"Golden" arithmetic

And now let's try to develop the "golden" arithmetic. Let's remember now the number system with an irrational bases (Bergman's number system):

 (1)

where ai is the binary numeral of the i-th digit in the number system (1); ti is the weight of the i-th digit; (the "golden proportion") is the base of the number system (1).

"Golden" addition and subtraction. There exists the following fundamental identity connecting powers of the golden ratio:

 ti = ti-1 + ti-2 (2)

Note that the expression (2) connecting the digit "weights" in the "golden" representation (1) is similar to the recurrent correlation for Fibonacci numbers. This means that rules of the "golden" addition and subtraction are coincident mainly with the similar rules for the Fibonacci addition and subtraction.

"Golden" multiplication. However, the golden ratio powers possess the following property, which is very important for the "golden" multiplication and division:

tn ´ tm = tn+m.

It follows from this property that the "golden" multiplication is performed in accordance with the following multiplication table:

 Table 1. 0 ´ 0 = 0 0 ´ 1 = 0 1 ´ 0 = 0 1 ´ 1 = 1
We can see that the given table coincides with the multiplication table of the classical binary arithmetic. As this takes place the "golden" multiplication is reduced to the classical binary multiplication, i.e. to the following rules:
1. To form the partial products in accordance with Table 1.
2. To sum the partial products in accordance with the rule of the "golden" addition.

Let's demonstrate the "golden" multiplication by using the following example. To multiply the "golden" fractions of A = 0, 0 1 0 0 1 0 and B = 0, 0 0 1 0 1 0.

Solution:

1. To represent the "golden" fractions of A = 0, 0 1 0 0 1 0 and B = 0, 0 0 1 0 1 0 in the form with floating point:
 A = 0 1 0 0 1 0 ´ t -6 B = 0 0 1 0 1 0 ´ t -6
2. To perform the "golden" multiplication of the A- and B-fraction mantissas:

3. To reduce the product to the minimal form

0 0 0 1 0 1 1 0 1 0 0 = 0 0 1 0 0 0 0 0 1 0 0

The result of the "golden" multiplication is the following:

A ´ B = 0 0 1 0 0 0 0 0 1 0 0 ´ t -12

"Golden" division. As is known that the classical binary division is reduced to the fulfillment of two elementary operations namely number comparison and subtraction. One may show that these operations underlie the basis of the "golden" division. As the number comparison is realized over the "golden" numbers represented in the minimal form it follows from here that a peculiarity of the "golden" division consists of the fact that the intermediate results are reduced to the minimal form on each stage of the "golden" division.