Binary Numeration System with Alternating Signed Digits and Its Graph Theoretical Relationship

Round 1

Reviewer 1 Report

This is a very nice short paper discussing the binary numeration system with alternating signing digits and connects it with paths on a certain tree-like graph with natural numbers as vertices.

The results are simple but beautiful and I enjoyed reading this paper. One cannot be 100% sure that something similar has not been done before(*) but the results presented here definitely were new for me. The paper is clearly written and mostly can be published as is. There are just several minor remarks:

1) formulation of Theorem 2 is a bit confusing and over-complicated: for one thing, it is very hard to understand what the word "form" is related to, and also why define s_a, s_b, etc only to say immediately after that they must be alternating +1 and -1, why not just
n = s 2^a - s 2^b + s 2^c ....  where s \in {+1, -1} 
or even simpler replace "every non-zero integer" with "every positive integer" and omit s altogether?
2) The formulation of Theorem 5 is a bit confusing for me, too. Since G is a tree, there is just a single n -> 0 walk. So, W_n a set of a single element? In turn, R_n is a set of all binary representations of n, which has two elements (differing in the last 2 digits, as described in the proof of Theorem 2). Surely, there cannot be a bijection between a 2-element set and a 1-element set? 
So, this cannot be true. I tried to reconstruct what the author meant, and it seems a bit tricky:
- it seems that only representations satisfying condition (d) of theorem 3 are counted (others have an end loop in their graph representation);
- then existence of bijection is trivial: there is always bijection between sets of 1 element; (it is still trivial if one things of bijection between set W of all W_n and set R of all R_n, since both of them are countable with index n)
- the statement in the proof is actually about the internal structure of the path W_n (i.e., an order sequence of visited n_i's) and the numbers obtained from R_n by dropping off first digits, and - contrary to the theorem formulation - is not trivial 
The author should rewrite the formulation to fit the proof.
 3) it will be nice to have a sketch of G_0(n) for some small n (maybe, 16 or 32).
4) there is a missing reference in line 39

Also, I wonder if the author knows of similar graph-theoretical interpretations of other numbering systems with not-strictly-positive digits, which he mentions in the introduction.

(*) Indeed, the objects studied here - normal binary system, alternating binary system, and a graph connecting each natural number n to numbers (2^k - n) for all k - are really basic, so they could have been studied (and the relations proven here spotted) at any time in the last, say, 300 years. Let me emphasize that this remark is not aiming to discredit the author - quite the opposite: it is pleasant to see such nice and basic objects in the center of the study. 

I am not a native-speaker, but I found English of this paper perfectly ok.

Author Response

Thank you for the careful work, observations, and advice.

As for the specific questions: (1) I corrected the text. (2) I detailed my interpretation. I hope it is now clear what I mean by the two sets. Both must be of size two. Of course, knowing this, a simple bijection can be established between them. The proof works without reference to the cardinality. Thus, it can be interpreted as the graph-theoretical analog of a fundamental theorem regarding number systems. (3) I inserted a diagram. (4) I corrected the typo in the reference.

Furthermore, I provided an example where the graph-theoretical approach is very useful. I believe additional examples can also be given.

Reviewer 2 Report

The work concerns an interesting topic. Of very great practical importance. However, the author does not refer to the practical (technical) application of the poorly presented idea.

Work, they do not meet the formal (technical) criteria for publication in MDPI.

Not all works from the bibliography are cited in the body of the work. There are also strange quotations, e.g. [?] - line 39

In my opinion, the work is incomprehensible in many places, e.g. 87-124. On the one hand, the analysis is obvious, but is it new? I don't think so.

While reading the article, I have doubts as to what is a scientific contribution and what is a literature review.

The conclusions have been recorded, but do they have any substantive value? In my opinion no.

A very serious drawback of the work is the lack of simple, understandable examples, especially technical ones.

Author Response

I've sorted out the references (crossed out two, cited the rest in the relevant places, and corrected the typo in the misspelled reference).

My research began with the observation of a graph theory proposition. The recognition that this is related to number systems was, for me, an idea, a mathematical creation. My initial reaction was that this must be known. At that point, I started an in-depth investigation, contacted experts, and found no trace of this graph theory fact and its related number systems. In the conclusion section, I described what is available in traces. These are either statements without motivation or comments without proof. 

I added a new section emphasizing the importance of the graph theoretical view. I introduced an alternative version of the Zeckendorf numeral system. It also seems to be new.

The future impact of my results (along with many other publications) is uncertain. Nevertheless, introducing a new number system raises questions (such as natural algorithmic problems). This opens up further research opportunities and could serve as a basis for thesis topics. I also mention that in the case of earlier systems, authors like Donald Knuth and Martin Gardner point out the beauty of the system (see my references). I think there is beauty in the suggested system in my paper.

The introduction of examples (despite the technical simplicity of the article) is a good idea. In the graph theory section, I drew a new diagram, I also gave further examples.

Thank you for the constructive suggestions and honest feedback.

Reviewer 3 Report

Positional number systems were introduced by Arab-Indian mathematicians. Their development and diffusion of the base 10 number system is a major achievement of ancient/medieval mathematics and is fundamental in mathematics. The advent of computers gave particular importance to the base 2 number system. These number systems encode natural numbers as a finite sequence of digits. Positions in a digit sequence are digit-weighted local values. The sum of these weighted values is a number, encoded by the representation. In this essentially summary paper, the Author addresses the problem of representing relative numbers on a digital basis.

The article is well written and satisfies the reader's curiosity. For this reason this papere can be accepted for publication.

Author Response

I appreciate the positive and encouraging opinion.

Round 2

Reviewer 2 Report

1. I believe that the article abstract still does not meet the expected requirements. There is a lack of scientific purpose and practical purpose. Nothing has been written about the test results.

2. Work seems to be better.

But practical (technical) examples are still missing.

I think the article is interesting. But it should be clearly said: I am a practitioner, not a theoretician.

Beauty alone, especially the beauty of numbers, is not sufficient for publication.

Author Response

Thank you for the comments.

I agree that as a practitioner, the utility of the results is questionable. Here, I am not proposing a new algorithm for a new problem that can be tested and compared with existing ones. I am addressing a very old, classic question: how to represent integers. There have been many answers to this question, and new ideas often attract attention with their elegance. However, these new number representations were not sufficiently motivated. The path by which they were created was not clear. The article presents a method that starts from graph theory research and leads to a number representation. The article lies at the intersection of mathematics and algorithms, leaning more towards mathematics than results related to applications. The lack of testing is due to this.