Role of the number systems in mathematics progress
Bergman's discovery has a certain relation to such the oldest mathematics branch as number systems. To access properly the importance of his discovery we should tell briefly about the number systems history and evaluate their role in mathematics progress. This history dates back to the ancient period of mathematics development. The discovery of the positional principle of a number representation is considered as the highest achievement of the ancient elementary arithmetic This discovery was made in the Babylonian mathematics. It is well known that the sexagesimal number system of the ancient Babylonians emerged about 2000 BC was the first of the familiar number systems based on the positional principle. We use the decimal number system in our daily life. It is well known that the "father" of the decimal number system is the Hindu number system emerged about the 8th century. The famous French mathematicians Laplas (18-19 C.) expressed his enthusiasm about the positional principle and decimal number system in the following words:
"The idea to represent all numbers by the 9 numerals giving them, in addition to the form value, still the position value is looked as much simple that just from behind this simplicity there is difficult to understand how much it is astonishing. How it was difficult to come to this method we can see on the example of the greatest genius' of Greek's learning Archimede and Appolonius for whose this idea was hidden".
Leonardo Pisano (Fibonacci) in his work "Liber abaci" (1202) came forward as the convinced supporter of the new enumeration. He wrote:
"The nine Hindi numerals are the following: 9, 8, 7, 6, 5, 4, 3, 2, 1. Using these numerals and the numeral of 0 called "zephirum" in Arabian, one may write some number".
Here Fibonacci used the word "zephirum" for designation of the Arabian word "as-sifr" being by the literal translation of the Hindi word "sunya", that means "empty" and serves for designation of 0. The word "zephirum" originated the French and Italian word "zero". On the other hand, the same Arabian word "as-sifr" was designated as "ziffer", and from where the French word "chiffre", the German word "ziffer", the English word "cipher" originated.
However modern computers are based on the "binary" number systems. Who discover the binary number representation and binary arithmetic? The discovery of the binary representation of numbers is ascribed to the Chinese Impair Fo Gi who lived in 4 millennium BC. However the famous German mathematician Leibnitz elaborated the rules of the binary arithmetic in 1697.
Leibnitz did not recommend the binary arithmetic for practical calculations instead the decimal number system but predicted that "the calculation with help of the deuces, i.e. 0 and 1, as an compensation of its duration, is basic for science and generates new discoveries, which prove to be useful afterwards even for the number practice and especially for geometry. The reason of that is the fact that there arises the wonderful order after the reduction of all numbers to the simplest elements, 0 and 1".
Brilliant Leibnitz's prediction was realized only in 2,5 centuries when the outstanding American scientist, physicist and mathematician John von Neumann suggested applying just the binary number system as the universal method for computer information coding ("John von Neumann principles").
Thus, many outstanding mathematicians claimed that the discovery of "positional principle" of number representation is one of the epochal mathematical achievements for all mathematics history and the same positional number systems played a revolutionary part both in the theoretical development and in the practical applications of mathematics.
In Greek's mathematics achieved the high standard for the first time the mathematics is divided into two parts, the "higher" mathematics, to which geometry and number theory belonged, and the "logistics", the applied science about the technique of arithmetical calculations ("school" arithmetic) and methods of geometric measuring and construction. Already since Plato's time the "logistics" was considered as the "lowest", applied discipline, which did not belong to the range of the scientist and philosopher education. The scornful attitude to the "school" arithmetic and its problems and also an absent of some serious need in creation of new number system in computational practice, which was satisfied fully by the decimal number system and by the binary number system (in computer science) is a reason why mathematics did not attract its attention to development of number systems. And in this field modern mathematics did not move ahead far in comparison to period of its origin.
However, situation changed sharply after appearance of modern computer science! Just in this field an interest in new methods of number representation and in new computer arithmetic appears. That is why appearance of Bergman's number system, which turns over our ideas not only about number systems but also about number theory in general can be considered as the revolutionary mathematical discovery of the 20th century! And we will tell about this discovery at the next pages of our Museum. Follow us!