Ternary mirror-symmetrical representation

Conversion rule. Let's consider the representation of natural number N in Bergman's or "golden" number system:

(1)

The abridged notation of (1) has the following form:

N = anan-1 ... a1a0,a-1a-2 ... a-m.(2)

We will use so-called "minimal form" of (1). This means that each binary unit ak = 1 in the "golden" representation (2) would be "enclosed" with the two next binary zeros ak-1 = ak+1 = 0.

Let's consider now the following identity for the powers of the golden ratio:

tk = tk+1 - tk-1.(3)

The expression (3) has the following code interpretation:

(4)

where `1 is a negative unit, i.e. `1 = - 1. It follows from (4) that the positive binary 1 of the k-th digit is transformed into two 1's, the positive 1 of the (k + 1)-th digit and the negative `1 of the (k - 1)-th digit.

The code transformation (4) might be used for conversion of the "minimal form" of (2) into the ternary "golden" representation.

Let's consider the "golden" representation of number 5:

(5)

The representation (5) is abridged notation of the following sum:

Let's convert the "minimal form" of (5) into the ternary "golden" representation. To do this we apply the code transformation (4) simultaneously to all digits being the binary 1's and having the odd indices (k = 2m + 1). We can see that the transformation (4) could be applied for the situation of (5) only to the 3rd and (-1)-th digits being the binary 1's. As the result of such transformation of (5) we get the following ternary "golden" representation of number 5:

(6)

We can see from (6) that all digits having even indices are equal 0 identically but the digits with odd indices take the ternary values from the set {`1, 0, 1}. This means that all digits with the even indices are "non-informative" because their values are equal to 0 identically. Omitting in (6) all the "non-informative" digits we get the following ternary "golden" representation of the initial number N:

(7)

where b2i is the ternary numeral of the (2i)-th digit.

Let's introduce the following digit enumeration for the ternary "golden" representation (7). Each ternary digit b2i is replaced by the ternary digit ci. As a result of such enumeration we get the expression (7) in the following form:

(8)

where ci is the ternary numeral of the i-th digit; t2i is the weight of the i-th digit; t2 is the base of the number system (8). With regard to the expression (8) the ternary "golden" representation (6) takes the following form:

(9)

The latter has the following numerical interpretation:

Representation of the negative numbers. Like the ternary symmetrical number system the important advantage of the number system (8) is a possibility to represent both positive and negative numbers in the "direct" code. The code of the negative number (-N) is got from the ternary "golden" representation of the initial number N by means of application of the rule of the "ternary inversion":

1 ® `1, 0 ® 0, `1 ® 1.(10)

Applying this rule to the ternary "golden" representation (9) we get the ternary "golden" representation of the negative number (-5):

Mirror-symmetrical property. Considering the ternary "golden" representation (9) we can see that the left-hand part (1`1) of (9) is mirror-symmetrical to its right-hand part (`1 1) regarding to the 0-th digit. It is proved that this property of "mirror symmetry" is the fundamental integers property arising at their representations in the ternary "golden" system (8). Table 1 demonstrates this property for some natural numbers.

Table 1.

i3210-1-2-3
t6t4t2t0t -2t -4t -6
000 0 0. 000
100 0 1. 000
200 1`1. 100
300 1 0. 100
400 1 1. 100
501`1 1.`110
601 0`1. 010
701 0 0. 010
801 0 1. 010
901 1`1. 110
1001 1 0. 110

Thus thanks to this simple investigation we discovered one more fundamental property of integers, the property of "mirror symmetry", which appears at their representations in the ternary "golden" system" (8). That is why the ternary "golden" system" (8) is called the "Ternary Mirror-Symmetrical Number System".

The base of the ternary mirror-symmetrical number system. It follows from (8) that the base of the ternary "golden" representation (8) is the square of the "golden ratio":

This means that the number system (8) belongs to the number systems with irrational bases.

The base of the number system (8) has the following traditional representation:

t2 = 10.

Now let's discuss the result obtained above. We discovered the highly unusual number system. First of all this number system is the ternary symmetrical number system with the ternary numerals {1, 0 and -1}. Secondly, it has the unusual base, the square of the golden ratio. But the most unexpected result is the property of mirror symmetry arising at the representations of integers.

What practical importance has the property discovered above? It is clear that this property is the fundamental distinctive peculiarity of integers (positive and negative) and we can use the property of mirror symmetry in computers if we represent integers in this form. But the most unexpected result is the fact that this property can by used in computers for control of arithmetical operations. We will tell about this at the next pages of our Museum. Follow us!