Mirror-symmetrical addition and subtraction The following identities for the golden ratio powers underlie the mirror-symmetrical addition:
where Let's prove the identity (1). In fact, we can do the following transformations over the right-hand part of (1): t The identity (1) is proved. The identities (2) and (3) follow from (1). The identity (1) is the mathematical basis for the mirror-symmetrical addition of two single-digit ternary digits and gives the rule of the carry formation (Table 1).
The principal peculiarity of Table 1 is the addition rule of two ternary 1's with equal signs, i.e. We can see that at the mirror-symmetrical addition of the ternary 1's with the same sign there arises the intermediate sum Table 1 describes operating the simplest ternary adder called the
As the carry from the Let's describe the logical operating the mirror-symmetrical full single-digit adder 4å. Note first of all that the number of all the possible 4-digit ternary input combinations of the mirror-symmetrical full adder equals 3
The sum (4) takes the values from the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}. The operation rule of the mirror-symmetrical full adder 4å consists of the following. The adder forms the output ternary code combination -4 = `1`1; -3 =`1 0; -2 =`1 1; -1 = 0`1; 0 = 0 0; 1 = 0 1; 2 = 1`1; 3 = 1 0; 4 = 1 1. The lower digit of such 2-digit ternary representation is the value of the intermediate sum The multi-digit combinative mirror symmetrical adder realizing the addition of two (2
As an example let's consider the addition of two numbers 5 + 10 in the ternary mirror-symmetrical number system: Note the sign « marks the process of carry spreading. It is important to stress that the result of the addition (the number 15) is represented in the "mirror-symmetrical form"! As was noted above the important advantage of the ternary mirror-symmetrical number system is a possibility to sum all integers (positive and negative) in the "direct" code, i.e. without using notions of the inverse and supplementary codes. As an example let's consider the addition of the negative number of (-24) with the positive number of 15: It is important to stress that the result of the addition (the number -9) is represented in the "mirror-symmetrical form"! And how is about the subtraction? The subtraction of two mirror-symmetrical numbers N
It follows from (5) that before the subtraction it is necessary to take the ternary inversion of the number of Let's discuss the result obtained above. We have discovered the highly unusual method of the number addition-subtraction. Firstly these arithmetical operations are performed in the "direct" code! And we do not have a necessity to compare numbers before the subtraction! And we can forget about the inverse and supplementary codes for representation of negative numbers! Secondly these arithmetical operations are controlled according to the property of mirror symmetry! This means that the property of mirror symmetry is invariant about the addition and subtraction. However we do not know yet: is the property of mirror symmetry the invariant about the multiplication and division? You will know this at the next page of our Museum. Follow us! |