The following trivial identity for the golden ratio powers underlies the mirror-symmetrical multiplication:
The rule of the mirror-symmetrical multiplication of two single digits is given in Table 1.
The multiplication is performed in the "direct" code. The general algorithm of the two multi-digit mirror-symmetrical number multiplication is reduced to forming the partial products in accordance with Table 1 and their addition in accordance with the rule of the mirror-symmetrical addition. For example let's multiply the negative number - 6 =`1 1 0 1, 0`1 by the positive number 2 = 1`1, 1:
The multiplication result in the above-considered example is formed as the sum of the three partial products. The first partial product 1 0, 1 0`1 is the result of the multiplication of the mirror-symmetrical number - 6 =`1 0 1, 0`1 by the lowest positive unit of the mirror-symmetrical number 2 = 1`1, 1, the second partial product 1 0`1, 0 1 is the result of the multiplication of the same number - 6 =`1 0 1, 0`1 by the middle negative unit of the number 2 = 1`1, 1 and finely the third partial product 1 0 1, 0`1 is the result of the multiplication of the same number - 6 =`1 0 1, 0`1 by the higher positive unit of the number 2 = 1`1, 1.
Note that the product -12 =`1 1 0`1, 0 1`1 preserves the property of mirror symmetry. As its higher digit is the negative unit `1 it follows that the product is the negative number.
The mirror-symmetrical division is performed in accordance with the rule of the division in the classical ternary symmetrical number system. The general algorithm of the mirror-symmetrical division is reduced to the sequential subtraction of the shifted divisor, which is multiplied by the next ternary numeral of the quotient. Note that at the division of the integers N1 : N2 the result of the division is expressed through two integer numbers, the quotient Q of the division and the rest R of the division. But the numbers of Q and R are integers always and this means that the division results Q and R are controlled according to the property of mirror symmetry.
Thus we have discovered the highly interesting number system allowing representing numbers and performing all arithmetical operations in the "direct" code. And the most important that all arithmetical transformations over numbers are controlled according to the property of the mirror symmetry. At the next pages of our Museum we will tell about technical realization of the pipeline mirror-symmetrical adder. Follow us!