> why is the letter m used when talked about an angle?

This isn’t the best example, because a simple Google search gives you immediate results for that as well, and arguably more nuanced ones. You’ll learn, for example, that m is also the standard letter for slope, which may or may not be related.

Also, if a teacher finds that question “obscure nonsense”, I’d argue that they are a bad teacher.

Hmm, it's not the best example yea. It's just that ChatGPT understands math notation a lot better than Google. If needed, you can always let ChatGPT do a Bing search.

You'd love a very old book named Mathematics for the Million. The author was a committed Marxist and an Engineer, and due to this combination, it places a strong emphasis on demystifying concepts, the historical context of mathematical ideas and its practical application.

Of course, some of it is a bit strange for modern audience, the way lots of things in calculus are taught nowadays is completely different. If you are looking for rigorous formal proofs, you won't find them on this book.

But it is a quite funny and quirk book that gives you a different perspective on math concepts and will satisfy your hunger for knowing the people and the historical context behind it.

Is it? I didn't bother to check as it wasn't too relevant. I wasn't stuck anymore. Worst-case it's a memnonic that is untrue. Best case: it is. For actual derivations I check the work.

This is what it wrote by the way.

I posted an image and asked:

Why is this m

m for angle mangle?

ChatGPT:

In geometry we often need to distinguish the angle itself (a geometric object) from the numerical size of that angle (a real-number measurement, usually in degrees or radians). To keep the two ideas separate we put a little “m” in front of the angle symbol:

m∠B reads as “the measure of angle B.”

* ∠B refers to the actual wedge-shaped region determined by the two rays that meet at vertex .

* m∠B is the single number—e.g. \90∘—that tells you how “wide” that wedge is.

So in the worked example the author writes

m∠A + m∠B + m∠C + m∠D = 360∘

because they are adding the sizes of the four interior angles of a quadrilateral, not the angles themselves.

If you dropped the m you’d be adding geometric objects, which doesn’t make sense; the m reminds the reader that what’s being added are real-number measures. (Some textbooks use ∣∠B∣ or simply write the letter with a bar over it, but “m∠” is by far the most common in U.S. and many international curricula.)

So the mysterious “m” isn’t another variable—just a shorthand for “measure of.”

---

Edit: this took quite some work to write well as ChatGPT answers don't copy/paste neatly into HN. So I won't be doing this for the other example.

ChatGPT is right, although I'm not sure how historical the notation is.

∠ is traditionally a function from points to axiomatic geometric objects. ∠ABC is the angle at B oriented so that we start at A, go to B, then to C.

Your text seems to be using ∠ either as a kind of type annotation (indicating by ∠B that B is an angle) or (perhaps more likely) is just suppressing the other letters in the triangle and is short for something like ∠ABC.

Since ∠B is an axiomatic Euclidean object, it has no particular relation to the real numbers. m is an operator or function that maps axiomatic angles to real numbers in such a way that the calculations with real numbers provide a model for the Euclidean geometry. Why call it m? I'm not aware of it being historical, but almost certainly it comes from measure, like the μ in measure theory.

Obviously ∠ is a graphical depiction of an angle, and my guess is it probably evolved as a shorthand from the more explicit diagrams in Euclid.

Traditionally angles are named with variables from the beginning of the Greek alphabet: α, β, γ. Then we skip to θ presumably to avoid the Greek letters that look nearly identical to Roman letters.

"I'm not sure how historical the notation is."

I conflated this with another ChatGPT conversation where it gave 3 possible historical sources for another symbol that I fell over and then had trouble proceeding.

See slope: https://en.wikipedia.org/wiki/Slope

It isn't customarily used for angles (those get Greek letters).

The m stands for mystery.

Edit: ah, but I see that this prefixed m for "measure" is also used sometimes. It appears at https://en.wikipedia.org/wiki/Angle#Combining_angle_pairs though I'm not sure why it's necessary. Maybe because you want to clarify absolute magnitude is meant, to avoid adding negative values.

yes, it's the equivalent of |AB| for a line segment and is not uncommon in high school maths texts in some parts of the world

I thought it was a different kind of nonsense, but it still has a subtle error. Goes to show even more how risky it is to repeat LLM answers as factual.

Could you point out what you mean? It's really hard to follow you. You say it's nonsense but it is not clear why. Then I write down a response that took me quite a while to format correctly (and to look up) and you then say "I thought it was a different kind of nonsense." Then you say it still has a subtle error.

It is really hard to follow you if you don't explain yourself.

I'm not saying it's factual. The reason I showed that answer was simply to verify to see if it was what you thought it was (hence I asked "is it?"). It turns out that it wasn't fully.

> This is a great example of why using LLMs to learn is perilous. This is nonsense.

These type of answers from teachers, co-students, web communities, blogs etc. are – I would assume – why people ask LLMs in the first place.

It is a problem that people who are unwilling to perform some basic research resort to 'learning' the output of LLMs. No one is entitled to answers.

"No one is entitled to answers" feels very definitive, defeating and tiring. Especially because you don't explain your own thought process.

Could you please assume a good faith discussion?

I have. I was criticized for pointing out spurious nonsense in LLM slop by someone who claimed people wouldn't have to resort to it if other people made an effort to explain things better.

But I don't believe anyone is entitled to an explanation. I find things out by looking up books and testing things. Any explanation someone deigns to give me is a bonus and doubted until corroborated.

I don't know why anyone would think they are owed a custom explanation for their specific questions and thinking like that will get you in trouble when you come to depend on what anyone (or anything) is willing to chew up for you.

Maybe I was terse but I don't think I was rude or illogical.

> I was criticized for pointing out spurious nonsense in LLM slop

I can see that you experience it as such but I think it's more of a spectrum. Often times, LLMs give good answers. Often enough times they don't. One needs to keep that in mind. In my example, given that it was just a symbol, all I needed was knowledge at the level of a memnonic which would, on average, at least somewhat also point directionally to the truth. But that's a bonus. I could make up a memnonic myself, but I like having that bonus.

Given that ChatGPT is directionally towards the truth, but not fully (on average), I'd need to test it or verify it if I want a better level of knowledge than that. If that's the case, then ChatGPT is basically acts as a sort of cache as it's quicker to ask a question to ChatGPT than to research on one's own. One can experience a cache hit or cache miss. Such a thing will happen in the verification stage. Specifically, for math this is quicker, in my experience.

But anyways that's my experience. Your experience is that it's spurious nonsense slop. And I suppose you therefore find it a problem. I don't see the issue as there are different levels of knowledge and different time commitments you need to give to them. A lot of my knowledge is based on trust anyway and sometimes it's broken (e.g. replication crisis in psychology, I felt betrayed having studied the field).

> I don't know why anyone would think they are owed a custom explanation for their specific questions

I'm not sure if anyone said anything like it. Regardless of that, the need still exists. People will still act on that need. I suspect you see that as a problem. I'm neutral on it.

> thinking like that will get you in trouble when you come to depend on what anyone (or anything) is willing to chew up for you.

IMO teaching and learning is a 2 way street. It's the teacher's job to explain it well enough. It's the student's job to do their best to understand it. Math Academy offers exercises and explanations. Sometimes I find their explanations a bit lacking. So I use other sources to augment it.

> Maybe I was terse but I don't think I was rude or illogical.

Reading/writing text is tough, which is why I stated how I felt. It'd probably have been easier in an actual conversation. I didn't mean to imply you were being rude or illogical.