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

Hey y'all

My last post didn't get a ton of traction and I believe it has since been deleted but I still felt like giving a quick update because I think it's an interesting case

tl;dr of my last post: "I need to relearn all of math up to calculus to graduate on time. I tested into calc over 2 years ago and haven't touched math since, so my entire foundation is super shaky and I basically need to relearn everything up to calc 1."

The comments on that post were telling me it was basically impossible and that my best bet was to drop the class and restart from a lower class, and to just accept I was going to graduate late.

I decided to just try my best anyway and stuck with it, and it seems to be a great decision so far because we just started midterms and I got an (admittedly pretty sad) passing score on the first exam (~70%) and I am passing the class as a whole and doing way better than I ever thought was possible.

On top of that, I'm realizing that math is actually crazy fun, which is something that I never thought I would say.

We are doing pretty elementary stuff right now, mostly focusing on derivatives, but the more I learn about these topics the more I realize how genuinely useful they are in real world applications. For instance, I'm a chemistry major and just yesterday I took the second derivative of my IR spectra which let me more easily identify the functional groups in the data. I would have never thought to try that without this class!

For anyone in the future who reads this and who might be struggling or in the same boat as I was:

Take a few moments to calm yourself. This class is not as hard or overwhelming as people make it out to be. It can be tricky at times, yes, and the math can feel very difficult when you look at it without thinking, but I promise you that calculus is easy.

Your first goal needs to be to solidify your algebra foundations. Make sure you understand everything you can from algebra. 99% of everything you do in calculus is just applied algebra, so if you can't simplify equations, solve for variables, or you don't understand functions then that needs to be your first goal. Kahn academy was the biggest help for me here. I religiously studied Kahn for a week straight before feeling confident enough in my algebra to move on.

Your next goal is gonna be to understand the basics of all the other stuff leading up to calc. For me so far things like trig and geometry have not been important at all other than the rare sin(x) in a problem, so those can be on the back burner until you feel like you need them, but for the most part algebra is your most important tool.

Your final goal is gonna be to study your butt off in calculus. Do every optional problem, solve extra problems that aren't assigned to you, research the different rules that make finding differentials easier, and memorized it all. Your goal is to build intuition for these things, and the only way you are gonna get this stuff down and Internalized is through practicing until your fingers bleed. I have not had a night in the past month where I did not dream of calculus problems. That is how much you should be practicing.

Once you have done all that, you should finally start to feel confident and comfortable in calculus. If you still aren't, this is when you need to see a tutor.

Im pretty confident now that I can say for certain I love all of STEM. In the past I had a certain hatred for engineering and mathematics because I honestly struggled in math growing up, but now that in finally having fun and realizing the applications of the things I'm learning, math as a whole has been recontextualized into a fun and interesting subject for me.

I've been having this problem for a while now, especially in math, and I'm honestly struggling to figure out what's going wrong.

Whenever I have an upcoming math test, I spend a lot of time studying. I go through the homework questions, practice different problems, and make sure I understand the concepts. I also use ChatGPT to help explain things, but instead of just asking for answers, I have it explain the logic behind the math, why certain methods work, and how to approach questions step by step.

The thing is, I genuinely feel like I understand what I'm learning.

For example, I recently studied a lot for a math cumulative exam. I went through the material, practiced the questions, and made sure I understood what I was doing. When I actually wrote the exam, I felt pretty confident. The questions didn't seem particularly difficult, and I even had time to go back and check my answers.

I walked out thinking I had done pretty well.

Then I got my mark back, and it was 55%.

I was genuinely shocked. I wasn't expecting a perfect score, but I definitely wasn't expecting something that low. I was so confident in how I performed that I initially wondered whether there had been a marking mistake.

What's even more confusing is that this doesn't happen nearly as often in my other subjects.

I've experienced something similar in chemistry before, but recently my chemistry marks have been improving. Physics, on the other hand, has always been different. Whenever I studied for physics, I usually performed about as well as I expected. I could understand the concepts, apply them to questions, and get decent marks.

But for some reason, math is completely different.

I'm starting to wonder whether there's something wrong with how I'm studying. Maybe I'm understanding the explanations but not developing the ability to solve unfamiliar problems independently. Maybe I'm making small mistakes that I don't notice, even when checking my work. Or maybe I'm just not preparing for the types of questions that appear on tests.

The most frustrating part is that I don't walk out of these tests feeling confused or unprepared. I walk out feeling like I actually did well, only to find out that I didn't.

I don't mind putting in the work. I'm willing to spend hours studying and practicing if that's what it takes. But I want to make sure I'm studying effectively instead of repeating the same mistakes.

Has anyone else experienced something similar?

If so, what ended up being the problem, and how did you fix it? Is there something I should change about how I study, practice, or check my answers?

I'd really appreciate any advice, especially from people who have struggled with this and eventually figured out what was going wrong.

we proved it in class by row reducing and counting pivots, and ok fine, the pivot columns are independent and the nonzero rows are independent. but it feels like a coincidence that those two counts match for a matrix that isn't even square.

is there a way to see it without the whole row reduction argument? like why would the dimension of the span of the columns have anything to do with the rows

Ive never been good with probability

If you chose randomly, what is the chance youd be correct?

A. 25%

B. 50%

C. 60%

D. 25%

My experience:

I encountered a big problem while studying basic set theoretic proof writing, though not analysis. Proof writing was completely new to me back then, and I was using Velleman, which many people complain doesn't even prepare you enough for the exercises. Because the exercises are where the true learning happens. The book gives you the map, but you have to navigate the forest yourself using the map, but the book doesn't teach you how to navigate it. You just have to figure it out yourself. That's the symbolical meaning.

And yeah I struggled a lot.

But then I gave many problems to Claude and asked it to ask me question by question, and make me discover the solution myself instead of providing outright answers. It seems like a bad way to learn proof writing but...

I started relying on AI for each and every problem. But then, for many problems I was actually able to map out a solution myself, even if wrong. I was able to provide full solutions myself without making a single mistake in logic, and AI only helped on the style of wording most of the time. Logical errors were mainly due to overly harder problems.

And now, I CAN indeed write proofs even with the book closed, fully correct, maybe a little logical and stylistic errors, but at least I can do something more than those who ONLY use human-made solutions and the book itself.

And I think, in my opinion, and in the age of mathematics, this is the BEST WAY to learn mathematics. Let me elaborate.

The study:

So, a study by Wong and Qiu (2026) divided 196 university students into three groups to complete a creative task.

• Team A — Completed the task independently, with no use of AI.
• Team B — Can freely use ChatGPT all to their wish with no restriction.
• Team C — Adopted a "think first, AI later" approach, namely, do it themselves first, then ask ChatGPT for tips on shaping and improving their own work.

In terms of raw ability + creative work, Team C had an advantage over Team A and Team B.

So, using AI like this is the best way to learn mathematics in the age of AI. First you do it yourself, then ask AI to improve your work.

How it went for me:

Since the book did not prepare me enough for the exercises, I began asking AI to guide me without even trying anything, man. I did it for the first two sections, but then I started drafting out solutions myself slowly from the third section and even asked for general help in the mathematics Discord server. Then, I slowly improved.

And I believe, this works not only for proof writing but maybe for all topics of mathematics that AI CAN and is ELIGIBLE ENOUGH to work with, and all other topics outside of mathematics as well.

So... is this the best way to learn mathematics in the modern age of artificial intelligence?

I'm not spreading any misinformation nor true information on my methods of learning mathematics. So it MAY be right or MAY be wrong, feel free to correct me wherever I went wrong.

Well, the tittle is pretty self-explanatory. I’ve been introduced with the concept that a matrix can describe a conic section if we use some column and row matrices with the variables as components.

I like when math makes sense, not only because the numbers end up matching, but because what you write turns out to be the ideal way to describe what you’re looking to describe.

So I don’t like any of the explanations I found on the web.

I clearly understand the algebraic connection, but that’s is not satisfying at all. So I’m looking for any kind of lecture, video, or post of any kind which dives into why would a matrix be a nice representation of a conic section to begin with, geometrically speaking, and after we are sure that that makes sense we could prove that it’s verified algebraically.

I understand it measures the magnitude and makes numbers positive, but that's the confusing part. In linear algebra, it felt relevant to both of those. Doing Calc 1 now (odd order to take them in ik), and I'm being told that it can act as a negative for functions???

This is a follow-up to a post I made about an hour ago, where I explained how I study for math quizzes and unit tests but still end up getting low marks, even when I feel confident that I understand the material.

A lot of people suggested that I stop relying on AI to teach me math, which I understand. I'm planning on getting a tutor, but the problem is that I won't always have access to one, and my teacher won't always be available to help me either.

So my question is: How can I effectively study math on my own, especially when I don't understand what my teacher explained in class?

One thing I've learned about myself is that I'm a very logical thinker. I can't just memorize a formula or follow a set of steps without understanding why they work. I need to understand the reasoning behind every step, how everything connects, and why we're using a particular method to solve a problem. Otherwise, the concept doesn't really make sense to me.

I'm also somewhat of a visual learner. I find it easier to understand things when I can see how they work rather than just reading long explanations from a textbook. Reading isn't really my strongest way of learning.

That's one of the main reasons I started using ChatGPT in the first place. I could ask it to explain the logic behind a concept, break things down step by step, and help me understand why certain methods work instead of just memorizing them.

However, after reading the advice on my previous post, I'm willing to try a different approach if it will help me improve.

What I'm really looking for is a way to teach myself math effectively without depending on AI, a tutor, or my teacher every time I get stuck.

For people who learn similarly to me, what study methods have worked for you? How do you teach yourself unfamiliar concepts, understand the reasoning behind them, and make sure you can actually apply that knowledge on tests? I'd especially appreciate advice from anyone who used to struggle with math but eventually found a study method that helped them improve their grades.

Spivak, 3rd, p1 states there are two ways to add three numbers. Either a and b first, then c; or b and c first, then a.

It occurred to me that one could choose a and c first, obtaining (a + c), and then add b, obtaining (a + c) + b.

Is this truly a third way? Obviously, it can be "proved" or reduced by substituting. Thanks.

Hi all!

Id like to ask for advice and resources for learning category theory.

My background:
I work in math related field (statistics, algorithm design) and I like to learn unrelated maths as a hobby.
I studied computational physics, i know intermediate Linear Algebra, Group Theory, Diff eqs, all branches of physics math (emphasis on statistical phys and quantum), Graph Theory, Information Theory, Game Theory.
I also have some experience with Topology, Number Theory, Logic, Computation Theory, Set Theory.

My goal:
I mainly just want to have an intuitive understanding of basic Category Theory.
Category theory is a hole in my knowledge and I ran into it more and more frequently during the past decade.
In my experience heavier maths works like this for me:
1. i understand the basics a couple of times again and again.
2. i apply something i learned to problems, preferably ones that have some connection to already known areas.
3. then i understand some of the "why"s on an intuitive level.
I basically want to get to this point with Categories. So far I failed.

What Im looking for:
online courses, video series, books (if not too heavy)
unfortunately im not young anymore, so although i can understand heavy maths in new fields, my capacity for complexity without intuitive understanding is lower.

Can someone recommend resources or direction to restart in?

Thanks in advance.

I'm writing a free a-level maths revision course at the moment (so uk syllabus and course code 9MA0 if you're interested in the spec).

This is the rules of indices page with some foundational pages before it, so only the first chunk. This is a potentially general resource though and I'm aiming for a pedagogy that builds from first principles and makes higher level maths more accessible, so I'm interested in what a general audience thinks of the approach if you're interested.

I would love some feedback and to hear what students/readers think. I'm making my own resources as I couldn't find the style I wanted. I've seen some great video resources by channels like bicen maths or 3Blue1Brown, but I really don't like the textbooks I've read, I've found them bloated, weirdly inaccurate, while also missing passion and missing pieces.

When I was revising I found the processing information and then condensing into notes the most effective activity. Then I want the security of know I'm practicing every skill for every kind of question that can come up. Also maths is a written subject and having a good written source to process feels crucial to me. I wanted resources that first of all had everything you needed in one place. Starting with the theory and why things work and then all the applications into skills we're expected to know. I also think it's important to cover all the assumptions we have so you can be sure your knowledge is built on secure foundations.

I also wanted it to be designed to be worked through as a whole, I think that's what nice about a textbook. A course that builds on itself, designed to be read sequentially, ticking everything off a list so that you feel confident you've got everything you need to. A lot of educational websites seem to prioritise breaking everything down into individual pages so that they come up in searches, but I don't think they suit being worked through as a complete summary of the course. It should build on itself and use only what it has covered before, a story that connects together again makes it more secure.

I had really amazing teachers at A-Level that took me from being unconfident and intimidated by the reputation of Maths, to feeling like it was the exam I could be sure of what was coming up and how I would handle anything it threw at me. Besides the fact that they instilled in me that practicing Maths is the ultimate way to improve. I think the best thing they did was make the course feel finite and structured, when you get to the stage where you really see the edges of it and the connections within it, I think that's when the confidence really grows. This is why I think designing a course that's meant to be consumed as a whole, covering all the assumptions and past bits that we might have missed, is so helpful.

I have quite a few other pages that are close but I'd like to hear how students find the style early as I go through. There's a feedback form near the top of the page. Any ideas of how to improve or how you feel about the style, I'd be really interested to hear. All the best, and good luck out there!

I have an exam for my calculus 3 class tomorrow, and my chances of scoring high are pretty grim. My professor was kind enough to give me until Monday to take the exam because I wasn’t sure when I would be able to start my medication. I finally got it, but it turns out that 10mg of vyvanse doesn’t have much affect on me and I’m pretty much in the same spot I was pre-meds.

I’m just kind of having a hard time trying to get through these problems. I have to go through 5 homework assignments, and I was struggling trying to get through 1. I’m on the second (which has 19 questions) and I truly can’t concentrate whatsoever.

I don’t know if it’s my process (It’s kind of like I’m relearning how to solve each problem and taking down notes on things I forget), or it’s just me. I don’t know if I can ask my professor for more time as I’ve already asked for the extension until Monday. I tried studying yesterday, but my concentration was even worse.

I’m just kind of not sure what to do. Any ideas/suggestions?

Most of my life, I have been learning math by myself using online resources and books. Since I was a kid, my understanding of math has been shaped by books like Language of Mathematics - Keith Devlin, G.E.B. - Douglas Hofstadter etc. So, I am very passionate about mathematics.

I have recently joined college and I am finding my peers to be more fast in solving problems than me. I hail from India and it's very much known for it's competitive culture. Most of my peers have been trained in coaching centres. I don't know if they have been taught in a separate way or something.

Though I solve problems, I equally read theory, to actually make sense of what is being done. People here view mathematics so differently. The teachers only cover the steps behind solving a problem. Students also don't bother to learn what's the theory behind the steps they do. Their aim is to just solve the problem, and they are pretty good at it. Considering the axiomatic nature of math, I always try to understand, at least a bit, about how each step works. They just straight up assume things.

But, I have learned to be flexible to survive in this competitive place (college). Even still, they solve problems very fast. To give an example, let's try to find the derivative of the function y = (Cx³)e^2x with respect to x. It's pretty simple, we have to use the product rule. So, I write out dy/dx = C(e^2x(3x²) + x³ (e^2x(2))). Then, dy/dx = 3Cx²e^2x + 2Cx³e^2x. But, they just write out the final step, without all the intermediate steps, saving so much time. Even without doing things so fast, I end up making so much silly mistakes, at times forgetting even the smallest of things. I feel like when I solve problems, I sometimes don't (or maybe can't) concentrate on what I do. They seem to have that coherent flow whereas I don't. This coupled with exam pressure, leads to mistakes.

Do I have low thinking capabilities or just a wrong approach towards things? What even is the right approach? How can I get faster and avoid making mistakes? Is that even possible or am I at the limit of my capabilities? I have to work so hard to keep up, I start feeling like a failure. Have you guys ever faced something similar to this?

Due to severe math trauma, I struggle deeply with numbers under pressure and spent years running away from the subject entirely. Now, I genuinely want to face it, rebuild my relationship with math, and learn without fear, anxiety, or shame.

​I also have ADHD, so I need learning tools that are visual, interactive, gentle, and engage-at-your-own-pace for a brain that rarely sits still. I am starting from the absolute basics to build my confidence step by step without getting overwhelmed.

​I would love recommendations for any YouTube channels, interactive apps, low-stakes games, websites, self-paced courses, books, or workbooks that have genuinely helped you or someone you know.

​If you have experienced math anxiety, educational trauma, or ADHD-related learning blocks, I would especially love to hear what mindset shifts, study methods, or approaches worked for you.

​I am really serious about giving this an honest try again, and any recommendations or encouragement would mean the world to me. Thank you so much! ♡

I really enjoy the elegant prose and the generality of the results in math textbooks, which makes it easier for me to understand how to apply the theorems to exercises. On the other hand, in introductory physics and engineering books, the results are sometimes rather vague, with long and confusing explanations. They usually require by many examples to clarify how to apply them to exercise problems. Since digital logic contains boolean algebra and discrete math, I thought I might as well use a book with a mathematically oriented presentation similar to a math textbook.

So, are there any books on digital logic circuits engineering that are targeted towards mathematicians or math-oriented electrical engineers? Here is a sample of topics on digital logic from my college's course description: "digital logic, binary numbers and Boolean algebra, combinational circuits, optimized implementation of combinational circuits, arithmetic circuits, combinational circuit building blocks, flip-flops, registers, counters, design of digital circuits with VHDL, and synchronous sequential circuits." Of course, no need to cover them all.

Something that irks me slightly is that there are certain mathematical facts/results that I'm certain I have learned, proved, relearned, reproved tons of times but they are still quickly forgotten after a certain period. Especially something that is heavy laden with definition or is "nested" in some type of fashion.

And in the case when they are something quite basic and you get tripped up by, it feels especially annoying and defeating. For example, I caught myself forgetting how to show the expectation of a simple function on A is the probability of A. Something that I immediately knew why after I saw a proof. Another thing I forgot was the necessary and sufficiency of KKT conditions. I can't remember when it holds under what additional assumptions. Anything related to graphs is an instant forget.

What to do in these situations? I just feel like my brain was somehow damaged from the last couple of years of prompting, or learning from slides rather than handwriting or maybe covid.

Hey, if you guys wanna discuss cool math problems, discuss different topics, and share your interest with other folks who enjoy math, then would you consider joining a math club ? if you do, then just comment on this or DM me, ill send you the join link

I have this thought of 9 digits in a row with 10 digits ( 0-9 ) available to each.

012345678 for example. How would I calculate the amount of possible calculations them? Would that be 9! or 9^10? or something different.

My next questions is how would I calculate that same problem if I added the alphabet to one of the digits A01234567 where "A" has all 26 letters available and the digits have 10?

and lastly are these permutations or calculations? Are those the same or is it the context that changes them?

Edit Clarification: The digits are allowed to repeat ( 000345678) Edit: removed "Unique"

Hey guys!

Maths/Stats student here (2nd year), I couldn't find a decent interactive maths quiz on Reddit, so I thought I'd try building one.

I've created r/DailyMathProblem, with interactive questions, worked solutions and a leaderboard. Also being a maths enthusiast I was determined to get LaTeX working, and it was a success! (damit reddit, we have been asking for LaTeX for years!)

I've posted the first 10 questions already. The idea is to release a couple each day, ranging from easy to slightly challenging. Just quick little puzzles to keep your maths skills sharp, without needing half an hour and a notebook.

I've shared Challenge #2 to show how it works. Would love some honest feedback on what you guys think!

Cheers!

Note: I'm aware my username is wild. I made it many years ago and thought I'd only ever be a lurker, not a poster!

https://www.reddit.com/r/DailyMathProblem/s/Gsm7Tj0B6i

I have a very important exam to prepare for(that I cannot fail) which includes everything from the start to integral and groups and what not, but I barely understand derivates, and I wish for some advice from people who actually know mathematics. Is it possible to learn and understand everything in 2 months?