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by earthboundkid
1028 days ago
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Yes, I clicked the link and read that. It wasn’t actually very relevant and did not help me understand the original article. I eventually figured it out on my own by, yes, examining the two examples they gave and proving to myself how it works. The article has a bad explanation. Untouchable numbers are a very simple idea. Any elementary school student who knows about divisors and prime numbers can have it explained to them in about ten minutes. The Wikipedia page for the article should be pitched so that a layman can understand it. I understand that if you’re writing a cookbook, you have to decide if you’re pitching it first time chefs who need “how to boil water” explained to them or professional chefs who just need references for some ratios. What serves one audience wastes the time of the other. If you want to know how init works, you probably already know what Unix is. If you’re looking up a particular species of mushrooms, you probably know how genii and species work. Untouchable numbers is a simple concept. It should not be pitched at professionals first. It should start with an explanation that a middle schooler can understand and then move on to advanced explanations in later sections. |
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• That there are no concrete examples on the page: this is not true, the page has a couple of concrete examples (the numbers 4 and 5), and I added a third one (the number 6) earlier today.
• That “proper divisor” and “sum of all the proper divisors” aren't explained: but these are explained on the pages (divisor and aliquot sum respectively) linked from the first two sentences of the article, in keeping with the nature of an encyclopedia.
So what is the problem exactly? What explanation/wording would you suggest? It may be easier to understand the general statement “Math wiki pages are so bad”, if there was an answer to “How would you suggest improving it?” for this specific wiki page.
[I have a theory, that this has to do with “depth”: to understand this concept one needs to understand, in order, (1) the idea of divisors, (2) the definition of a proper divisor of N as any divisor of N other than N itself, (3) the idea of the sum of all the proper divisors of N, (4) finally, the idea of an untouchable number, as a number not achievable as such a sum for any N. Each step is individually easy, but because of this ordering requirement, overall there is “depth”. In fact I actually think that any middle-schooler would have no trouble learning the concept from Wikipedia as it stands, if they read the relevant sentences from the relevant Wikipedia articles in order, e.g. the following from https://en.wikipedia.org/wiki/Aliquot_sum seems clear enough for (3):
> For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
It's just that no single page explains all four ideas in order. So the middle-schooler doesn't need to be an experienced mathematician, just experienced at reading Wikipedia and following links. (Arguably this is part of mathematical maturity, i.e. understanding definitions as needed until you can understand the original definition.)
The process you describe: “I eventually figured it out on my own by, yes, examining the two examples they gave and proving to myself how it works” seems to suggest things are working as intended; it's just the nature of mathematical concepts that they require a little bit of thought! As you say, “Any elementary school student who knows about divisors and prime numbers can have it explained to them in about ten minutes.” (I agree, and one doesn't even need prime numbers): about 5–10 minutes is how long it takes, but readers complain about mathematical articles because it's uncomfortable that reading a handful of sentences should require several minutes to understand, when actually this is inevitable. This reminds me of the “monad tutorial fallacy” https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti... — after a bit of struggle to understand, when it eventually clicks, the reader says “ah it's so simple” and thinks that the earlier explanations were bad. But that's just my theory, and if you have a concrete suggestion for how the article could be improved, that may be revealing.]