# Binary Numeration System with Alternating Signed Digits and Its Graph Theoretical Relationship
## 1 Introduction  
Positional number systems are fundamental in mathematics (see [1]). The development and spread of the number system with base 10 is a major achievement of ancient/medieval mathematics. The advent of computers gave special importance to the number system with base 2. These number systems encode natural numbers as a finite sequence of digits. The positions in a sequence of digits are local values weighted by the digits. The sum of these weighted values is a number, coded by the representation. In the number system with base 2, the digits are 0/1 (they are called bits), and the values of the positions are \(\ldots\), 16, 8, 4, 2, 1. This basic theorem of the binary system can be formulated as follows. **Theorem 1**. : _Every natural number can be written uniquely as the sum of distinct powers of two, where the order of the members of the sum does not matter._  
First note that the empty sum is also considered; its value is 0. The value of a sum with one term is the value of the only term. Second, we eliminate the freedom in the order of the terms by listing them following increasing order, that is, writing down the number \(n\)  
\[n=2^{\kappa}+2^{\beta}+\ldots+2^{\omega},\quad\text{where}\quad\alpha>\beta> \ldots>\omega.\]  
The digits 0 do not contribute to the value; hence, we do not include them in the sum. The ternary number system can be introduced in a similar way. In the number system with base 3, the digits are 0, 1, and 2, and the local values are \(\ldots\), 81, 27, 9, 3, 1. The decimal number system can be discussed similarly. All number systems are suitable for the unique encoding of natural numbers. If we want to consider integers, the usual method is to introduce signed numbers. Following a +/ - sign, we encode the absolute value of a number. The mathematical beauty is somewhat compromised: +0 and \(-0\) are two codes for the same number. It helps if you use negative digits. Then, the positive/negative numbers can be handled homogeneously. This option was first published by [PERSON] [2]. Later, independently, [PERSON] [3] also discovered this possibility. A survey on the subject was published in Kvant [4]. If we have digits \(\{-1,0,1\}\) (or more generally, a subset of the integers that is closed under taking opposite), then the sign change will be the sign changes of the digits. This advantage was also noticed by [PERSON] [5]. When designing multiple computer architectures, this advantage was considered [6; 7; 8; 9; 10; 11; 12; 13]. The digits \(-1\), \(0\), and \(1\) can be called _trits_. Their technical realization is possible. Two early architectures build on this [12; 13; 14]. Many have looked in more depth at the ternary computers [15; 16]. Currently, binary architecture--thanks to its explosive development--is the single dominant technology. We highlight two possibilities where integers are represented by a finite sequence of \(-1/0/1\) digits. The first is the balanced ternary system. In this number system, the digits \(-1\), \(0\), \(1\) are powers of \(3\) of the local values [17]. The advantage of this number system is that all integers can be represented and the representation is unique. [PERSON]'s well-known statement, in his excellent book [18], describes this system as \"Perhaps the pretiest number system of all is the balanced ternary notation,...\"  
The second is the negabinary. Here, the number system is based on \(-2\). The local values are the powers of \(-2\), and the digits \(0\) and \(1\). We can expand \((-2)^{\alpha}\) and consider it as \(2^{\alpha}\) with a possible sign. For further systems, see [19; 20; 21; 22]. In this note, we propose several new numeral systems based on our main theorem, Theorem 2. We present six options to produce a scheme for a unique representation of integers. In spite of their simplicity, I was able to find only one of them in the literature [20]. We call the new number system the alternating binary number system.