\(|\alpha|\) is the length of string \(\alpha\), and \(x,y\) are any digits):  
\[|\alpha|<|\beta|==>\alpha<\beta\]  
\[\alpha<\beta==>\alpha x<\beta y\]  
\[\alpha<\beta==>xa<x\beta.\]  
We call this enumeration the LXS (Lexicographic X-decimal System). The LXS is a positional system, where digits contribute to the value of the denoted number according to their positions. A complete enumeration system based on orders and periods is an Archimedean Enumeration System (AES). An AES is monotone if its numerals are non-empty strings over a finite set of symbols, and any numeral \(\alpha\) followed by a digit \(x\) is a numeral too, such that the number \([[\alpha]]\) denoted by \(\alpha\) coincides with the number of orders before the order where \(\alpha x\) occurs. Such a system has period \(p\) if its initial order has \(p\) numerals. The XDS, as well as the usual decimal positional system, the D0S, are Archimedean natural monotone enumeration systems.