Wonderful properties of Bergman's number system

The mathematical expression of Bergman's number system has the following form:

 (1)

where A is some real number and ai are the binary numeral, 0 or 1, i = 0, ±1, ±2, ±3, ti is the weight of the i-th digit, t is the base of number system (1).

At the first glance, there douse not exist any peculiarity of the expression (1) in comparison with all traditional positional number systems but this is only at the first glance. The main peculiarity consists of the fact that Bergman used the irrational number called the "golden section" or the "golden ratio" as the base of his number system (1). Bergman called his number system (1) the "Tau-system".

Let's consider the "Tau System" since computational point of view. Its base t determines all unusual properties of the number system (1). We know from the preceding pages of our Museum the following property of the golden ratio:

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

Let's consider the representations of numbers in the "Tau System" (1). It is clear that the number A notation in the "Tau System" (1) has the following form:

 A = anan-1 ... a1a0, a-1a-2 ... a-m. (3)

We can see that the number notation of A is the binary code combination, which is separated with comma into two parts, the left part anan-1 ... a1a0 corresponding to the "weights" tn, tn-1, ..., t1, t0 = 1 and the right part a-1a-2 ... a-m corresponding to the "weights" with negative powers: t -1, t -2, ..., t -m. Note that the "weights" ti (i =0, ±1, ±2, ±3, ...) are given with the mathematical formula (2).

For instance, let's consider the binary combination of 100101. It is clear that this one represents the real number

 A = 100101 = t5 + t2 + t0. (4)

Using Binet's formula we can get that the number of A given with the expression of (4) is equal to

Note that the number is the irrational one. This means that we represented some irrational number A in the "Tau System" using the code combination 100101 consisted of the finite number of bits!

It is clear that the base of the "Tau System" is represented with traditional way:

But we know from our preceding experience that it is impossible to represent some irrational number by using final totality of numerals. That is why the possibility to represent some irrational numbers (the powers of the "golden ratio" and their sums) in the "Tau System" using the finite number of bits is the first unexpected result, which is in contradiction to our traditional ideas about positional number systems!

There arises a question about representation of natural numbers in the "Tau System". For that let's consider one more the identity (2). On the code level we can interpret this identity as the following code transformation:

 100 = 011. (5)

How we can use the code transformation of (5)? For that we consider the "golden" notation 100101, which represent the number of (4). If we apply to the higher three digits of (4) the code transformation of (5), we can get the new "golden" notation of the number (4) namely:

A = t5 + t2 + t0 = 100101 = 011101 = t4 + t3 + t2 + t0.

Note that for this case the golden ratio power t5 is replaced by the sum of the two next powers t4 + t3 according to the fundamental identity (2). Note that this does not change the initial number of (4). On the code level such transformation corresponds to the following code transformation:

 100 ® 011. (6)

This code transformation is called a "devolution". The back transformation

 011 ® 100. (7)

is called a "convolution".

Thus, numbers have many-valued representation in the "Tau System". This is the second unexpected result following from the "Tau System". Using the above-introduced code transformations of "convolution" and "devolution" one may get two extreme representations of the same number in the "Tau System". For instance, let's consider the code combination 0111111. If we perform in it all possible "convolutions" (7) we will get the first extreme representation called the minimal form:

0111111 = 1001111 = 1010011 = 1010100 ("minimal form").

Note that in the minimal form two binary 1's near do not meet.

Let's consider the code combination 100000, which represents the irrational number Performing in it all possible operations of "devolution" (6) we can get the second extreme representation called the maximal form:

100000 = 0110000 = 0101100 = 0101011 ("maximal form").

Note that in the maximal form two binary 0's near do not meet.

Let's show how one may get all "golden" notations of natural numbers by using the operations of "devolution" and "convolution". We start from the number of 1. It is clear that the number of 1 can be represented through the golden ratio in the following manner:

1 = t0.

But by using the expression of (1) we can represent the number of t0 = 1 as the following:

 1 = t0 = 1,00. (8)

Note that in the "golden" notation of 1,00 the comma separates the 0th digit from the digits with negative indices.

Then, using the "devolution" we can represent the number of (8) as the following:

 t0 = 1,00 = 0,11 = t -1 + t -2. (9)

Now we add the binary numeral 1 to the 0th digit of the "golden" notation 0,11. As result we get the "golden" notation of number 2:

 2 = 1,11. (10)

Applying the operation of the "convolution" to the higher digits of the "golden" notation of (10) we get the new "golden" notation (the "minimal form") of the number of 2:

2 = 10,01 = t1 + t -2.

Adding the binary numeral 1 to the 0th digit of the "golden" notation of 2 we get the "golden" notation of the number of 3:

3 = 11,01.

Applying the operation of the "convolution" to the higher digits of the "golden" notation of the number of 3 we get the new "golden" notation (the "minimal form") of the number of 3:

2 = 100,01 = t2 + t -2.

The "golden" representations of the numbers of 4 and 5 have the following forms:

4 = 101,01 = t2 + t0 + t -2;
5 = 1000,1001 = t3 + t -1 + t -4.

Continuing this process one may get the "golden" notations of all natural numbers in the "Tau System". It means that each natural number could be represented always as the final sum of the "golden ratio" powers! This result is the third unexpected result following from the "Tau System".

And now let's turn back 2,5 millenniums ago and let's imagine the Pythagorean reaction to this statement. The main Pythagorean doctrine claims: "Everything is a Number". This means that natural numbers and their ratios underlay the Universe. But we shoved just now that every natural number can be represented through the "Golden Proportion". It follows from here the new fundamental doctrine, which could be proposed by the Pythagoreans if they knew about our new result: "Everything is the Golden Proportion"!

But the Pythagoreans were shocked one more if they could know that natural numbers possess one more unique mathematical property called Z-property. And we will tell about this property at the next page of our Museum. Follow us!