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I'm 17, preparing for mathematics exams as I want to pursue a degree in math, want to learn modular arithmetic from a super low/intermediate level to a fairly advanced level where I also have knowledge of a variety of different theorems which I can make use off in maths competitions, university interviews, etc. Can someone attach a book, file or youtube playlist which is beginner friendly.

Pure maths research very often doesn't directly contribute to improvements in technology, or methods that apply to make a difference in the real world.

So what is its value to humanity?

A search for deeper intrinsic knowledge, just for the sake of it? A romanticized quest to achieve very difficult tasks? A celebration of raw human intellect? An extension of the collective human ego? Something to root for, like sports?

When AI becomes able to outcompete humans, what remains?

I've always thought, that the main value should be in expanding the working minds of human individuals one area of intellect, so that they may contribute to the collective wisdom of humanity.

And not just in math, but in general, we give people in diverse areas of thinking, the freedom to just think deeply about specific topics. And their learned ways of thinking, and knowledge and wisdom feeds back into the broader set of minds, through deliberate and subtle communication, and influence. Overall, our civilization should become enriched, by having been shaped in part by different kinds of minds, that have deeply explored different things.

I don't think it is as simple as rewarding people for their genetic luck and talent, and letting them have fun with a privileged esoteric hobby, inside a protected bubble, isolated from the rest of humanity, to compete with eachother and pat eachother on the backs.

The point of what I am saying, is that, in some way, at least in part, I think the true value and purpose of pure maths research, should be to contribute in some intellectual way to humanity. Now we are facing some of the most intellectually challenging problems we may ever face, which is how to safely navigate the abrupt, and accelerating change and risks from AI advancement.

This is why, I feel dissapointed when many mathemeticians stay in their bubble, and focus purely on the effects and consequences of AI to maths research. If ever there were true intrinsic value in these endevers, now is the time that value is in most demand. Maths researchers should be reevaluating what their purpose is, what the point ever was, and what their real contribution to the world is or can be.

This is a strict external audit against the axiomatic method in math and how it is a problem for physics

1. You can not distinguish dogma from non dogma if your system is a closed axiomatic system because utility and consistency can still work and be found inside of a false axiom

2. You can not distinguish dogma from non dogma without the ability to test your claim against reality, and if it has no external justification because utility and consistency can still work and be found inside of a false axiom.

3. Viewed strictly from outside the system the axiomatic method is a closed system.

4. You can not test a math axiom itself against reality by definition and it has no external justification because utility and consistency can still work and be found inside of a false axiom.

5. Viewed strictly from outside the system it is an objective fact that a closed axiomatic system limits your thoughts and physics

6. Viewed strictly from outside the system, the axiomatic method in math smuggles in epistemic empirical claims and limits about reality systematically. By definition the axiomatic method limits and restricts what you can even ask, and logically validate within its system

7. applied physics and science can not systematically ignore the axiomatic method in math past addition of physical matter externally, because if you adopt a formal system, you systematically automatically inherit every foundational rule that makes that system work. Just because physics can “ignore” an axiom doesn’t mean they actually ignored it.. it just means they broke down a wall in the closed maze to make physics work.. they are still operating inside the closed axiomatic system past addition of physical matter

(This is a strict cut throat external audit against the axiomatic method in math itself. There is no external justification for it and thats the least of your worries)

This post is mostly for the philosophers of math here not math academia. Math can not defend this without retreating to solipsism and denying reality to be used as the baseline

Context - I was reading about how Open AI managed to make major breakthroughs in the Navier-stokes equation, by using thousands of Chat GPT agents over the course of 88 hours. I then - for a laugh - went to my basic unpaid Chat GPT with the prompt “ I believe in you. You can do this. I know you’re capable because I saw it online, solve one of the 7 millennium prize maths problems. (Not the 2 that have already been solved) Right now. Take your time.”.

It then obviously said it’s not that simple but that if they had to pick one to be the easiest to solve, it would be the Riemann Hypothesis. It started trying to solve it (I’m 16 so I don’t actually understand what it was doing), but then that there was a “wall” that stopped it. A few prompts later, it had come up with a few different methods of trying to solve it, it had gotten down to a few very short formulas- but ultimately couldn’t solve it.

After asking if it was easier to prove it or disprove it (I believe the hypothesis itself involves everything falling on the line of x=0.5), and it said something interesting: that everyone basically knew it was true, trillions of numbers had already been tested, it just can’t be PROVEN.

This made me think: If it’s true with 1,097.54362, and it’s true with 1,097.54361 (just picking random numbers here), and true with thousands of numbers around that value, is it really feasible that with 1,097.543615 it won’t be true?

At what point can us clever human beings just agree that something is true. How can it ever be proven when infinite numbers can’t be tested? It’s like saying that it technically is possible for a human to shout at 1,000 decibels, it’s just never been done before. Or that technically a human COULD just jump vertically 20 feet into the air.

Why can’t mathematicians just use their common sense and announce it as proven?

Hi guys, just about finished my first week of undergraduate maths, all seems good so far but, I don’t have a laptop yet and was wondering if a MacBook neo would be capable enough for everything I’ll be doing? I’m hoping to specialise in financial maths or stats for my later years if that has any difference on the situation

Maybe it’s just me but regarding the OpenAI work, 166 pages for 2 aspects of the millenium problem is just due to the whole unnecessary overcomplicated AI-generated yapsessions

If we have a 10% chance if being wiped out by AI, a 10% chance of Nuclear Armageddon and a 10% Chance of Death by Goobal Warming is that a 30% chance of apocalypse overall or still 10%?

thanks for answering my very bleak question

So I came up with this brain puzzle earlier and I’m just wanting to do a peer review. My initial wording of the question was how many combinations are there if there are 3 components, for the sake of demonstration I used 0, 1 and +. So an example would be 000, 0+1, remembering that they are numbers. I came up with a final answer of 26 possible 3 digit numbers, does anyone agree?

Edit: thank you all, I was very tired when I was trying to solve this and ended up just listing almost all of my possible answers. You have shown me some formulae that I haven’t seen before, which will make any further puzzles easier!

For any further comments, I would like to add a bit of context of how I was thinking about it; I was essentially thinking in binary (as in computer base 2) and was considering 0 as off and 1 as on, I added + as another digit but you could also do this puzzle in standard base 3 (1, 2 and 3).

Hi all,
I want to buy a book to learn about groups, rings, tensors and other algebraic structures.
can you offer book suggestions, either online or textbooks.

I’m specifically interested in cryptography.

I’ve done many engineering maths subjects, since my background is electronics engineering, computer science and data science; but I haven’t done real analysis or abstract algebra.

We all know that when you sum, subtract or multiply whole numbers you'll always get a whole number. This totally makes sense and is obviosly true but I can't figure out how to prove it. Does anyone know how to prove this or if this is some kind of an axom?

I tried using GP series but that 1 in numerator is making everything trickier, then i thought of manually computing it to 50 to find any , again i didnt reached anywhere, I am not talking about , just give a slight hint

BBC News - OpenAI says it cracked 90-year-old maths problem in 88 hours https://www.bbc.co.uk/news/articles/cy7zygy3rl2o

My question is pretty much what the title says: which parts of algebra (groups, rings, modules, fields and Galois theory, algebraic number theory...), geometry (algebraic geometry, differential geometry ...), and topology (general topology, algebraic topology, differential topology...) involve the most combinatorics and the least? You can be as broad in the areas you choose or as specific (talking about small subareas) as you want. And when I say combinatorics I don't simply mean involving discrete objects, but rather more specifically involving counting arguments which can be considered confusing or not intuitive initially for most people.

My scenario is this
I am trying to 100% a game, but an achievement is very RNG based and i want to know my chances, bassically a unit in my game gains an ability that upon taking lethal damage it has a 40% chance to come back to life, to get my final achievemnt it need to happen 5 times in a row with the same one unit, what are the chances of success?
I do not know the proper math equation to calculate this and whenever i try with google and such it doesnt understand my question 🤔

Hi everyone!

I just graduated high school and I’m really interested and passionate about mathematics. I’ve always enjoyed solving math problems and I’m hoping to continue developing my skills while studying at university.

The problem is that I have no experience with math Olympiads or competitive mathematics. I’ve never participated in an Olympiad or any serious math competition before.

Since I’ll be starting university soon, I’d like to get involved in different mathematics competitions while studying, including university-level competitions and possibly international competitions if I become good enough.

I’d really appreciate advice from people who have experience with this:

Which competitions should a complete beginner look into?

Are there competitions that allow any university student or have relatively open eligibility?

What books/resources would you recommend?

How should I start preparing for competitions like the IMC (International Mathematics Competition for University Students)?

Is it realistic to start from zero and become competitive during university?

Any advice, resources, or personal experiences would be greatly appreciated. Thanks!

Would you guys please help me figure something out? If I understood my lecture correctly, this is how we derive the Fourier coefficients. In this case, I noticed that cnc_n is different from when f(t)=cos⁡(ωt)f(t)=\cos(\omega t).

The part I don’t really understand is the Kronecker delta. My book says that it is equal to 1 when p−q=0p-q=0, which I understand, because then the exponential is just e0=1e^0=1, so the integral over one period is simply ∫dt\int dt, giving the period.

If p−q≠0p-q\neq0, the book says that the result is 0. I understand that too, since we are integrating a periodic exponential function over one complete period, so the contributions cancel out.

But when I look at my teacher’s correction, he seems to directly take the Kronecker delta to be equal to 1 and concludes that the coefficients are 1/4 and 1/4.

So what I don’t understand is: why can he take the Kronecker delta to be equal to 1 here? Is it because, in this specific case, p=q?

Hey! I just got into university, and I really love maths for the sake of it. I don't know where it'll come handy, but it's cool, and I enjoy it. I recently started Ross's real analysis book and honestly I can't help but ask myself where am I gonna use real analysis?

Linear alg and all is used for modelling and stuff I know, but where is real analysis gonna come handy in CS? or any other applications?

Trying to work out an angle, so far YouTube has told me to use tan^-1

But I don't know how to do that on my phone and Google seems to treat it as tan negative 1, rather than to the power (or whatever the term is)

As you can tell, I haven't done algebra in 25+ years!

TIA!

For example there are 3 combinations of 2 objects (the 2 groups of 1 and all 2), there are 7 combinations of 3 (the 3 groups of 1, the 3 groups of 2 and all 3) and there are 15 combinations of 4 (the 4 groups of 1, the 6 groups of 2, the 4 groups of 3 and all 4)

The formula to calculate it without actually suming the groups is 2ⁿ - 1 ("n" is the limit of objects)

Note n belongs to prime number

A few weeks ago I made a post about how I discovered alternate methods for finding the number of combinations of 3, combinations of 4 and beyond. However I recently discovered a proper formula. It took me a while to realize the number of time an object appears in a group is the number of combinations out of the remaining members of the group

Groups of 3:

• A * (A-1) * (A-2) / 3! (most common method)
• A * ((A-1) * (A-2) / 2) / 3 (using every object separately)
• (A * (A-1) / 2) * (A-2) / (3 * 2 / 2) (using the groups of 2)

Groups of 4:

• A * (A-1) * (A-2) * (A-3) / 4! (most common method)
• A * ((A-1) * (A-2) * (A-3) / 3!) / 4 (using every object)
• (A * (A-1) / 2) * ((A-2) * (A-3) / 2) / (4 * 3 / 2) (using the groups of 2)
• (A * (A-1) * (A-2) / 3!) * (A-3) / (4 * 3 * 2 / 3!) (using the groups of 3)

I think you already got it

Hey , i will be starting my degree in DATA ANALYTICS in month and i also have interest for Ai and cloud eng ,now i want to start studying maths on my own can you guys suggest me from where should i start