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

 S = {1, 1, 1, …} (1)

be the infinite set of the standard line segments of the length 1. Then the natural number N is determined as some line-segment that may be presented as the sum of the standard line segments taken from the set of S, that is,

 N = 1 + 1 + 1 + ... + 1 (N times). (2)

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 A is some mathematical objects, which are given in the following manner:

 (3)

where ai Î {0, 1} and i = 0, ±1, ±2, ±3, ... .

The number definition of (3) has the following geometric interpretation. Let

 B = {2n}, (4)

be the infinite set of the standard line-segments of 2n (n = 0, ±1, ±2, ±3, ...). Then all mathematical objects, which can be represented as the final sum of the standard line-segments of (4) in the form of (3), are called the "constructive" real numbers.

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 "constructive" real numbers represented by the final sum of (3) and "non-constructive" ones, which can not be represented as the final sum of (3). It means that all traditional irrational numbers (for example, p, , the golden ratio, Euler's number of e, etc.) and a part of rational numbers (for example, 2/3, 3/7, etc.) are "non-constructive" within the framework of the number system of (3). Note that each "non-constructive" real number could be represented in the form of (3) approximately and the approximation error D would be decreased as the number of the terms in (3) increases, however D ¹ for the "non-constructive" real numbers.

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

 (5)

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:

 B = {tn} (6)

where n = 0, ±1, ±2, ±3, ..., and the golden proportion degrees are connected by the following identity:

 tn = tn-1 + tn-2. (7)

Then the "constructive" real numbers in sense of (5) are called all mathematical objects, which can be represented by the way of the final sum of (5) consisting of any set of the "standard line segments" taken from (6).

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 generalized golden proportion or the golden ð-proportion (ð = 0, 1, 2, 3, ...) introduced by us above. And we can use "Newton's number definition" to introduce the following unusual number definition.

Let's consider now the infinite set of the standard line-segments based on the golden p-ratio of tp:

 (8)

where n = 0, ±1, ±2, ±3, ...; are the golden p-ratio powers connected by the following identity:

 (9)

The set of (8) generates the following constructive method of the real number representation:

 (10)

where ai Î {0, 1} and i = 0, ±1, ±2, ±3, ... .

Note that first the number definition given by (10) had been introduced by Prof. Stakhov in his article "Golden Proportion in Digital Technology" published in the Journal "Automation and Computer Technology" (1980, No 1, pp. 27-33). In this article the name "Codes of the Golden Proportion" was introduced for positional representation (10). Stakhov's book "Codes of the Golden Proportion" (1984) was devoted to statement of the new theory of positional number systems given by (10).

 Stakhov's book 'Codes of the Golden Proportion' (1984) and the book review written by Prof. Guljaev in the Journal 'Electronic Simulation'

However let's consider now the positional representation (10). The real numbers represented as the final sum of (10) are called the "golden" p-numbers.

Since the sum of (10) is reduced to the sum of (3) for the case p = 0, and also to the sum of (5) for the case p = 1 and to the sum of (2) for the case p = ¥, it means that the expression of (10) is a wide generalization of the classical Euclidean number definition of (2) underlying the classical number theory, the number definition of (3) underlying the basis of constructive mathematics and modern computer science and Bergman's number system of (5), which can be put to the basis of new computers.

The expression of (10) divides all real numbers into two parts namely the constructive "golden" p-numbers, which could be represented as the final sum of (10) and the non-constructive real numbers with respect to the sum of (10).

It is clear that all the golden p-ratio powers of the kind of (i = 0, ±1, ±2, ±3, ...) can be represented as the final sums of (10) and respectively as the final totality of "bits". For example

It means that all real numbers of the kind of (the powers of the golden p-ratios) are the constructive "golden" p-numbers with respect to (10). It follows from the definition of (10) that all real numbers being the sums of the golden p-ratio powers can be represented as the final totality of "bits" with respect to the sum of (10). For example the real number is represented as the following binary code:

= 100,101.

For the cases when the parameter ð in the expression (10) is between 1 and ¥ (1 £ ð < ¥), the number systems of (10) possess the unusual property; namely, their "radices" are some irrational numbers of tð ("golden ð-proportions") being the roots of the following algebraic equation:

xp+1 = xp + 1.

Thus, the new number definitions given by (10) for the cases 1 £ ð < ¥ are the "number systems with irrational radices". Note that first the notion of "number system with an irrational base" was introduced in 1957 by the American mathematician Gorge Bergman. Although Bergman's number system is the partial case of the number system (10) corresponding to ð = 1 but this case is especially important since practical point of view.

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 Z-property) can be found only at their representation in the number system of (5) or (10). And before appearance of Bergman's notation of (5) and its generalization of (10) these properties simply could not be found! And therefore one can suppose that the "Codes of the Golden Proportion", assigned by (10) hides in themselves many new number-theoretic results, which could be found hereafter.

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!