The influence of the place-value structure of the Arabic number ...

interexponential structuring principles. The only exception is presented by the combination of
positional and multiplicative structuring which is just logically impossible. A number system
in which the quantity at a certain power level and this power level itself are represented by
two distinct symbols (i.e., multiplicative organization) cannot comply with the limitation of
coding the value of a number symbol not only by its shape but also by its position within the
string of number symbols (i.e., positional structuring). In a positional system the unit sign and
the power sign used for multiplicative coding of one power level would rather reflect two
adjacent power levels, thereby making the combination of multiplicative and positional
coding impossible.
Table 1: Intra- and interexponential structuring and their combinations in known numerical notation
systems
Interexponetial Structure
Additive
Power levels are summed up to
obtain total value
Positional
Value of power levels multiplied
with positionally coded multiplier
(e.g., hundreds) before
summation
Intraexponential structure
Cumulative
Sum of signs per power
value indicate total value
Ciphered
Total value represented
by only one sign
Multiplicative
Total value computed by
multiplication of unit
sign(s) and a power sign
Roman numerals
678 = DCLXXVIII
(500+100+50+10+10+5+1+1+1)
Greek alphabetic
678 =
(600 + 70 + 8)
Traditional Chinese
678 =
6 100 7 10 8
Babylonian cuneiform
678 =
(10 + 1) x 60 + (10 +8) x 1
Arabic number system
678
(6 x 100 + 7 x 10 + 8)
Logically excluded
From the fact that the Arabic number system is by far the dominating numerical
notation system one may conclude that “the evolution of written numeration converges [in the
Arabic number system] […] because place-value coding is the best available notation” as
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