Replies (2)

• @FishFace@ioc.exchange 2026-08-02 13:10

  @bemmesr@mathstodon.xyz I hadn't realised you were the same person as was asking about partial fractions :) That kind of thread truly makes me despair, as it's a high school topic that a high school teacher *should* be able to explain :/ There's a curiosity that he is very focused on linking the same topics again and again. Whether this is just to draw people to his own threads, or because those are just the topics he brings everything back to, I'm not sure. But in that case, your question *was not about* pronumerals so it's quite weird that that's what he brought up. In unrelated maths topics he relentlessly brings up order-of-operations as an example. My strong suspicion is that it's an unhealthy obsession that he can't let go of... and you know, I can't criticise him too heavily for that, can I? The positive thing that came out of my past interactions with him is that I've been motivated to watch out for opportunities to explain maths. Cajori is a source that SmartmanApps has read a lot (but not all of it - there's actually a fantastic section on factorials which points out several places where expressions like "n · n - 1 · n - 2 · ..." were interpreted as n! rather than n^2 - n - 2n - 3n etc. I love this because it's a situation where the intended interpretation is obvious even though it's not strictly in accordance with the rules as written.) The sentence you quoted has a reference to Lennes' textbook. As far as I can work out, the textbook never makes this explicit, but it can be determined from examples. In Example 155 on p212 of the book (https://archive.org/details/highschoolalgeb01lenngoog/page/212/mode/2up?ref=ol&q=order) you can see that he is performing all multiplications before the divisions. SmartmanApps is on the record as saying this textbook is just wrong, so it won't convince him. When I challenged him that he simultaneously thinks "all textbooks agree" and "Lennes' textbook is wrong" I believe this was the point where he said that, since his sentence was in the present tense, he had not meant to include Lennes. I never did work out exactly what timespan of publication he includes under "all textbooks agree." Perhaps it's a vacuous truth, unless at the very moment of utterance, a texbook on the subject is being written. Or printed? I saw he brought up his "e-calcs" thing again. He needs to be reminded that they behave exactly the same as that Casio fx-110!

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• @bemmesr@mathstodon.xyz 2026-08-02 14:59

  @SmartmanApps@dotnet.social I have been looking for the textbooks as you requested. I've found numerous sources which say that there is such a discrepancy, though I have yet to see actual examples of it. Meanwhile, while I continue to search, I'd suggest you read https://math.berkeley.edu/~wu/order5.pdf as it essentially makes the same argument I'm making, perhaps with more clarity and conviction. It's not an uncommon idea. In ‘Mathematics From the Birth of Numbers’ by Jan Gullberg, on page 122 under ‘Operational Precedence’, Gullberg says, ‘we also note the left‐to‐right rule of precedence for multiplication and division: 24/2×6... but as this principle of precedence is not universally respected, we recommend, for the first case: (24/2)×6...’ Now, this has turned out to be somewhat of a research task, so I'll at least give you that there are not *many* textbooks published in the last 150 years which differ on the order of operations, but I won't yet concede that there are *none*, since of course we have no reason to believe this to be the case. I'd also like to add that the reason I say that the order of operations is arbitrary is because arithmetic is essentially just an axiomatic system, and of course the interpretation of a given sentence in such a system depends entirely on your definitions and axioms.

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