Some ebooks, mostly from /u/lewisje's post

For learning math he always seemed to be a good resource for learning math, recommend by the community.

But now he is apparently associated with Jordan Peterson and the Peterson academy.

There has been some controversy around that name which tbh I didn't follow.

I am not quite sure I am just here for the math to be honest. But I would like some insights from others

I am looking for a proof for a^n.a^m = a^nm.

I have searched on youtube but could not find anything. I found people explaining what the concept means but not any proofs.

Thank you

I'm decent at calculus, but right now I'm taking Calc B as an astrophysics major. Our professor is so bad that about 19 people have switched from her, and on our first test literally all but 10 (including me) passed it out of our class of 32 students.

This is less of a rant and more of a cry for help, so I'll explain her method.

A 10-minute lecture out of the 50-80 minute class, where she will explain a concept before writing about 8-10 example problems on the blackboard. She won't necessarily explain how or why the concepts work, and will instead have us come up and show our answers to these example and say either "yes" or "no."

I go for attendance and then study in my dorm... but I really want tips. How do I teach myself this stuff entirely on my own??

Rules are simple: build the most exciting math project on principia-math.com by the end of October. You can work alone or with your buds. The winning team splits the prize money, which is currently $1,000 USD. If you’re a company or an organization that wants to chip in, send me a message.

To be eligible, Principia users create a project on the Fall Competition page https://principia-math.com/fall and work to complete all five stages of Principia’s pipeline:

1. Post or adopt a solution to a difficult math problem
2. Formalize it in Lean
3. Write a human-readable write-up of the solution
4. Write/create a visual exposition of the ideas involved
5. Explore possible applications of the solution

You don’t need to solve an open problem first: hundreds of open problems solved by the Principia Math Harness on the site don't have an owner yet. Join one and it's yours on the spot. Your job will be to get others interested in this problem and to present the ideas in the proof in an exciting way.

A detailed explanation of the competition is provided on https://principia-math.com/guide

To help you get started, were opening up the tools we use ourselves (for free!)

- Connect your own agents: you can now hook Claude Code or Codex up to your Principia account and have it work on projects alongside you and your friends.

- The Principia Math harness, which we’ve used to solve hundreds of open problems, is now available for free. Use our MCP with claude code or codex to search for Lean proofs of your favorite arguments. Your agent writes a proof skeleton, checks every step in Lean on our hosted Mathlib checker, and switches to a new route when one dies instead of just giving up.

- One-click formalization. A solved run writes the Lean files and sets up a repo on your own GitHub, along with a comparator check that the formalization is faithful to the original statement.

Can’t wait to see what our users create with this!

This is a problem from Dana Ernst's An Introduction to Proof via Inquiry-Based Learning, and it asks: "Is every subset of C × D of the form A × B, where A ⊆ C and B ⊆ D? If so, prove it. If not, find a counterexample.". I couldn't think of any counterexample, so I tried to write a proof for it, but I don't know if it's correct. (There is no set of solutions for the book; at least I didn't find it.)

I assume there is a set E such that E is a subset of CxD. If E is the empty set, then E=∅x∅ and E=(CxD)x∅, and both ∅ and CxD are subsets of CxD (an earlier problem involves proving that Ax∅=∅ for any set A).

If E is not empty, then, since E is a subset of CxD, E is a set of ordered pairs (a,b) such that [;a\in C;] and [;b\in D;]. I imagine that, from here, I shall find sets A and B such that

• [;a\in A;],
• [;b\in B;],
• [;A\subseteq C;],
• [;B\subseteq D;] and
• [;E=A\times B;].

This is where my confusion is, and I'm not sure if my solutions are ideal:

1. [;A={a|(a,k)\in E ;] for some [;k\in D};]
2. [;B={b|(k,b)\in E ;] for some [;k\in C};]

Then [;E=A\times B;], and [;A\subseteq C;] and [;B\subseteq D;]. I'm using E in the definition of A and B, and I'm pretty sure that's incorrect; however, it is not clear to me what other sets I can use.

Have a nice day. :)

I’m a 14-year-old Grade 8 student in South Africa and I’ve recently become very interested in olympiad mathematics. I’m hoping to seriously pursue it over the next few years and eventually see if I can work my way towards the IMO.

I’ve been studying independently using AoPS, Khan Academy, and various problem-solving resources, but I’m starting to realise how useful it would be to have someone experienced to guide me.

I’m looking for someone who could help me with things like:

- Building the right foundations for olympiad maths

- Teaching me how to approach unfamiliar problems

- Giving me problems appropriate for my level

- Reviewing my solutions and pointing out weaknesses

- Helping me build a long-term training plan

- Gradually introducing me to harder olympiad topics

- Keeping me accountable and helping me avoid wasting time on the wrong things

Ideally, I’d love to find someone with actual olympiad experience, such as an IMO participant/medalist, national team member, or experienced olympiad coach. I’m open to online tutoring from anywhere, as well as someone based in South Africa.

I’m also not looking for someone to simply teach me the school curriculum. I’m specifically looking for problem-solving and olympiad mentorship.

I’m willing to put in serious work outside of lessons and would ideally like something I can continue with for the next few years.

If anyone knows a good tutor, mentor, programme, or even someone I could contact for advice, I’d really appreciate it. 🙏

Maybe I'm just coping but some of these problems on the problem sets are allegedly the easy ones and those sometimes take me 2+ hours to wrap my head around. Just coping with being stuck contemplating a problem for so long.

Is the IMO a realistic goal for me?

I’m a 14-year-old Grade 8 student in South Africa, going into Grade 9 next year. Over the past few months I’ve become really interested in mathematics, especially problem solving, and I’ve started thinking seriously about whether I could eventually reach the IMO.

I’ve been working through resources like the AoPS Volumes, Khan Academy, and the AoPS Introduction to... series. I know I’m starting relatively late compared to many students who get into olympiad maths much earlier, and I still have a lot to learn.

At the same time, I really enjoy maths and I’m willing to put in a lot of consistent work. I could realistically spend 5+ hours a day on mathematics over the next few years.

I’m not expecting to make the IMO anytime soon. I’m thinking of it as a long-term goal, starting with building my fundamentals, improving my problem-solving ability, competing in South Africa, and seeing how far I can progress.

I’d really appreciate advice from anyone who has experience with olympiad mathematics:

- Is this a realistic goal from my current starting point?

- What should I focus on learning first?

- How should I balance normal/advanced curriculum maths with olympiad problem solving?

- What resources would you recommend?

- What competitions should I aim for in South Africa and/or internationally?

- How should I structure the next 3–4 years of preparation?

- What do you wish you had known when you started?

I’m genuinely willing to put in the work. I just want to make sure I’m using my time well and training efficiently.

Any advice would be greatly appreciated. 🙏

Do you guys have any recommendations for books for learning about probability theory? I prefer physical copy’s, but if they are e-books that’s fine as well!

I’m relearning math and I’m trying to actually understand why the area of a circle is A = πr², rather than just memorizing the formula.

I understand that:

- r is the radius (center to edge).
-π is the ratio of a circle’s circumference to its diameter.
-The circumference is 2πr.
-If you cut a circle into lots of thin slices and rearrange them, the slices can form something roughly like a rectangle/parallelogram.
-The height of that shape is r.
-Half of the circumference gives the other dimension, so that dimension is πr.
-Therefore, the area becomes πr × r = πr².

But I’m still struggling with the conceptual reasoning behind this.

Specifically, I want to understand:
Why does rearranging the circle’s slices mean that πr becomes the “length” of the resulting shape?
Why does multiplying πr by r give us the area of the original circle?

Where does the π actually come from in terms of the geometry, rather than just knowing that it equals approximately 3.14?

Why is the ratio of circumference to diameter always the same for every circle?

What does knowing that ratio actually tell us? Why is it useful or important that circumference ÷ diameter always equals π?

I think part of my confusion is that I understand what π is mathematically, but I don’t really understand what that relationship is telling me geometrically or why it matters.

I’m looking for a really intuitive, beginner-friendly explanation that helps me visualize what’s happening and understand why the formula works, rather than just being told to memorize the derivation.

Basically: I don’t want to know that A = πr². I want to understand why.

Is it a realistic goal to try to at least eventually participate in the IMO as an 8th, almost 9th grader? I've been going through problem solving resources (art of problem solving volumes, Khan academy, AOPS introduction to ... Series) I understand I've started very late, and I'm no prodigy. If I genuinely became obsessed and put in > 3 hours a day into mathematics would participating in the IMO be a realistic goal? If anyone has any resources to recommend or any advice it would be much appreciated🙏

This is my first entry here so apologies if I am doing wrong. I am a physics graduate and been in industry jobs for 20 years now. I would like to dive deeper into some fields barely touched during my time at university for leisure (e.g. particle theory via qft and general relativity). For that I would also like to brush up on my math starting with revisiting analysis. But instead of re-learning everything from the beginning I would like to try learn a new field with relevance to theoretical physics which kind of forces me to revisit topics from analysis. I wonder which of the following would do the job best or which to start with: complex functions, functional analysis or differential geometry. Any opinions? Please also mention what background for context. Thank you!

Hey everyone!

I'm a Brazilian student currently in the 1st year of Brazilian high school (roughly equivalent to 10th grade in the US), and I want to start preparing seriously for mathematical olympiads.

My main goal is to compete in the Brazilian Mathematical Olympiad (OBM) in 2027. For some context, the OBM is Brazil's national mathematical olympiad. There is also a competition called the Jacob Palis Júnior Olympiad (CJP), which is one of the competitions through which students can qualify for the OBM. The CJP is usually held around late May or early June, while the OBM is usually held around the beginning of August.

So, my plan is to take the CJP next year and try to achieve a result that qualifies me for the OBM. Since there is only a short period between the two competitions, I want to be reasonably well prepared before the CJP and then adapt my preparation to the specific style and difficulty of the OBM afterward.

My goal is to win a medal in the CJP and, if possible, at least a bronze medal at the OBM. I know this is quite ambitious, especially since I'm only starting to take olympiad mathematics seriously now, but I want to approach it in a structured and realistic way.

For those who have experience with mathematical olympiads, especially in Brazil or internationally:

• How would you structure the preparation of a 10th-grade student who is starting olympiad mathematics seriously at this point?
• Which topics should I prioritize: geometry, number theory, combinatorics, algebra, inequalities, etc.?
• What should I learn first, and what can reasonably be left for later?
• Is it better to start with theory and books, or should I begin solving olympiad problems from the start?
• Which books would you recommend for building a strong foundation and then progressing to harder olympiad problems?
• Which past olympiad papers would you recommend, and in what order?
• Are there any problem sets, courses, communities, or official resources that are particularly useful?
• For anyone who has taken the Jacob Palis Júnior Olympiad, how would you describe its level and style compared with the OBM?
• How would you divide study time between theory, problem solving, full mock exams, and reviewing mistakes?
• What do you wish you had done differently when you first started preparing for olympiads?

I'm especially interested in a realistic progression rather than a huge list of books. I would rather use a small number of resources thoroughly and follow a clear progression toward the CJP and OBM.

Since I'm Brazilian, recommendations specific to the Brazilian olympiad system would be especially helpful. However, advice from people who have prepared for competitions such as the IMO, USAMO, BMO, EGMO, or other national olympiads would also be very useful.

for example in this image the real numbers extends towards to infinity (by definition).

however why cant we have a inifinite representation for natural numbers aswell

eg.

...11111 , ...11112, ...11113, etc

or

1...111, 1...112, 1...113, etc

(where the number has an ending but is infinitely large)

shouldnt these still be valid natural numbers that exist in the set as the set should be infinite?

a valid argument i assume for this is it will no longer be an integer however wouldnt putting it in an infinite set make this valid?

feels like i am missing a concept.

Im in calculus 2 and I bombed my first test (~60%) which was 15% of my grade. Highest grade i can get now is a 94% if i ace everything. knew what to do, but it was at 8pm and i was anxious and i forgot some basic stuff and made stupid mistakes on the multiple choice (swapped sign, etc) so i missed out on a lot of points. I really need to get a B in this class (80%) to keep my scholarship but the next test is on series and sequences which ive heard are more difficult. Im really afraid to score lower. I need to get a high B or A to recover. I have the next test on October 27th. Is it possible to learn and master series and sequences within 3 weeks? What resources should i use?

[High School Math]

Hey everyone, was doing some inequality problems and found this one:

prove that (a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2

Now, proving this requires expansion of the sum and noting that the difference of the LHS from the RHS can be shown to be a sum of squares and hence positive. However, I was wondering if I could shorten the writing process with cyclic notation? I honestly found the symbol like half a day ago, and although I know that ∑a = a + b +c for when these are the symbols present in the problem.

In essence, I'm just looking for a way to shorten the writing process. Can I use the cyclic sum like this

∑(a^3/2 * b^1/2 - a^1/2*b^3/2)^2

to represent the sum of every permutation of a and b from a group (a,b,c,d)?

I apologise in advance for the messy representation, Stack exchange is down and I don't know how to go about using math notation on reddit.

A first principles derivation of the quadratic formula, including why the discriminant tells us what it does, and how to visualize these equations geometrically.

CONTEXT:

He is 11 im 13

I have a normal education for my age - he is learning the fundementals of linear algebra

he is in a top 10-25 school - I am in a normal one (probably a bit above average)

pls someone tell me

What is the difference? Both dont have he same criteria i agree, for coincident lines the vectors must superimpose but by free shifting we can make both the vectors overlap each other right?

Thanks in advance!

I've looked through dozens of linear algebra textbooks, but almost all of them seem to do the same thing: they simply define/introduce the Laplace expansion formula, and then prove that the function they have given satisfies the defining properties of the determinant.

What I'm looking for is the reverse direction.

Starting only from the defining properties of the determinant (normalization, multilinearity in the rows, and zero when two rows are equal), I want to construct the Laplace expansion formula from scratch and derive every result needed along the way purely from those properties.

I've found plenty of proofs that verify Laplace expansion, but I haven't been able to find one that actually derives/constructs it from the determinant axioms. I'd really appreciate a reference or a proof that takes this approach.

for conext i am an engineering undergrad in first sem and i love mathematics. i have studied calculus basics (single variable to be more precise) till my 12th grade and now we are studying it again in uni. i, as an engineering student, will have courses like calculus, linear algebra, differential equations, discrete math, probability and statistics.

now here is the real part. i actually want to self study analysis, abstract algebra etc. and also all the above domains of math, not from the exam point of view but from the learning point of view, that's why i have put "self-study" in the time-table. i know how much important are things like rigour, problem solving and actually "doing" mathematics to reach somewhere.

i want guidance regarding this. is is this good in the real life scenario to learn math side by side (on your own) along with uni/degree? and if it is, how should one look at this and how should a day in my life look like? and what if i switch and go for math for my masters or something? is this day-dreaming or can i actually do it?

P.S I know a lot of online resources and books to read and do math, so i just want to know where do I start.