Discovering Productive Numbers
Audience: high-schoolundergraduategraduate
Tags: number-theoryinductionabstract-algebrafactorizationlatticesrecursion
The website is focused on explaining, from scratch, the main ideas of an alternative way of axiomatizing the natural numbers call productive numbers. Beginning with the basics of factorization and logic, the book very gradually builds up to the formal definition of productive numbers. After exploring some examples, it proceeds to give an intuitive proof of the first theorem: an elegant bijection with the natural numbers. The following sections introduce some binary options, a partial order and prove these form a distributive lattice. While the actual content of productive numbers probably not going to be particularly useful to students, the goal of the book is to give an engaging portrayal of how new mathematical concepts are discovered. There is a very deliberate emphasis on intuitive explanations throughout as well as some personal anecdotes about how and why certain decisions were made. Many curious students are put off by the strict formality of mathematical writing, leading them to believe they could never understand things in such a way (or at least I have met many such people). My greatest hope for this book is that the reader comes away thinking “that’s so easy! I could have thought of that”
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I picked up the graph representation of number based on their prime factors. Two logically consistent operators (and the two locations you thanked me for reading all of it). I appreciate the code but it took a very long time to load.
Very well written article exploring an interesting idea very systematically and clearly
Very new and interesting topic.
The writting itself is a bit over the place, some simple ideas are explained while more complex ones are not (Basic set operations are explained but Q^+ being a group is not). For the level it is presented (undergrad / grad) there is too long of an introduction, which does not seem too relevant even to the topic itself.
The book goes into an astonishing amount of detail, as a math undergraduate who recently had a discrete math course, this has broadened my horizons for where and what kind of structures can emerge. I definitely need to read through the whole book some more times to completely appreciate the material.
The motivation for productive numbers was amazing, but then it was dropped in favour of the formal definition; an explanation of why equation 14 is the correct thing to do from our intuition would be nice. Cool niche investigation though.
This book and webpage is so well-ordered. I’ve bookmarked it, and aim to comb through the details you’ve thoughtfully paced and placed on productive numbers. I’m glad I found your content, I enjoyed your rant and encouragement. So I hope you decide to make more content. Your Ranking score I’ve rated is an average of these individual scores, good luck: Motivation: 9 Clarity: 9 Novelty: 5 Memorability: 8