Ternary mirror-symmetrical representation
The abridged notation of (1) has the following form:
We will use so-called "minimal form" of (1). This means that each binary unit Let's consider now the following identity for the powers of the golden ratio:
The expression (3) has the following code interpretation:
where `1 is a negative unit, i.e. `1 = - 1. It follows from (4) that the positive binary 1 of the The code transformation (4) might be used for conversion of the "minimal form" of (2) into the Let's consider the "golden" representation of number 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 (
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
where Let's introduce the following digit enumeration for the ternary "golden" representation (7). Each ternary digit
where
The latter has the following numerical interpretation:
Applying this rule to the ternary "golden" representation (9) we get the ternary "golden" representation of the negative number (-5):
Thus thanks to this simple investigation we discovered one more fundamental property of integers, the
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: t 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 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! |