Mirror-symmetrical addition and subtraction

The following identities for the golden ratio powers underlie the mirror-symmetrical addition:

2t2k = t2(k+1) - t2k + t2(k-1);(1)
3t2k = t2(k+1) + 0 + t2(k-1);(2)
4t2k = t2(k+1) + t2k + t2(k-1),(3)

where k = 0,  1,  2,  3, ... .

Let's prove the identity (1). In fact, we can do the following transformations over the right-hand part of (1):

t2(k+1) - t2k + t2(k-1) = t2k+1 + t2k - t2k + t2(k-1) = t2k + t2k-1 + t2k-2 = t2k + t2k = 2t2k.

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).

Table 1.

`101
`1`1 1`1`10
 0`1 01
 1 0 1`1 1`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 sk equal to the unit with the opposite sign and the carry ck equal to the unit with the same sign. However the carry from the k-th digit spreads simultaneously to the next two digits, namely to the next left-hand, i.e. (k + 1)-th digit, and to the next right-hand, i.e. (k - 1)-th digit.

Table 1 describes operating the simplest ternary adder called the single-digit half-adder. The latter is the combinative logical circuit having two ternary inputs ak and bk and two ternary outputs sk and ck and functioning in accordance with Table 1 (Fig.1-a).

Mirror-symmetrical single-digit adders
Figure 1. Mirror-symmetrical single-digit adders: (a) a half-adder; (b) a full adder.

As the carry from the k-th digit spreads to the left-hand and to the right-hand digits this means that the full mirror-symmetrical single-digit adder has to have two inputs for the carries entering from the (k - 1)-th and (k + 1)-th digits into k-th digit. Thus the full mirror-symmetrical single-digit adder is the combinative logical circuit having four ternary inputs and two ternary outputs (Fig.1-b). Let's mark through 2å the mirror-symmetrical single-digit half-adder with two inputs and through 4å the mirror-symmetrical single-digit full adder with four inputs.

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 34 = 81. The values of the output variables sk and ck are some discrete functions of the algebraic sum S of the input ternary variables ak, bk, ck-1, ck+1, i.e.

S = ak + bk + ck-1 + ck+1.(4)

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 cksk in accordance with the value of the sum (4), i.e.

-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 sk and the higher digit is the value of the carry ck, which spreads to the next (to the left-hand and to the right-hand) digits.

The multi-digit combinative mirror symmetrical adder realizing the addition of two (2m + 1)-digit mirror-symmetrical numbers is the combinative logical circuit consisting of the 2m + 1 ternary mirror-symmetrical full single-digit adders of 4å (Fig. 2).

Ternary mirror-symmetrical multi-digit adder
Figure 2. Ternary mirror-symmetrical multi-digit adder.

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 N1 - N2 is reduced to the mirror-symmetrical addition if we represent their difference in the following form:

N1 - N2 = N1 + (-N2).(5)

It follows from (5) that before the subtraction it is necessary to take the ternary inversion of the number of N2 and then to perform the mirror-symmetrical addition.

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!