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Introduction

Volume II is devoted to the discussion of two fundamental problems of science and mathematics, the problem of measurement and the problem of numeral systems, their relationship with the development of science and their historical role in the development, first of all, of contemporary mathematics and computer science by taking into consideration the contemporary achievements in mathematical theory of measurement and numeral systems.

As it is known, a set of rules, used by the ancient Egyptian land surveyors, was the first measurement theory.From this measurement theory, as the ancient Greeks testify, there originated the geometry, which takes its origin (and title) in the problem of earth measuring.

Already in the ancient Greece, the mathematical problems of geometry (that is, earth measuring) were the main focus of ancient mathematics. The science of measurement, related to geometry, was developing primarily as a mathematical theory. It is during this period that the discovery of the incommensurable segments and the formulation of Eudoxus’ exhaustion method, to which the number theory as well as the integral and differential calculus go back in its origin, were made.

By basing on these important mathematical discoveries, which had the relation to measurement, the Bulgarian mathematician academician Ljubomir Iliev, the leader of the Bulgarian mathematical community, asserted that “during the first epoch of its development, from antiquity to until the discovery of differential calculus, mathematics, by studying primarily problems of measurement, did created the Euclidean geometry and number theory” [137].

In 1991, the Publishing House “Science” (the main Russian edition of the physical and mathematical literature) has published the book, Mathematics in its Historical Development [102], written by the outstanding Russian mathematician academician Andrey Kolmogorov (1903–1987). By discussing the period of the origin of mathematics, academician Kolmogorov pays attention to the following features of this period:

“The counting of objects at the earliest stages of development of culture led to the creation of the simplest concepts of arithmetic of natural numbers. Only on the basis of the developed system of oral numeration, the written numeral systems arise, and gradually methods of performing four arithmetic operations over natural numbers are developed …

The demand for measurement (the amount of grain, the length of the road, etc.) leads to the appearance of the names and symbols of the most widespread fractional numbers and the development of methods for performing arithmetical operations over fractions. Thus, there was accumulating material, which is added gradually to the most ancient mathematical direction, arithmetic. Measurement of space and volumes, the needs of construction equipment, and a little later, astronomy, cause the development of the beginnings of geometry.”

By comparing the views of the academicians Iliev (Bulgaria) and Kolmogorov (Russia) on the period of the origin of mathematics, it should be noted that these views mostly coincided to and were reduced to the following. At the stage of the origin of mathematics, two practical problems influenced the development of mathematics: the counting problem and the measurement problem.

The study of the counting problem ultimately led to the formation of such an important concept of mathematics as the natural numbers and to the creation of the elementary number theory, which solved important mathematical task in studying the properties of the natural numbers as well as solved the problem of creation of the elementary arithmetic that satisfied the needs of practice in performing the simplest arithmetic operations. The study of the problem of measurement led to the creation of geometry, and within this direction, the existence of the irrational numbers, the second most significant fundamental ancient mathematical discovery, which caused the first crisis in the foundations of mathematical science is proved.

By discussing the origins of mathematical science, we should not forget about another outstanding mathematical discovery of ancient mathematics, Eudoxus’ exhaustion method, which, on the one hand, was created to overcome the first crisis in the foundation of mathematics, associated with the introduction of the irrational numbers into mathematics and, on the other hand, underlies the Euclidean definition of the natural numbers, which represents the same natural number as the sum of the “monads” . It follows from these arguments that the Eudoxus’ exhaustion method attempted to unite the two ancient problems that underlie the ancient mathematics: the problem of counting, which led to the natural numbers, and the problem of measurement, which led to the irrational numbers.

Volume II pursues two goals. The first goal is to set forth the foundation of the new mathematical measurement theory, the Algorithmic Measurement Theory, worked out by Alexey Stakhov in his doctoral dissertation, Synthesis of Optimal Algorithms of Analog-to-Digital Conversion (1972). The second goal is to show that this new mathematical theory of measurement is the foundation of all traditional positional numeral systems by starting with the Babylonian positional numeral system with base 60 and by ending with decimal, binary, ternary, and other traditional numeral systems. But the most important is that this new theory of measurement generates new, earlier unknown positional numeral systems, such as the Fibonacci p-codes [6, 16, 55, 56, 58, 60, 97], the golden p-proportions codes [6, 19, 53, 97], which are a generalization of the classical binary system, and finally, the ternary mirror-symmetrical system and arithmetic [72, 94, 97], which are a generalization of the classical ternary system and arithmetic.

Volume II consists of seven chapters, which can be divided into two parts. The first part includes Chapters 1 and 2.

Chapter 1 provides the introduction to the Algorithmic Measurement Theory [16], which is based on the constructive approach in modern mathematics.

Chapter 2 is devoted to the Fibonacci measurement algorithms, which generate the Fibonacci p-codes for mission-critical applications.

The second part of Volume II consists of five chapters (Chapters 3–7). Chapter 3 provides a brief statement of the most interesting facts in the history of traditional numeral systems (Babylonian numeral system with the base of 60, Mayan numeral system, decimal, binary, and ternary systems).

Chapter 4 is devoted to the description of the unique positional numeral system with irrational base (the golden ratio), proposed in 1957 by the American mathematician George Bergman [54], and following from Bergman’s system the “golden” number theory, where the new properties of the natural numbers (the Z- and D-properties) are represented.

Chapter 5 is devoted to the description of the unique ternary arithmetic, the “golden” ternary mirror-symmetrical arithmetic, which opens the new direction in ternary computers.

Chapter 6 is devoted to a study of the Fibonacci p-codes and Fibonacci arithmetic, which are the new scientific results for computer science and can lead to designing of the Fibonacci computers for mission-critical applications.

Chapter 7 is devoted to the study of the general class of the redundant numeral systems. The classical binary system is the partial case of the codes of the golden p-proportions (p = 0), the remaining “golden” codes, corresponding to the cases of p = 1, 2, 3, …, are a generalization of Bergman’s system (p = 1), and for the general cases of p = 1, 2, 3, …, they represent the general class of numeral systems with the irrational bases, which have a fundamental importance for mathematics (as the new definition of the real numbers) and also for computer science (as the basis of “golden” computers) and for the digital metrology (as the basis of the new theory of resistive dividers).