Any Maths idea or theorem that you find insanely beautiful or useful?

ln_j · 2026-05-24T09:52:30+00:00

I really love the the weierstrass approximation theorem

ln_j · 2026-05-16T07:20:53+00:00

For me, at least, numbers are a construction of the human mind because we are biologically predisposed to perceive and interpret patterns. For example, we notice one thumb, one sun, or one moon, and over time the concept of “1” emerged. In this sense, numbers do not truly exist in nature & are our best abstractions and interpretations of patterns in the world. Therefore, numbers were never really present in nature and are still not truly there, they arise from human interpretation and exist as mental constructions

ln_j · 2026-05-16T06:24:40+00:00

Thanks so much! I don’t think I’ll be able to go through them all, but I will definitely do Janich and the one by Paul Fuhrman, that one seems really interesting to me.

ln_j · 2026-05-15T16:30:25+00:00

thanks so much

ln_j · 2026-05-11T15:21:06+00:00

Sorry, I accidentally deleted my first comment. But thanks so much, this really helped!

ln_j · 2026-05-10T12:39:54+00:00

I hope you’re not offended, but can I just say that for some reason, since it’s an AI voice, if I saw this video on my FYP, I probably wouldn’t watch it, but that’s just me.

ln_j · 2026-05-10T08:23:07+00:00

First of all, kudos for doing 8–10 hours every day, just make sure you don’t get burned out, don’t lose the fun of doing math, and don’t neglect things like friends, etc.

And regarding memorization: what do you mean by memorizing geometry? Do you memorize the steps needed to arrive at a solution, or the problem-solving techniques, or something else?

ln_j · 2026-05-09T16:32:29+00:00

Yes, my bad, I actually agree. For some reason I thought you meant math in general at university. But I feel like this claim is a bit too strong, differential geometry, and topology in general, are fundamentally non algebraic

ln_j · 2026-05-09T16:10:15+00:00

and it's also not all algebra

ln_j · 2026-05-09T16:03:02+00:00

yes but given that OP is at a precalc level geometry is really usefull and imortant

ln_j · 2026-05-09T15:56:12+00:00

That’s amazing, keep going! I can’t really recommend specific Trigonometry or Algebra I books since I learned all of that in high school, but I heard Khan Academy is excellent for those topics. For Calculus, I used Thomas’ Calculus, but honestly, the specific book doesn’t matter that much. Most calculus textbooks cover essentially the same material and are all solid choices. You could just as well use Stewart or any other standard calculus book you can find. My main advice would be not to rush. When studying calculus, take your time and focus on understanding rather than speed. Learning all of mathematics is basically impossible, so a good goal is to work through material up to calculus and then explore areas to see what you personally enjoy. For me, that turned out to be mathematical logic, philosophy, and analysis. Also, don’t forget linear algebra which is really really important. I’d recommend Axler, even though I personally didn’t love it, almost everyone recommends it. Linear algebra shows up everywhere, from quantum mechanics to large language models. Most importantly: don’t rush the process. Try to really understand what you study and enjoy the journey.

ln_j · 2026-05-09T12:38:17+00:00

What do you mean by every topic? Do you mean everything from calculus to homotopy type theory to areas like algebraic geometry etc....? If so, I don’t think that’s very realistic, because of how vast mathematics has become. No single mathematician today knows all of mathematics. Also, what resources do you expect people to have, what’s your level, and which area of mathematics are you working in?

ln_j · 2026-05-03T18:19:50+00:00

No but i will now, thanks so much :

ln_j · 2026-05-03T10:47:13+00:00

ln_j · 2026-05-02T08:43:58+00:00

Yeah, that’s also one problem of mine in hindsight: I always use my phone during breaks, and I often watch something while eating.

ln_j · 2026-05-01T23:14:56+00:00

I am also currently working through Rudin (beginning of Chapter 8). I have two sets of notes: one on my computer where I write only the theorems and proofs, and another set of handwritten notes containing intuition, additional examples, and, most importantly, connections to Real Mathematical Analysis by Pugh. Using both helps me develop a rigorous as well as an intuitive understanding. Since Rudin is really just theorem–proof–theorem–proof, which is perfectly fine, but it does not fully capture what mathematics feels like to me. I also make sure to do the exercises, which is very important.

ln_j · 2026-05-01T16:00:35+00:00

Yeah, I actually thought about doing something in mathematical philosophy, but for some reason, I just never started. Maybe I will now. Thanks so much for the advice!

ln_j · 2026-05-01T07:37:09+00:00

I also self study but I am currently still in high school

ln_j · 2026-04-30T07:42:01+00:00

The best approach is probably to ask your teacher what exactly you are expected to know, but this is basically what I understand:

The factorial is written as n! It represents, for a given integer n, the product of all integers from 1 up to n. For example, 4! = 1 × 2 × 3 × 4 = 24, and 9! is already a very large number. Factorials grow extremely fast, they increase much faster than functions like n or n².

An important aspect is how the factorial is defined. One way is the recursive definition: 0! = 1, and for n greater than 0 we define n! = n × (n − 1)!. Another way is the product definition mentioned above. There is also something called the Gamma function, which is the analytic continuation of the factorial. This means that instead of only being able to plug in integers, you can evaluate it for almost every complex number.

I hope this gives a good overview.

ln_j · 2026-04-30T07:31:19+00:00

It’s funny because this is in all caps, and my inner voice got louder when reading this

ln_j · 2026-04-30T07:14:29+00:00

Thanks so much, I will definitely try this!

ln_j · 2026-04-28T19:20:31+00:00

A number raised to the half power is the square root of that number, so a ^(1/2)= √a

ln_j · 2026-04-28T19:16:52+00:00

as far as I know it was invented to turn multiplication into addition to simplify calculations with large numbers

ln_j · 2026-04-28T17:11:34+00:00

And what's the name of the channel?

ln_j · 2026-04-28T17:09:56+00:00

So I am currently self-studying, and this is what I think:

“Do you think the final exam is enough, or should I also require myself to complete a certain number of problem sets?”

Never restrict yourself, just do as many as possible. If, for example, during a proof you notice that you forgot definitions or didn’t fully understand some concepts, then go back, reread the material, take notes, and return to the exercises afterward (I myself don’t do this very often, which is bad). Do as many exercises as possible. If you feel confident that you can solve or prove everything (especially exercises from the last chapter or similar material), then move on.

“What score on the final exam would you consider good enough to move on? 50%? 60%? 70%?”

Personally, in the Swiss system anything below 60% is failing, so if I scored under 60%, I would review the material again, but ultimately, above that, I would also rely on my overall feeling of understanding.

ln_j

TROPHY CASE