Algebraic properties of the Golden Proportion

What is the "miracle" of the nature and mathematics, the concern to which not only does not wither in time, and on the contrary - increases with each century? For the answer this question we tender to the visitor of our Museum to strain all mathematical knowledge and to be loaded in the world of mathematics; only by such way you can enjoy by wonderful mathematical properties of the Golden Proportion and through these mathematical properties understand and evaluate all beauty and harmony of the Golden Proportion

Let's begin from algebraic properties of the Golden Proportion. It follows from the equation of the Golden Proportion

(1)

the next surprising property of the Golden Proportion. If we substitute the root t (the Golden Proportion) instead of x in the equation (1), we will get the following identity for the Golden Proportion:

(2)

We can be convinced, that the identity (2) is true. For this purpose it is necessary to execute some elementary mathematical manipulations above the left-hand and right-hand parts of the identity (2) and then to show that they coincide.

Really, we have for the right-hand part:

On the other hand,

It means that the identity (2) is true.

The identity (2) can be represented in the following form:

(3-a)
or
(3-b)

Let's analyze, for example, the identity (3-b). It is known, that each number of a has the inverse to it number of 1/. For example, the fraction of 0,1 is the number, inverse to 10. The conventional algorithm for obtaining of the inverse number of 1/a consists of division of the number 1 by the initial number a. This is a rather complicated procedure. Let's try, for example, by division to get the number inverse to the number of a = 357821,093572. This can be made only with the help of the modern computer.

Let's consider the Golden Proportion How to get from it the inverse number of 1/t? The expression (3-b) gives a very simple answer to this question. For this purpose it is enough to subtract the number of 1 from the Golden Proportion t. Really, on the one hand we have:

On the other hand, as follows from (3-b), the inverse number of 1/t can be got from t by the following way:

Let's prove now one more surprising property of the Golden Proportion being grounded on the identity (3). If we substitute to the right-hand part of (3) instead t its value assigned (3), we will come to representation of t in the form of the following "multi-storied" fraction:

If we continue such substitution many times in the right-hand part of (3-b) as outcome we will get the "multi-storied" fraction with infinite number of "levels":

(4)

The representation (4) is called in mathematics as "continuous" or "chain" fraction. Let's note that the theory of "chain" fractions is one of the relevant parts of modern mathematics.

Let's consider now once again the identity (2). It can be represented in the following form:

(5)

If we substitute now in the right-hand part of the identity (5) instead of t its expression assigned (5), we will get the following representation for t:

(6)

If we substitute in the right-hand part of the identity (6) again the expression (5) instead of t and we will repeat this operation an infinite number of times, we will get one more "remarkable representation" of the Golden Proportion in "radicals":

(7)

Each mathematician intuitively aims to express the mathematical result in the simplest, compact form. And if he found such form, this delivers to him "aesthetic enjoying". In this respect (tendency to "aesthetic" expression of mathematical outcomes) the mathematical creativity is similar to creativity of the composer or poet, the main problem of whom consists of obtaining the perfect musical or poetic forms giving us "aesthetic pleasure". Let's note, that the formulas (4) and (7) produce also "aesthetic enjoying " and invoke a feeling of a rhythm and harmony, when we begin to think above infinite repeatability of the same simple mathematical elements in the formulas for t assigned (4), (7).