I was reading Terence Tao's essay "A Close Call" and was really shocked to learn what happened during his PhD general oral exam at Princeton.

Although he was a child prodigy, he played video games, attended classes that he enjoyed, dropped out of the ones he did not, and barely studied for two weeks.

During the oral exam, he couldn't answer even basic questions that were not related to his strong area. His advisor Elias Stein bluntly told him that he was disappointed in his performance.

And this near-failure made him realize that disciplined study is needed, and that relying only on raw talent will not take him far.

It really makes me wonder: how can someone with that level of natural talent almost fail? Does every 'genius' eventually hit a wall where raw ability stops working and discipline has to take over?

For those studying math here: at what point in your journey did raw intuition stop working, and how did you learn to actually study?

Hi guys,

I am just writing this as I have found interest in learning in general. I haven’t used my brain for anything technical or practical… maybe just intuitively haha.

I really want to learn mathematics to improve in learning sciences and engineering.

Any suggestions and advice on how to lock in?

Currently working a full time labor job that doesn’t require much math or thought.

8.5 hours of work and the rest is sleep…

Main topics of math I want to learn:
Differential Equations (Partial) and Trigonometry

But all math would be useful!

Want to become a math person by the end of the year.

In advance: Thanks for all your help and support!

So hi all, I hope this is allowed. If not a direction to a more fitting sub would be appreciated. I'm bad at math. Always have been. My interest is more in the humanities, and as such when I was in school I just sorta suffered through it.

As an adult though, my chief special interest, philosophy (mostly continental) has shoved me in the direction of wanting to learn the stuff. My main problem is I'm not only bad at math, I also just have learning difficulties on the whole. I've found I do best when I have some kind of visuals, something I can relate to (which makes learning more abstract forms of math like calculus incredibly difficult), and some form of applicability or demonstration.

Now, that said, outside the school system I've found I've learned things a lot easier. But with this subject I'm not fully sure where to start. The last things that really stuck in my craw were elementary algebra, but I'm really interested in learning calc because it collided with my special interest.

Any suggestions about a good YouTube channel where a guy like me might learn a bit?

Hi everyone. I have a question or better said I’m seeking help. I’m 34 years old, I graduated high school back in 2011 in Eastern Europe. Now I’m in USA and math here is a bit different from what I’ve learnt. Any advice how I can improve my learning so I pass this course ?
Thank you

At the start of school, I was actually one of the top students in my class. I was genuinely excited to learn new math concepts and loved the subject.

Unfortunately, burnout hit me hard, and things completely fell apart around the COVID-19 pandemic. A mix of burnout and bad experiences with teachers turned my enthusiasm into total resentment. I stopped trying, fell behind, and now my grades are red across the board.

I’m sick of struggling and want to change this. I want to build a solid routine, fix my math foundation, and actually understand what I'm doing again.

I'm looking for advice on where to start filling in the missing gaps from past years without getting overwhelmed. What structured websites, YouTube channels, or textbooks are best for self teaching from the ground up? Also, if anyone has experience using AI as a math tutor without relying on it to just solve the problems, I'd love to hear how you structure your prompts. Lastly, how do you overcome the mental barrier and frustration when opening a math book after hating the subject for years?

When I do some counting problems I always miss some cases that seem so obvious in hindsight. When doing casework what method do you use to make sure that you got all of the cases down. Do you just have to ponder on the question for a long time making sure that you hit all of the cases.

Why does a negative exponent turn into the reciprocal of a number with a 1 in the numerator? (a^-n = 1/a^n)

I understand that, for example:

2^-3 equals 1/2³

but I don't understand where the 1 in the numerator comes from. And why does the exponent suddenly become positive? I just want to understand.

f: R --> R

f(x) = 1/x

What would be the domain of this function? By def:

f: A ---> B , then Domain(f) = A .

Then according to this definition, the domain of the f(x) = 1/x should be R but f(0) is not defined. In the textbooks, it is written that the Domain of this function is R - {0}.

It is really confusing.

I have a physical copy of this esoteric LA book just sitting around called "Linear Algebra an inquiry based approach" with a link here, but everyone usually recommends Strang or something else for getting started. It's a pretty large book so I don't want to get started in the wrong direction. Is anyone familiar with this book and is it good?

I’m just about to start the Graphing Trigonometry Functions section in my Advanced Functions study and I’m looking for any advice or tips on creating a scale using either graph or lined paper.

Graphing has historically been a roadblock for me in my math journey and I find setting up the graph before even plugging any data in to be the most difficult part.

Any advice is appreciated.

I am thinking of a graph with a line y=x+1. The problem is that my line fluctuates at some points because of either an increased rate or the probability of this not occurring. Basically my question is how would you graph a non-determinable equation like this? I feel like it’s impossible due to the chances being infinite. My highest level of knowledge is Algebra 1 btw so do not get angry.

Can anyone tell me how far reaching are the consequences of Henstock integral over measure integral? I took a course on Henstock integral in one semester but I really couldn't understand it (it felt dry for some reason) and so didn't like it. On the other hand I found measure theory to be one of most beautiful things in math. I didn't really remember much but Henstock integral involves intervals I guess, so my question is if I'm handed with an abstract space where there is no concept of distance or intervals, then what would be my first choice to do integration Henstock or measure? When I was an undergrad student of 3rd semester I attended a seminar on integration where the professor told us that Henstock integral is the most powerful integration theory till now, although at that time I didn't even have concepts of Riemann integral.

Let's just say I've been out of high school for way too long so when my daughter comes home with algebra homework and ask for my help I am somewhat clueless. Ask me to do the quadratic formula I got you for some reason I remember that but ask me to rewrite this random basic formula into standard form and I have no idea please help.

An open top box is formed by cutting squares out of the corners of a 3-foot by 5-foot piece of cardboard then folding the sides. The volume V(x) can be aiven by V(x) = (3-2x)(5-2x)(x). Rewrite this equation in standard form.

V(x) = (3-2x)(5-2x)(x)

I am doing an engineering degree part time and got through Calc 1 with a B.

Calc 1 in my school finished with integrals/IBP/U-sub, and slicing by volume for cones/cylinders.

The reason I mention it is I recently applied for a new school (my current one changed the semesters to quarters and I didn’t appreciate the pacing) and I’m feeling out of my groove. I got married this summer and took the summer off of doing math courses. I’m a part time student/full time worker and it’s been about 8 months since I did my calc 1 class. Looking back at the practice tests, I feel like I’ve lost every ounce of knowledge I had prior to that.

Does anyone have any suggestions? I’m leaning towards retaking calc 1 even though I already passed.

I find that I am the absolute slowest learner of math because I insist that I should be able to always understand the logic behind the solutions, even when I would already be able to get the correct answers through applying correct formulas. Otherwise solving problems feels like a waste of time.

It's why I always found math incredibly frustrating in school. I had to spend a lot of time on it but didn't really understand what I was doing and why I was doing it that way.

For example, right now I am studying factorials and trying to understand why exactly there is a multiplication happening. I understand that n! will give us the number of different ways in which n can be organized, but I don't understand how and why the multiplication of all parts achieves this. This happens to me with many different types of math problems.

I know I am dense when it comes to STEM subjects, but I'm just wondering how I should best deal with this? Do I just move on or do I continue to battle, trying to understand the underlying logic?

Hey y'all! I need some help knowing what to review from my Calculus I & II courses for my next math class. It's been about a year and a half since I took a math course, and I'm super rusty. What are the biggest things I need to review so I can succeed in this course?

The fun thing with learning Math is that you will not forget the concept once you understood it. Even 5-6 years on you can still recall all the important points just by glancing at it. Isn't it?

I had to drop a class and now I have extra time to learn things. I do plan to transfer at the end of next semester so I should focus on applications, but I also figured I would learn a couple of things now instead of waiting. I just wanted some advice on what path I should take and maybe any resource suggestions.

To start, I was thinking about reading “how to prove it” by vellemen and “basic mathematics” by lang to help solidify fundamentals and learn proofs before transferring. I’ve been at CC for a year now and I know most learn proofs around this time (to clarify I’m in my third semester).

Since I find physics interesting, I was going to try and start learning physics. I take physics next semester (it’s a long story that deals with my scholarship and how it operates) which is the semester I plan to transfer. I was going to read halliday and resnicks book on physics while watching “the fundamentals of physics” lectures on MIT OCW.

Which leads to my final path of learning computer science foundations since I’m still interested in coding (and competitive coding). I’m interested in machine learning and see it as a potential career so I figured it wouldn’t heard to learn some concepts now and figure out if I want to pursue it. They posted a pathway and each course is about 6 weeks. I can attach the link if you would like to view it.

I know I said that was my last path, but since I want to go to grad school, I figured learning how to do research is also an option. Stuff like learning how to properly read research papers, learn LaTeX, read past papers to get an idea, learn how to write papers, etc. This can probably be done while reading how to prove it and basic mathematics though.

Which do you recommend doing? Should I combine any of the paths I wanted to take?

Edit: Forgot to add another thing I wanted to do. I finished calc 1 last semester and was thinking about doing a review since I might have to take advanced calculus 1 when I go to university. I remember a little from a calc 2, but felt like I need more practice and to delve a little deeper since the course was accelerated. I’m sure there were a couple of things that had been left out so that’s I wanted to review to make sure.

Im 18 and im currently struggling with absorving easy math stuff. The way i judge if i undertood something or not is asking myself if i could come up with the solution without decorating the formula. For example, right now im really struggling with combination without repetition; if i got 4 elements and i had to combine 3 of them without repetition, why do i have to divide it by 3! ? Like, i tried to visualize it and i simply cant. I did wrote all possibilities and tried to undertand what does 3! have to do with all the combinations starting with 1 - repetitions = 3 and starting with 2 = 1 and the pattern it follows if i add elements or number of elements i want to combine. I could never write a formula for that and i dont get it

How do i solve this? And what kind of problem do i have? Am i just slow and i have to train work memory or is this linked to another math concept that i havent searched? Please help

I came to realize yesterday that, like the complex numbers, other finite ring extensions may be represented as matrices over the base ring being extended. It was honestly shockingly trivial once I took into account that a finite extension ring is also a finite dimension R-module, and that left-multiplication forms a linear map, I'm shocked I wasn't taught this before.

This did make me wonder how much theoretical utility the equivalence offers though. I was able to draw up a few simple conclusions, for example I believe that isomorphism of extension rings corresponds to similarity relations of matrices, though I could only prove one direction. What I suspect is more powerful is that I noticed the determinants of these matrices do correspond wonderfully to the norms defined on the complex numbers, as well as over extension rings of square roots over the integers. Given that the mapping is a multiplicative mapping back onto the base ring I figure it permits a study of these extensions through this determinant? E.g. unit determinant iff unit in extension, zero-divisor/zero determinant iff zero-divisor/zero in ring, irreducible determinant iff irreducible in ring. I'm not familiar enough with matrices over non-fields to exactly prove these, and even then a premise included is that the inverse/annihilator will exist IN the subring, not just the ring of matrices itself, but it is very suggestive, especially as I know these relations to be true in the extensions of integers I mentioned.

Either way, I'm curious if my suspicions about this determinant are true (or inversely if these determinants are a special case of a property more advanced and detached from matrices), and more generally if this matrix representation lends itself more readily to certain theorems concerning extension rings/fields, or if anything which you may prove with it may just as easily be proven using the typical polynomial factor ring approach to finite extensions?

I think the prof will most likely expect intuiton, examples, demonstration of understanding what's happening. However, I have holes all over Abstract Algebra and they're very hard to fix because, well, they're abstract.

I thought I just had math anxiety or impostor syndrome, but I am genuinely behind my peers. I have an undergraduate degree in math, am beginning a graduate degree in math next semester, and have presented at multiple math conferences. I am part of multiple math meetups, and so on and so forth..but I always felt off.

I always felt like the people around me were way smarter than me, and I finally figured out why.

I decided to prepare for my next semester of grad school by reviewing math topics I learned as an undergraduate, but it was so difficult that I ended up decreasing the difficulty of the topic being studied until I reached high school level math. I am studying from an Art of Problem Solving book right now and it's kicking my butt.

I'm scared that I even got this far while being so bad at math. I think it's possible because my problem solving skills are poor but I have an above average memory, which can compensate sometimes.

Do my recommendation letter writers know I'm below average? Does my grad school research supervisor know I'm below average?

Is it possible for me to catch up before the semester starts? I have 16 weeks (including this one) and plan to read 13 books, but this AoPS book (written for actual children!) is taking me so long to get through...

What would be the best way to spend this time preparing for the next semester, knowing how behind I am? (Please don't say it's hopeless...)

Hi all, I did A-level maths > 10 years ago, which is probably my last formal education in maths. I’m in the medical field, and only use basic maths and probably a bit of statistics in my day-to-day practice. Trying to relearn maths again - if you have any good book recommendations please let me know, thank you.

I am doing maths major in my undergraduate going through linear algebra I don't want to see the proofs and any hint possible. Just want to solve on my own if I can . But I am struggling in it , can someone give me tips who have gone through that In past .

Like why is 7⁻² = 1/49 and not -7 * -7

Ai: this is a discussion wanna lemon 42 republican spark what do you think quantum physics furries