TL;DR
This paper offers a comprehensive, well-structured exposition of the properties of fractions, highlighting overlooked propositions and providing historical context on generating the set of all fractions.
Contribution
It introduces a systematic and properly ordered presentation of fraction properties, addressing gaps and misconceptions in existing number theory literature.
Findings
Identifies missing fundamental propositions in fraction theory
Provides a historical overview of generating all fractions
Proposes a clearer, more systematic exposition of fraction concepts
Abstract
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set. Some concepts are being introduced without motivation, some proofs are unnecessarily artificial, and almost invariably both seem to be understood as related to specific structures rather than to the set of fractions in general. Surprisingly, there are essential propositions whose very statement seem to be missing in the number theory literature. This article aims at improving on the said state of affairs by proposing a general and properly ordered exposition of concepts and statements about them. In addition, historical remarks are made on generating the set of all fractions -- a much older discovery than it is widely believed.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · History and Theory of Mathematics · Mathematical and Theoretical Analysis