New geometric definition of the Number As is well known the Number is the most important concept of mathematics and the number theory is one of the most ancient mathematical theories. There exist a number of different definitions of the Number concept. The most known from them is the Euclidean number definition, which has the following geometric interpretation. Let
be the infinite set of the standard line segments of the length 1. Then the natural number
In spite of the utmost simplicity of such definition this one played a great role in mathematics and underlies the basis of many useful mathematical concepts, in particular the prime and composite numbers, and the concept of divisibility, which is one of the main concepts of the number theory. It is well known the constructive approach to definition of real numbers. According to it the real number
where The number definition of (3) has the following geometric interpretation. Let
be the infinite set of the standard line-segments of 2 Note that the number of terms in (4) is final but unlimited potentially (the constructive notion of potential feasibility). The definition of (3) divides all the real numbers into two parts namely the During many millenniums mathematicians developed and stated more precisely the concept of the Number. In the 17th century, that is, in the period of new science and new mathematics origin, the methods of "continues" mathematics are developed and the notion of the "Real Number" comes out ahead. The great mathematician Newton in his "Universal Arithmetic" gave the new definition of the "Real Number": "We understand over numbers no as much the set of the units as the abstract ratio of some value to another one of the same kind, which we use as the unit". This formulation gives us the common definition of the real number both the rational one and the irrational one. If now to consider the "Euclidean definition of number" (2) from the point of view of "Newton's definition" then the Euclidean "monada" plays here the role of the "measurement unit". In the binary notation (3) the role of the "measurement units" is played by the number of 2, which is the radix of the binary number system (3). Let's consider now Bergman's number system
since Newton's point of view. Clearly, the notation (5) quite corresponds to "Newton's definition", but its main feature consists of the fact that the "gold proportion", which is irrational number plays the role of the "measurement unit" in Bergman's notation! And then the number definition assigned with (5) has the following geometrical interpretation. Let's consider the infinite set of the "standard line segments" being the golden proportion degrees:
where
Then the Thus, Bergman's number system (5) is nothing as the new number definition corresponding completely to "Newton definition"! And just some irrational number ("golden proportion") but no traditional natural number (2, 10, 60 etc.) plays the role of the "measurement unit" and we can represent arbitrary real number by using Bergman's number system! And now we can ask: whether is there the more general number definition, which could join all above-considered number definitions? We can give a positive answer to this question. Really, such number definition is based on the concept of the Let's consider now the infinite set of the standard line-segments based on the golden
where n = 0, ±1, ±2, ±3, ...; are the golden
The set of (8) generates the following constructive method of the real number representation:
where Note that first the number definition given by (10) had been introduced by Prof. Stakhov in his article
However let's consider now the positional representation (10). The real numbers represented as the final sum of (10) are called the Since the sum of (10) is reduced to the sum of (3) for the case The expression of (10) divides all real numbers into two parts namely the constructive "golden" It is clear that all the golden It means that all real numbers of the kind of (the powers of the golden = 100,101. For the cases when the parameter ð in the expression (10) is between 1 and ¥ (1 £
Thus, the new number definitions given by (10) for the cases 1 £ But if we have new number definitions then each such definition corresponds to the new number theory! But then there exists the infinite number of the "theories of numbers" and the "classical theory of numbers" based on the "Euclidean definition" of (2) is the "degenerated" case of the general theory of numbers based on the general definition of (10). And for the example of Bergman's notation (5) we could be convinced, that the new number theories can generate the new and extremely interesting properties of numbers, in particular, the "natural" (in traditional sense) numbers. And these "exotic" properties (such as the It is important to note, that the new number definitions assigned by (10) is the essential element of the Harmony Mathematics concept and we will tell about it on the following page of our Museum. Follow us! |