Abstract: The paper discusses the information approach to the investigation of numeration systems aimed at recognizing the degree of their informational optimality. The method of investigation is based on the Benford’s Law and the principle of information cyclicity. The discovery of the unique optimality of commonly used decimal system is the end result of the investigation. Along with theoretical significance this is of practical use in applied metrology.

Keywords: probability, information, optimization, numeration system.
 
 

1 Introduction

A variety of numeration systems are practiced. The most widely used system of numeration is the decimal system, which uses base 10. The binary system uses base 2 and is important because of its application to modern computers. Two other systems are also used to this purpose, octal (base 8) and hexadecimal (base 16). Besides, the duodecimal system is practiced that uses 12 as a base and has some advantages. A system of base 60, used by the ancient Babylonians, still survives in our smaller divisions both of time and of angle, i.e., minutes and seconds. The enumeration of examples might be carried on. However, regardless of any technological or traditional reasons of application, it is the decimal system that is of particular concern to mankind.

In general, any integer z greater than one can be used as the base of a numeration system, and the system will employ z different digits. We use the decimal or other numeration system, which (to the author’s knowledge) has never been investigated in view of its influence on measurement and/or estimation information. Nevertheless, the quality of the information depends on the numeration system used. This happened to be true owing to the mathematical theorem known as Benford's Law [1] and the principle of information cyclicity [2]. The harmonization of measurements, for instance, demands the numbering system to be optimum; otherwise the additional loss or excess of measurement information is inevitable. The paper proves the information optimality of commonly used decimal system. Despite the brief presentation, the author believes that this paper may be of interest for the specialists in mathematics, metrology, qualimetry, and other academic fields.
 
 

2 Benford’s probabilities

According to the Benford’s Law, in any numeration system the probability P(d) of any number d from 1 to (z -1) is calculated as follows:

P (d) = log_z (1 + 1/d)

The probabilities P (d) form a complete group of independent events, i.e. their sum = 1, and a logarithmic sequence has obvious classification character. If, for instance, z = 10, then d = 1 ? 9 repeats for the subgroups: 10 ? 90, 100 ? 900, etc. owing to the first digits of each subgroup. The different probabilities of the numbers on the one hand, and the infinite diversity of potential numeration systems on the other hand suggest an existence of optimum numeration system that can be determined by examining the logarithmic sequences.
 
 

3 Optimization criterion

The minimum and maximum Benford's probability, i.e. P_min = P [d = (z - 1)] and P_max = P (d = 1) form the specific system of two components, which ratio indicates to the degree of optimality of a numeration system. In terms of information theory [3], the optimum ratio (P_min /P_max)_o can be determined with the recently discovered principle of information cyclicity. According to this principle, the approximate cyclical connection between the sufficient and maximum probabilities always exists in a system of independent events, characterized by the complete group of probabilities. The ratio of the sufficient probability to the maximum probability is called the accuracy coefficient r_o = 1/2p . The derivation of information cyclicity is briefly dealt with in the Appendix. From these considerations, the optimization criterion for Benford's probabilities may be expressed as follows:

(P_min /P_max)_o = P [d = (z - 1)] / P (d = 1) = 1/2p

4 Optimum numeration system

In accordance with the above-established criterion, the index of optimum numeration system z_o (that is the logarithm base) is determined as the rounding off number as follows:

z_o = arg min ? log_z [1 + 1/.(z - 1)]/ log_z (1 + 1) - (1/2p )? = 10

Therefore, the optimality of decimal system may be considered as proven.

The same result can be achieved with the equation [log_z (1+1)/2p = log_z [1 + 1/(z_o - 1)], from which it follows that z_o = 1+ [exp (ln2/ln2p ) – 1]^-1 = 10 (as the rounding off value). This value does not depend on z that in some way signifies the absoluteness of the information optimality of decimal numeration and enables to consider z_o = 10 as one of information constants (common with the golden section f_o = 0.618 as the mathematical measure of harmonious relation and the optimum accuracy coefficient r_o = 1/2p ) which rounding off product r_o* f_o* z_o = 1 may be called The Law of Information Constants.
 
 

5 Conclusion

The obtained result demonstrates the unique informational optimality of commonly used decimal system. Another words, any numeration system, other than decimal system, is informatively either insufficient or excessive.
 
 

Appendix: Information cyclicity
 
 

The derivation of information cyclicity is based on the equivalency between the entropy (-) of the system of n components characterized by probabilities (p) and the maximum entropy (ln j ) of the system of j informative components of the same probability (p_j = 1/j ). From the equality of these entropies, the number j? is calculated as j = exp (- ). The approximate (2% estimation error) cyclical connection: n/(n - j ) = p₁/ pj= 2p , called the principle of information cyclicity, was discovered when analyzing the formula for a linear diagram of probabilities (Dp_j = p_j – p_j+1 = constant > 0) proven as the best calculation model. This method contains the fundamental estimation uncertainty due to the inequality of probabilities in the redundant part (from j to n) of the linear diagram. This part, being considered as the separate subsystem of the complete group of probabilities, possesses its own redundancy. This local redundancy causes both the optimization insufficiency and uncertainty in the main informative part (j=1?j ) of the system of n components.

The more accurate calculation can be achieved when taking into account the redundant number of the subsystem components that is equal to (n - j)(1 - j/n). The same number of the weightiest subsystem components may be considered as the limit of possible addition to j or the interval of uncertainty in determining the informative components. The optimum numberj_o is within the limits from j to [j+ (n - j)(1 - j/n)]. By substituting the sum for integral and for n ® ?, the relative values j_o/n and the uncertainty of its estimation U (j_o/n) are determined as follows:

j_o/n = 0.5{1 + [(j_o/n)]²} = 0.840 = (1 – 1/2p );

U (j_o/n) = 0.5[1 - (j_o/n)]² = 0.015 = 0.1(1/2p ),

where: j= exp{- ; K₁ = 2/(n + 1);

(j_o/n) = {[(n + 1)/n²] exp (0.5 - ln 2)} = 0.824.

With 0.5% estimation error the ratio n/(n - j ) = 2p mathes the principle of information cyclicity. 

For the system of two components the derivation is simplified. In that case the goal is to find such a ratio between the two probabilities, for which the number of informative components calculated satisfies the following equation:

j = exp (- p₁ ln p₁ – p₂ ln p₂) = 1.5

This equation illustrates the most uncertain (50% confidence) situation about allowing or ignoring the lesser (by probability) of the two components. The calculation results in the conclusion that with 0.5% estimation error the ratio p₂/p₁ = 1/2p that is in agreement with the principle of information cyclicity.
 
 

References

[1] Malcolm W.Browne, Following Benford's Law, or Looking Out for N^o 1, The Standard, ASQ, 1998, Vol. 98-4, 18–20.

[2] D. Kisets, Optimum Traceability Type Hierarchies. OIML Bulletin, 1997, XXXVIII (2), 30–36.

[3] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379–423.