Hardest Math Course

We talk about what is the hardest math course, and how to succeed in this course. What do you think the hardest undergraduate math course is?
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Thank you:)

[@kshitizmangalbajracharya9451]

Real Analysis wrecked me during undergraduate years. All of that was mostly because the teachers usually skipped "Logic and Techniques of Proof" saying that it's not important for exams. I had to sit through the entirety of techniques of proof by myself. It took me a year (after I passed my exams) to fully understand what was going on. If you understand the techniques of proof, you have the power to understand anything in math. Real Analysis is not an exception. As far as what are we supposed to learn, it's mostly how to write proofs and why certain things work. Knowing how to work with math isn't always sufficient. It's necessary to know why they work, if you are a math major and want to become a mathematician.

[@therelaxlead676]

Hey , did you by any chance study in an Indian college ? your name seems Indian and the experience is similar to what my Brother went through when he was doing Bachelors of Science in Mathematical Sciences from an Indian College . If yes then what was the primary real analysis book you used ? Was it from M.D. Raisinghania and Shanti Narayan ( S Chand Publishing ) ? Thanks in Advance Fellow Indian on KZbin struggling through Undergraduate Math

[@kshitizmangalbajracharya9451]

@@therelaxlead676 I am Nepali. I studied in a Nepali university - Tribhuvan University to be particular. But yeah, Nepal and India are indifferent when it comes to highly exam oriented study. Many of the teachers have degrees which they got after memorizing everything for exams and that's what they tried to teach us. Never made sense to me, ever.

[@franciscoreyes7370]

My math undergrad math program required students take an "intro to proofs" course before allowing them to take upper division courses.

[@kshitizmangalbajracharya9451]

@@therelaxlead676 regarding textbooks, I followed the one by Prof. Dr. Prakash Muni Bajracharya (titled Real Analysis) for the first course. It was great enough (obviously after you understand it) to step up and do Apostol and Bartle on your own. If you are looking exclusively an Indian textbook, I recommend Elements of Real Analysis by Late Shanti Narayan and Dr. M. D. Raisinghania - the one you mentioned. Somewhat different would be "A Textbook of Mathematical Analysis" by Late Shanti Narayan and Dr. P. K. Mittal. There are several others, Malik and Arora, Dipak Chatterjee, etc but they're almost the same book with the same carelessness (if any) and with the same mistakes (if any). S. Kumaresan's "A Basic Course on Real Analysis" is a entirely different and a good one too.

[@kshitizmangalbajracharya9451]

@@franciscoreyes7370 that's the problem here. We have annual evaluation and logic and techniques of proof are a part of either Real Analysis course or an Abstract Algebra course - whichever they choose to teach first. But everything is exam oriented and just to save time, the teachers skip all the fundamentals. Probably because they themselves learnt only how to pass exams. Even I have requested many professors to introduce such a course. There's politics and some more politics which never allowed them to formulate it.

[@PhillipRhodes]

The hardest math course is the course you're currently taking!

[@TheMathSorcerer]

hahahahahahahaha

[@shemiahwalker]

Very true lol.

[@Occ881]

Agreed

[@MnkyKing777]

Words of wisdom

[@gregorywhite921]

Ha, you are spot on about real analysis. I did my undergraduate work in the 70s in electrical engineering. Back in the day at this University, they basically had three undergraduate tracks in mathematics. One was for engineers (some of this was a lot of plug and chug), one for math majors, and one for Honors College students. So I was in Honors college and math was easy, right? In one of my calc classes, I was getting really hammered. The other engineering classes I was taking were basically applied mathematics. So I was going under in this one math class. I went to talk to the prof about dropping out of his class because I didn’t think I could cut it. He was an exceptional teacher and explained to me his story. He was also an electrical engineering student as an undergrad. When he graduated, he said he really didn’t understand what a derivative was, or an integral. So he went to graduate school and took a year of real analysis. He said it was like Marine boot camp. He got in to it so much, he changed his major to mathematics and became a prof. To make a long story short, I stayed in his class and he really took the time to help me out. I ended up going to graduate school and got a PhD in mathematics, Not! Stayed in electrical engineering. But I always remembered that one math prof. Yes, real analysis is hard for everybody. Have you checked out the book by Jay Cummings, Real Analysis, 2nd Edition? A little slower paced than Rudin and others.

[@TheMathSorcerer]

Great comment! Yes, I have that book by Cummings, it's fantastic:)

[@kumardeepmukhopadhyay7393]

Was your professor Robert B. Ash by any chance as he has mentioned in his Real Variables book that he was trained as an electrical engineer but later became a mathematician. His book is great for a beginner.

[@gregorywhite921]

@@kumardeepmukhopadhyay7393 No, if I remember correctly, his name was Prof Joel Shapiro. This was at Michigan State in the 70s. Didn’t Ash also write a book on probability? Good book from what I remember.

[@kumardeepmukhopadhyay7393]

@@gregorywhite921 yes he did and one on abstract algebra as well.

[@martinhawrylkiewicz2025]

Papa Rudin.... It was the bane of my existence when I started to learn Analysis... And now after having some more mathematical maturity, I really appreciate it more.

[@kshitizmangalbajracharya9451]

You can think of it like this - every math you learned so far is like learning to talk as a kid. Real Analysis is the first time you actually hold a pen and write what you're talking in the form of letters and words and phrases for the first time during elementary school. Really hard in the beginning but not so much after you do it consistently.

[@benloud8740]

I got through Baby Rudin just fine, once i got the concept of compactness the rest is fairly straightforward. Right now I'm struggling much more with Complex Analysis. My self study life goal is to master Measure Theory and then hopefully eventually Functional Analysis but it seems absolutely daunting

[@dalelee7441]

If i could go back in time and counsel myself, here is the advice i would give. 1. Cut out all of the unhealthy non-academic impediments to learning. Toxic social dynamics, video game addictions, whatever. The endorphin feedback loop of overcoming a difficult math concept simply can't compete with the fleeting pleasures we crave to cover up the unpleasantness of a lousy lifestyle. When this happens, learning math becomes a chore that we struggle to discipline ourselves to maintain instead of intrinsically joyful for its own sake. Once math gets "hard", learning without motivation is impossible. 2. Make a study group of 2 - 4 like minded students and work together as much as possible. When you are taking a killer course, you are going to have some blind spots that are inaccessible even when you do the "right thing". Without a fresh perspective, the problem will never be resolved. The key is to make sure all members of the group are motivated by meaningful conceptual understanding. 3. If your study group made a serious attempt to understand something and you are collectively still stuck, use your prof's office hours. Profs and textbook authors are usually a lot smarter than students and not only will teach in a too sophisticated manner, but it also has the effect of making the student think there must be something egregiously wrong with themselves for being unable to follow along. I wish i had the confidence then to just shamelessly admit that i wasn't getting some seemingly basic concept and then request an oversimplified explanation. If you make the effort to clearly articulate what it is you do properly understand and where exactly your understanding falls apart, even the most intimidating profs will usually be willing to break things down to a more manageable level.

[@samuellundin5328]

In my school in Cameroon Africa, Our hardest Math Course we took as Physics majors was called Analysis 2. It was actually what is typically called real analysis. The main topic I can remember was metric spaces. It was all proof based. The difference with our system to the US is that we already took the US like Calculus in High school. Then our first year Calculus was an abstract introduction to Calculus with stuff like epsilon delta definition and proofs. This metric spaces course was quite abstract with theorems on compactness, closed balls etc. The later portions of the course was quite fun. We did sequences and series of function and integrals dependent on parameters. These were techniques I later found applicable and useful in my graduate Physics and engineering work.

[@yehuihe1825]

Great to hear. Sounds like a very standard Rudin analysis

[@avi184]

For me, the hardest math class was intro to proof writing. Before that class, all the math classes were just computational, but proof writing was really different. That class made me feel like I know nothing about math whatsoever, but after that class I took real analysis 1, and I found analysis to be not as hard as some other people say it is. I think it had to do with lack mathematical maturity and not being familiar with proof writing. But given time, anyone should be fine with those class. Sometime it takes while to understand some ideas but when it does, you really understand it.

[@nestorv7627]

My exact experience. We had weekly quizzes for our proof writing class. I failed my first quiz, and I had never failed a math quiz or exam until then. At that moment I learned that this math class was going to need more than intuition alone. I spent 5-8 hours each weekend reviewing the course lectures. Even still, I could only manage a B- for my final grade. Needless to say, my mathematical maturity improved a lot, so much that Real Analysis felt like a breeze

[@avi184]

@@nestorv7627 Indeed, I like that fact that we are always mathematically maturing. Like, before taking higher math classes like, analysis, algebra, the topics like measure theory, point set topology, seems something you would never be able to study, but once you take analysis, metric spaces, topology seems like just the continuity of the the previous classes, and feels very intuitive. But, right now, for me at least, looking at the actual research papers, and thought of having to publish my own research someday feels somewhat out of reach. But, I know in given time I will be mathematically mature enough so that those won’t be out of reach, though I know it would be hard.

[@saturn5732]

I am currently a grad student, but my hardest math course during undergraduate was functional analysis. I took it after real analysis

[@TheMathSorcerer]

Oh wow that's pretty cool that you took that as an undergrad!

[@kshitizmangalbajracharya9451]

Wow! Really, it is quite rare to hear that somebody took Functional Analysis during undergraduate.

[@danilojonic]

@@kshitizmangalbajracharya9451 why? not everyone is from the states. I am from Europe and here we have 3-4 Analysis courses during undergrad period. Real analysis as first year, and then on second/third there are complex analysis and functional analysis

[@expl0s10n]

@@danilojonicsimilar here… we got 4 basic analysis courses (Elementary real analysis i,ii, elementary complex analysis and metric spaces) and 4 advanced classical analysis (real, functional and complex of course)

[@shararan23]

Same, I had func analysis in my final year of undergrad!

[@diamartin6865]

I passed my Analysis I class this semester and here are some things that helped me: 1. Continuing learning even if i didn't understand a definition or theorem. It would sometimes take days, or several exercises for it to click for me, but as long as I push through, it clicks at some point. 2. Researching. I started off reading mathematics stack exchange forums to find solutions to my assignments, but most of the time i either couldn't find the exact question or i couldn't find a proof that i could understand. I used to feel really frustrated but now i see it as a blessing, because while looking for another question i would find really interesting new ways of solving things, and that is what helped me passed my exams when most my friends failed, because there were tricky/unorthodox questions that required further reading. 3. Watching videos. If you have the time, watching videos that are more on the popular science side makes it more fun. But there are also more university level animated videos that not only gives you definitions or the theorems, but also explains the idea behind it and illustrates it if possible. There are also many proof videos, they are less fun but still really informative. 4. Rewriting proofs. Like mentioned in the video, I also rewrote proofs i didn't really understand. I would copy the proof a few times and try to write it myself without any help. The line that i get stuck at when i am writing without help is usually where i struggle to understand the idea/logic and/or lack the intuition. So I would focus on that step, understand it better and try again. 5. Studying to understand. What I lacked during the semester was to sit down to understand, not to get assignments done or prepare for the exams. You can solve exercises and assignments without understanding the logic, which was what i did the whole semester. I would copy some solutions, derive some answers from textbooks and get okay points. But before the finals I had to understand deeply what was going on, so that I could generate solutions myself. I used to see it as a waste of time, and if you don'T actively try to understand and research i still think it would be a waste of time, but if you actually push yourself and use all available resources, understanding the theorem/topic will help you solve all fundamental questions.

[@jorymil]

Studying without understanding is one of the things I find infuriating about certain classes... there's simply not enough time to come to grips with everything, so you fall back on mnemonics and/or memorization. I'm taking organic chem right now and find that the memorize first, learn later aspect of it is driving me up the wall.

[@definitelynottigerwhitten5865]

Really cool thanks!

[@phnml8440]

For me i would say it was/is Algebra 1 (aka commutative Algebra). I‘m a mathematics student in Germany studying in my 4th semester in my bachelors degree and this course is by far the hardest yet. Our professor is himself a pretty well known algebraic geometer and is lecturing algebraic geometry 1 next semester and is using algebra 1 this semester to lay foundations for that lecture which makes this lecture pretty hard.. I‘m still motivated tho and am not doing as bad as i thought i would so i will keep on building my mathematics abilities and master this course. Edit: This course is about commutative Rings, Moduls and homological algebra

[@adamfarah3060]

im doing it right now it is hard

[@TRLBAN1.0]

Love your videos! You’ve really inspired me to take tougher courses at my highschool and I’ve really taken the time to study at home and even do some math in my own! Looking forward to where life takes the two of us, hopefully I can learn all that you have learned in so brilliant a way! Thanks again!

[@kazemattt]

Algebra, for me it the toughest one was algebra. Analysis was ok, not that it was easy - it wasn't, but at least I could get the idea what it was about, and with enough grind I could get through it. The abstract algebra was different - that part with group theory, rings, isomorphisms and stuff. I just didn't got it at all, up to the level of crying out of despair. Bit of context - 1994, a post-soviet country. Studied in an university with quite fundamental background. Teachers were not messing at all - you don't understand something - it's your problem. Soviet education system was not very "user friendly" to say the least ;) and that was just few years after Soviet collapse: everything including teachers, books were still quite Soviet. Teachers were not bad, some of them were really good. But they were bit like one of the Soviet math books you reviewed - they were not messing with you at all. You can always ask a question, but damn it before asking a question you'd need at least formulate the question, and even that was the hard part. Simple "I don't get it" somehow was not acceptable. First semester we had bit of linear algebra, matrices - that was not bad. And then they throw the abstract stuff on you, without any warning. And suddenly you need to pass an exam at the end of semester in Chinese, about China, taught by Chinese who speak only Chinese. That was the feeling. Level of abstraction was way above what 19 year old brain can handle. And I was not alone in that - many of friends complained about the same. I suffered from Algebra PTSD for many years, and only much much later started to appreciate it.

[@ahmedmalaq7410]

Same boat, here in Geneva we actually started first semester year 1 with Real Analysis which was probably our first introduction to the word of proof and also Linear Algebra which was about matrice, resolution of equations using Gauss Method, etc. First semester is already hard for a lot of people and the failure rate is around 70% for the real analysis exam and around fifty for the algebra. But now, second semester we’re having abstract algebra which is what you are describing : Group, Body, Ring etc. We’re laying the foundation to arrive at Galois Theory next year (which mean second year of Uni). And for me this part is harder than the real Analysis course. Even though real Analysis is way more rigorous than what we were use to, which is a bit frustrating in the beginning, we were already a lot familiar with the topic from high school. But to understand Algebra you need to have a completely new perspective of what you were used to. It’s really a level of abstract above. Like going from numerical equations to adding variable for the first time. And moreover the teacher is talking to us like we are familiar with groupes, isomorphism, homomorphism, Coset etc so she’s going pretty fast in the explanation and proofs

[@ahmedmalaq7410]

Here in Geneva we actually started first semester year 1 (Freshman Year for you guys) with Real Analysis which was probably our first introduction to the word of proof (the course was basically proposition-proof-proposition-proof and so on) and also Linear Algebra which was about matrice, resolution of equations using Gauss Method, etc. First semester is already hard for a lot of people and the failure rate is around 70% for the real analysis exam and around fifty for the algebra. But now, second semester (still Freshman year) we’re having abstract algebra which is is about Group, Body, Ring etc. We’re laying the foundation to arrive at Galois Theory next year (which mean second year of Uni, Somophore year I guess for you guys). And for me this part is harder than the real Analysis course. Even though real Analysis is way more rigorous than what we were use to (first time using epsilon proof…), which is a bit frustrating in the beginning, we were already a lot familiar with the topic from high school. But to understand Algebra you need to have a completely new perspective of what you were used to. It’s really a level of abstract above. Like going from numerical equations to adding variable for the first time. And moreover the teacher is talking to us like we are familiar with groupes, isomorphism, homomorphism, Coset etc so she’s going pretty fast in the explanation and proofs

[@peterbenoit5886]

My advisor forewarned me that real analysis was a "watershed course," but the epsilon-delta proofs seemed like the most natural thing to me. I finished the course with a 100 average, and then never looked back. All credit to the professor, whose lectures were a model of clarity.

[@TheMathSorcerer]

Awesome !!

[@kjerstiow]

I taught myself real analysis as a 2nd year math major. It was certainly not easy, as everyone else has said, it’s so much different from any other course. I spent so much time understanding each and every proof, and did every exercise in Understanding Analysis. When each problem can potentially take hours, it really makes you appreciate the beauty of what you’re actually doing.

[@spacetimemalleable7718]

Real Analysis is a difficult course. Unless you're a genius, you need to ensure you have the TIME to study and do as many proofs/problems and create counterexamples as possible. I recall spending 2-3 hours reading just ONE page trying to understand WHY a statement was "obvious" or "clear"

[@gregorywhite921]

Ha, did he say “it is obvious to the most casual observer”? I read that in one of my books long ago. Gotta love it.

[@lavieestlenfer]

Then it obviously isn't obvious. These types of textbooks are garbage.

[@Mnaughten601]

Real analysis was difficult for sure, but I found my abstract algebra course was way harder for me. There were only a few proofs towards the end of analysis that stumped me, where I was lost day 1 in abstract. So glad to be done with that stuff.

[@mouldyfart]

Learn logic first!! Pick up a book in symbolic logic. Don't sleep on predicate logic . Learn about the *rules of inference* (especially introducing variables for universal generalization and existential instantiation) that will guarantee that your next statement is correct.

[@kshitizmangalbajracharya9451]

100%. That's absolutely true. Not knowing quantifiers and techniques of proof will wreck you insanely hard.

[@LetsLearnNemo]

What I commonly find for students is that the "hardest" math course is the course students view as difficult before they start to go through it. I see a lot of people go into course such as real analysis thinking it's hard and immediately create a mental block on themselves, making a lot of topics that can be very intuitive if viewed through the right lens (such as Cauchy sequences or uniform continuity) but rather view them as "difficult" because they haven't spent the time to understand them. Historically in our learning career, long division of decimals and adding fractions with unlike denominators can be quite "difficult" when we first encounter them, but retrospectively, they look quite easy. Generally, every course especially in math can be difficult if you aim to understand every detail and concept of that course. Just my 2 cents :)

[@diamartin6865]

Also, here is a list of theorems/implications/lemmas that i noted before the exam, that i used religiously throughout my studying for Analysis I final: Sequences and Convergence - squeeze theorem - monotonic and bounded implies convergence - a_n, b_n convergent with all a_n

[@ansonpang8468]

I thought this was about Math 55 for one sec.

[@jackwatt8988]

Very glad that it's not. That was my thought as well.

[@mariotrevino4023]

lol saw this on criminal minds

[@CalBruin]

just curious, how is that? What episode and the scene that brought that class up?​@@mariotrevino4023

[@AhmadKhan-dn6yh]

I'm a second year mathematics major and enrolled in the analysis and geometry concentration so your video is going to help a lot! Fun fact! My ODE professor and Linear algebra professor said real analysis was the toughest course they took in their math journey as well!

[@vaughng.5522]

I'm a student at the university of waterloo in Canada , and I'm taking my introduction to real analysis course right now, but I'm on coop in a different city so I had to skip every class and teach myself the course material. got my exam in 8 days, worth 75%. pray for me.

[@audience2]

I'm glad to hear it is real analysis as that's the one I found hardest.

[@Extinguisher123]

The thing is most people overlook definitions ,in real analysis or math definitions are truly powerful they can help ease proofs because trying to do proofs without understanding the definitions makes it harder to memorize and harder to understand the solutions ,by understanding definitions you have already solved half of the problem.

[@douglasstrother6584]

Draw pictures: learn to visualize the situation. Construct a proof on the real line to see the logic of the argument.

[@RangerKun]

I was lucky enough to have taken enough of the prerequisites in my undergraduate to take the graduate level real analysis course (measure theory). Undergraduate real analysis was comparatively gentle. Proving stuff about Cauchy and convergent sequences wasn't too bad. The beginning of measure theory was fine, but I was *not* prepared for the tail end of measure theory. The first class where I just had no clue what was happening anymore.

[@expl0s10n]

The toughest course in my undergrad career perhaps is Representation Theory. I got a pretty weak foundation in abstract linear algebra (mostly because the professor of that course is just like Ms Google who read the book aloud without explaining) and our professor didn't provide a textbook recommendation that covered everything he taught, nor a set of lecture notes.

[@Douchemaster_McChest]

For me, my hardest Math Class as an undergraduate was Abstract Algebra I, which covered Sets, Group Theory and the first part of Ring Theory. I think the reasons it was the hardest for me are because of three things: 1) It was my first math theory course I ever took and I took it in my junior year instead of my senior year, 2) It was taken during the Fall semester during football season and since I was a starter on the team and busy with football activities, I probably didn't put as much time into it as I needed to. 3) In addition to Football, I had to work about 20 hours per week too that semester. I got a B in the course and it was the only B I ever got in a Math course during undergrad studies. To be fair, that math course wasn't the only course that I didn't get an A that semester. I think there was a General Electives course I coasted through and got only a B. I think I was taking 18 credit hours that Junior fall semester. When the second semester came around for the Spring Semester, I didn't have to worry about football taking time away, except for offseason workouts. Also, I wasn't strapped for cash either. So, I didn't have to work that semester. I took Abstract Algebra II (same professor, who was very good) that Spring, which covered the rest of Ring Theory, Field Theory, and other special topics. I aced that course, while also taking and acing Complex Variables, Prob & Stats II, Operating Systems, and a couple General Electives Courses (I think that semester was 18 hours also.) So, the time you have, the time you need, and the time you put into something really does contribute to the difficulty of any courses, in addition to how the course is being taught to you; ie. How good the professor is in teaching the subject matter.

[@ahmedmalaq7410]

Same boat, here in Geneva we actually started first semester year 1 with Real Analysis which was probably our first introduction to the word of proof and also Linear Algebra which was about matrice, resolution of equations using Gauss Method, etc. First semester is already hard for a lot of people and the failure rate is around 70% for the real analysis exam and around fifty for the algebra. But now, second semester (still freshman year) we’re having abstract algebra which is what you are describing : Group, Body, Ring etc. We’re laying the foundation to arrive at Galois Theory next year (second year of Uni which I guess is Somophore year for you guys). And for me this part is harder than the real Analysis course. Even though real Analysis is way more rigorous than what we were use to, which is a bit frustrating in the beginning, we were already a lot familiar with the topic from high school. But to understand Algebra you need to have a completely new perspective of what you were used to. It’s really a level of abstract above. Like going from numerical equations to adding variable for the first time. And moreover the teacher is talking to us like we are familiar with groupes, isomorphism, homomorphism, Coset etc so she’s going pretty fast in the explanation and proofs

[@parsatalaie9892]

I would highly recommend the channel "Bright Side of Mathematics"! Helped me tremendously throughout my analysis course.

[@sebastianrodriguez7157]

This is what a math undergrad takes as a first semester calculus course here in mexico, excited to see what I'll take in my next semester when i take real analysis.

[@umbraemilitos]

Real and Complex Analysis classes were really clear for me. Algebraic Geometry was the most difficult for me when I was in school.

[@EvilKarateClown]

I washed out of math undergrad a few decades ago and am now going back to study it in my free time. I'm working through Abbott's Understanding Analysis (after Hammack's Book of Proof), and it seems well-geared towards self-study, especially once you find the solutions manual and can check a lot of your own work. I really like the "complete this proof" exercises, which give me the nudge I would have gotten from recitations or bugging the professor. However, for a lot of proof exercises, it seems like there's a "How the hell did you come up with that step?" that seems to come out of nowhere. Not sure how to learn that, to be honest.

[@Apuryo]

having watched your videos, I was really afraid of proof writing, but when I took discrete math it was very easy for me. However, when I did Calc 2 I struggled because I did Calc 1 during covid and so I spent the whole time doing catch up😂. Luckily my university has a course on basic analysis before they cover real analysis, think Jay Cummings book for a semester and then Royden for 1 year. I think that really helped students first get used to how analysis worked, then jumped into the rigor

[@varun3282]

For me as a high schooler who's just finished school and about to enter college , it would be Analytic Geometry. About to become a math major,excited! :)

[@jorymil]

Hey... that ain't easy. I tried to derive the dihedral angle of a tetrahedron the other day for Chem class, and found I couldn't do it without help.

[@davidnelson4960]

It's been a while since i heard such good testimony. You rock

[@vonBottorff]

"...use more math books" Bingo. You pile up many, many references and eventually somebody explains it the way you'll understand, the light goes on! One great in-betweener book would be Tom Apostol's _Calculus Vol1..._

[@red1bk190]

In my university most people agree the hardest courses are galois theory, numerical analysis and PDE's (the pure math PDE's, not the engineering type). Real analysis was of course hard, but the professor that taught us real analysis would give exams that were relatively easy to pass. The course that most people failed was actually group theory, for the opposite reason: The teacher would give ridiculously hard exams. So i guess the experience of a hardest class really depends on how it is taught. I have heard of people jumping straight into Rudin for their very first analysis class, which is a TERRIBLE way to learn analysis.

[@jorymil]

The professor makes such a difference! To some, it doesn't matter to them whether their students understand or not--the class will be taught the same way. It doesn't have to be this way.

[@guidichris]

And 3, we love the smell of our books!!

[@markborz7000]

Analysis = Calculus + Topology. Topology, that's why analysis is so difficult for the beginner.

[@Manuel421]

Even though I despised Math in school I have to admit that now as an adult I can truly appreciate it more and actually find it more interesting for some reason so thanks for all these insightful and fascinating videos. I understand you mostly discuss textbooks but have you ever read books like Fertams Last Theorem or Flatland? Both clearly have Mathematical backgrounds and are interesting reads. Hope you enjoy your weekend.

[@ccbgaming6994]

I’m doing an honors maths sequence at my university where it’s all proof-based (kinda like an intro to real analysis) starting the first semester, where we go from first principles to real numbers to limits to calculus and then branch off from there. It’s tough but I’m sure it’s worth it, because you won’t be surprised or maybe even disappointed by how mathematicians actually work, after being 2 full years into a math major. It’s a great way of thinking even if you don’t want to pursue math research.

[@nitinmadan4009]

I did functional analysis at TU Munich, and that too without a math background. I have never been more lost in my life than in those lectures.

[@arg1051]

My hardest math course as an undergrad was topology, by far. Analysis was easy for me. My hardest graduate course was probably category theory.

[@Seyir_Kanali]

What determines whether a lesson is difficult or easy is whether there is a sense of curiosity. It is the teacher who awakens and develops this feeling.

[@DFivril]

Real analysis had me feeling that I had started learning "real" mathematics for the first time in my life. I was a bad student and spent very little time on it, but in my later years I have revisited my textbooks and its fucking beautiful.

[@dalitlegreenfuzzyman]

I always found abstract algebra MUCH harder than real analysis. But I do think I’m unique in that matter

[@kazemattt]

You are not unique, it was the same for me.

[@dalitlegreenfuzzyman]

@@kazemattt “unique” was poorly chosen. “In the minority” would’ve been better.

[@quasiscorner2589]

I went to a small college. When I took my 3rd Calculus(Multidimensional Calculus), I was the only person in the class. The professor gave me VHF tapes that he had made while teaching the class in the past. This is before the online stuff. I couldn't say that I was diligent about watching 5 hours of Calculus videos weekly. It was a summer course. I completed it with a C and took linear algebra, differential equations, and probability afterward, but that advanced calculus in the windshield terrified me. I felt like I had winged the last one. I was at a large state University now and all my math classes were me and 10 Asian guys. They were a lot more dedicated than I was. If I wouldn't have had to take that advanced calculus class I would have obtained a Math degree but I was terrified of it. Economics here I come.

[@BabdoSilentMovies-ku1jt]

As I heard before , the math inside actuary science is the hardest ever. Apparently it is not that hard to go to the university course but graduate is a different thing, a lot of student just abandon actuary during the years before getting the diploma.

[@Apuryo]

actuarial science im my opinion is more like mathematical statistics for insurance than a real math program. when I first did probability theory, some things didn't make a lot of sense until I did probability and measure. there's something to abstraction and definition which helps make more tangible concepts easier to digest. for example, the gamma distribution and beta distribution was very confusing for me, but after taking real analysis and looking back and those distributions, I was able to understand where the notions of chi square testing variance for statistical inference comes from. of course it is possible to simply use a calculator, but the proof based intuition provides a good base for depth of clarity in a topic. this is why I think most actuarial programs should focus less about drilling insurance mathematics and rather focus on theory of statistics, then things like generalized linear models, then stochastic processes, etc and work on a top down approach. at the same time, they should do an extremely bottom up approach with computation. learning how to do a hypothesis test for regression isn't too difficult, in fact they do this in high school statistics. I think a rigorous top down approach along with a practical bottom approach simultaneously will culminate in student epiphany moments towards junior year, and they will really appreciate what they are doing. right now, I think the programs are half asses as they try to do both but don't do either adequately

[@yassineghariani1505]

In my country, we don't do calc 1,2,3 and then real analysis. We directly do real analysis with only prior knowledge being highschool calculus . I think that maybe this is better since you start doing "real" math and proofs from the very beginning of your degree. Most of my classmates found topology to be really difficult but for me the most challenging undergraduate course has been measure theory although it ended up being one of my favorite courses. Functional analysis is also up there for sure, but our professor was one of the best in the country so that helped !

[@shyamatiwari8088]

Sir, you are a genius!! Intellectually- gifted by God with Mathematics and logical efficiency 😊 I fervently admire your quintessential teachings and knowledge. You may find it quite stupid; but I would reveal it to you:- I perceive your videos as the most soothing and comforting during times of depression I find a sense of positivity after exploring your channel as I too possess a pleathora of interest and good command over this subject. Felt like sharing sir ❤ Thank you! 😊

[@TheMathSorcerer]

Thank you so much for this comment:)

[@ShadowMind312]

I had a really tough time in Real Analysis, so i dropped it and retook it. It was a bit easier the 2nd time around, i got a B plus. But then i breezed through Set Theory and Topology, got an A and A minus in those courses.

[@johns.8246]

I felt Abstract Algebra had a bigger spike in difficulty midway through while Real Analysis felt more consistent throughout. I've heard the toughest class at my university was Statistical Inference. Luckily I never had to take it.

[@0marble8]

For me, out of the more "basic" courses, probability and statistics was definitely the hardest, I found it just so counter-intuitive

[@Taylor-rx4yb]

I'm in my first algebraic geometry course using schemes right now and it's so hard. It's so much effort with so little gain. I'm definitely getting better at it and I like it, but it's impossible to see myself as comfortable with the material right now. In undergrad analysis was tough too. I think for me it was really hard to adjust to how you need to think for analysis but after I figured that out it was ok.

[@simonservetar5391]

I had a terrible teacher for Real Analysis. Every class, he would pull a proof up from his notes-block text, single spaced, no visuals-and just read that shit in monotone with his back turned to the class. When I'd ask questions, which I did multiple times every class, he would struggle to understand, and he would blitz out incomprehensible answers without making eye contact. It felt like the class and he were divided by an impenetrable, plastic hazmat barrier. We could see each other, but we could not communicate. I consider it one of the biggest disappointments of my college experience.

[@Grassmpl]

Also you often prove something in analysis that ends up being a definition in topology.

[@chipdenman863]

Advanced Engineering Math was black magic.

[@newwaveinfantry8362]

Analysis was annoying at first, but it's definitely not the hardest class I've done in my bachelor's. Number theory, dynamical systems, differential topology, differential geometry, numerical methods, algebraic geometry, Galois theory, modules & categories were definitely worse. I think my university is at an unusually high level though.

[@wickz7119]

As a physics student primarily, a second course in algebraic topology covering cohomology and homotopy theory. The first algebraic topology course was fine since my knowledge of point-set carried, but learning singular cohomology without graduate level algebra was pure pain.

[@lavieestlenfer]

Real analyis made me quit math. My professor was worthless (being good at math and being good at teaching math are very different skill sets) and never once told me how to improve my proofs. The workload was insane. The textbook had a lot of "this is obvious" or "trivial" or "elementary" as a justification for skipping steps that I needed to see. It wasn't until I picked up Jay Cummings's book that I realized I actually can do this if it's taught well.

[@cjpwdesign6108]

My two favourite books are Elementary Classical Analysis by Marsden, and Advanced Calculus by Sokolnikoff. Also murray speigel advanced calculus (schaum book) is good for worked through problems. Finally richard courant's books. Fyi i am a practising structural engineer, not a maths major so not a maths whizz, but competent enough for doing structural and stress analysis!

[@nwosukingsley-pi8wn]

I don't think there is anything like hard course in mathematics it's just the way your brain can comprehend that determins the course that is hard for you , what you think is the hardest course might be a peanut for another person

[@andreysafin3224]

Studying functional analysis right now. It seems to me that it's harder - it's just that we are now familiar with the "setting" so to say, so we don't consider it frightening

[@jorymil]

I found real analysis hard because the fundamentals of logic, proof by induction, proof by contradiction, contrapositive, etc. weren't part of the math curriculum at my school. I took discrete math at a different school a few years later and light bulbs came on! So I feel a bit disserved by my original undergrad program, even 20 years later. I'd love to have the time to shore up my knowledge. So... young math major: take discrete math first! It won't make the proofs easy, but at least you'll be able to see which type of proof to use. I think there's a meta-problem here worth discussing: in every program, there's one or two classes where you feel like you're consistently behind, regardless of effort--it's like your brain doesn't absorb everything. But given a different professor or different textbook, things click. Self-teaching skills seem like something that could be taught: how to find the best resources when confronted by a mountain of different texts, websites, etc. Sometimes "just work harder" isn't the answer.

[@misraaditya9213]

IMO the hardest is the least familiar one. I blunted abstract algebra and real analysis with headstarts of self-study, but mathematical statistics got me working way harder than I'd anticipated - it was the least familiar.

[@alexcastrotello5165]

Mathematical stats is HARD. Is the hardest part for me. I’m taking a double degree, one in applied maths and one in Actuarial Science, and I think that Mathematical Stats is one, if not, the hardest. Greetings from Mexico City

[@deepkarmoker-fo8ew]

Love from Bangladesh

[@MaheshPatil-lr7wv]

Algebraic topology

[@jedediahjehoshaphat]

It's Field Theory and Galois Theory for me, the prerequisites alone bogs me down to frustration

[@greense65]

The baseline knowledge you should get out of a real analysis course is the following two things: 1. The definitions of the concepts you found yourself using a lot to solve problems back in Calc I, II, and III. 2. How to prove the theorems that you found yourself using a lot to solve problems back in Calc I, II, and III. I think these two things represent the personal reasons for taking this class--to understand why the techniques you were using in basic calc worked and to convince yourself that they indeed DO work. Beyond that, do whatever else you need to do to perform well in the class. But having a reason, other than "This class is required" should help provide some direction and motivation to get through what some people, incorrectly, see as a weird and pointless class.

[@masonr1666]

Numerical Analysis was a class I got a D in, and I accepted, and moved on with my life. I went to school to specifically be a High School Math Teacher. I was disappointed with how much Math was required, and how LITTLE "how to share information" was taught. I know I am not good enough to do Math at the Profesional level, so after I failed at teaching Math (High School Geometry), I got a job at McDonald's. Either go back to school, and learn a different field of study and waste more money, or cut my loses, and just start earning money. I went with the second option. I had Calc 1,2, and 3, Liner Algebra, Modern Geometry, Projective Geometry, Introduction to Higher Math [aka how to do proofs], Numerical Analysis, Modern Algebra (groups rings and partially ordered sets), Statistics with and without Calculus, Applied Calculus, Pre- Calculus with Trigonometry, Math for Education Majors (I may be missing a couple of Math Classes) So, I make minimum wage now. (For about 20 years.) At least I know the proper angle to flip the burger at.

[@jorymil]

But I bet you can empathize with people well, and might do well as a tutor, even if the classroom teaching experience wasn't your bag. A master's in education might also be a route, as might a master's in library science to be a math/science librarian: both of these deal with how to communicate.

[@dominikriegler]

How about Royden's Real Analysis book? I wanted to get into Real Analysis just for fun. I have my A-levels in math (test taken years ago) and stopped going further in different A-level subjects so I didn't finish the other one. Since I did my math A levels exam, I got deeply interested in math. Topics we covered in this course were Integral and differencial calculus, Probability and Statistics (binominal distribution), curve sketching, bacterial growth rates (e^x) (how much medication would you need to take and at which timestamps if you're ill), functions and graphs (prerequisite to curve sketching), area under a curve, and some equations. Would I have rested, I probably would have forgotten it all. But I didn't rest, my curiousity got sparked and I always wanted to go further in math by learning it myself. You really make excellent math book presentations and I'm really thankful for that. Not only are you providing some insights in various books, but you also provide some videos to help people by explaining the most important things many people are struggling. Knowing how to learn math is essential to get any further. By watching your videos I really gained some valuable advice and insights regarding which math topics are coming up in university and which books are excellent material for learning it. I think that when I really dedicate myself some minutes or an hour every day to use that time wisely and learn some math instead of wasting it every day, I will be able to understand more and more about it and this will open the door to understanding more and more topics in math and get good at it. My belief is that if I use the time to dedicate myself enough time to understand something, I can use that knowledge from there on and apply it in different ways in my life. My mind wants to be challenged and I know that only by doing the hard things, the challenge will be beneficial. Sure I could keep doing easy math problems but in order to grow, it's necessary to get a little bit outside the comfort zone. I want to accomplish that by reading "Baby Rudin" and I also have ordererd Royden's Real Analysis book. My inspiration for math started long ago with watching "Numberphile" (my favourite videos are about topics like Grahams Number, Conway's Soldiers and those phi numbers (evil twin), and Hilbert's Hotel, where infinity isn't even enough. Mindblowing, right? But with those infinite busses of guests and infinite rooms, at some time, the hotel is full.) I love things like that. It makes us understand the concept of infinity a little better. Anyway, I realized that just by watching those fun math videos, I won't get much better so I have to start doing math again by myself. Those books are my starting point, because I really like the fact that you can present a given statement in a way such that it's impossible to deny it. By writing proofs. Should it really be too hard for me, than that's not a problem either. There are many other books to be read on different topics in my bookshelf as well. For example "Concrete math" by Knuth, Graham and Patashnik. As of now, I only watched videos about one method of proving something. It's by contradiction. I wonder: are there other different ways to prove something? I don't know that yet. I will find it out myself when I start reading Baby Rudin or Roydens Analysis book which both should arrive soon. In the meantime the only thing I can do is getting into this topic more by watching math videos on youtube, especially proof or Analysis videos.

[@GreenxWine]

Just to be sure, you do know that Royden's book require analysis on the level of baby Rudin as a prerequisite, right?

[@dominikriegler]

@@GreenxWine I know now. Thanks for that. Let's see if I will survive Baby Rudin first by learning it myself. If the answer is yes, then I know what comes next. Thanks for the info. I really appreciate it.

[@xizar0rg]

I had a really good prof. for real analysis... still the hardest class I took in grad school. Even the shit I did independent study (had a couple I needed to get done in the off-semester to get my ms on time) went more smoothly.

[@tassotzobiko1135]

What about discrete math ? Is it supposed to be hard ? Advanced ?

[@ModuliOfRiemannSurfaces]

I’m in grad algebra 2 because he said he’d cover sheaves. Derived category is definitely one of the roughest constructions I’ve seen. Still, even through this, real analysis is still worse to me because I still am uncomfortable with ε-δ constructions.

[@davesynth3860]

It’s funny how you mentioned the teacher matters. I have friends that became engineers and they scored the worst in Geometry PAP in high school. It was a weed out course and the teachers would just test and quiz people vague things from the textbook…that’s all it took to weed people out lol.

[@franciscoreyes7370]

There are no hard math courses, only hard math professors.

[@okohsamuel314]

😂😂😂😂

[@TheMathSorcerer]

:)

[@kanalbostur7190]

the truest words I've ever heard

[@claytonbenignus4688]

This course ruined me. Though I actually got my Math Degree, I was mistaught on my first pass through it. The course was dumbed down on my second pass because my school was trying to run up their graduation count. This played into my desire to finish my degree quickly. As far as Math was concerned, I was brain damaged.

[@nekkinekki4721]

When I joined uni as a math undergrad, the 1st year's, 1st sem's 1st class was Real Analysis. So long story short, majority of my class failed, a lot of them changed majors. And failed, not by some marks, majority got straight up 0. Also, the teacher would just recite the theorems and its proof and then move on without explaining. She was a phd student and most of the time, she would not end up coming to class because she had her own work.

[@Nezha-Ishibashi]

Real Analysis we proof theorem's and stuff like that on calculus I, II... right? So learning how to proof things is something we already have to know or my teachar will give lessons on how to proof things? Im doing physics rn, but i really like demonstrations and things like that

[@STEAMerBear]

It’s whatever is taught by the least comprehensible professor. You did address this, and Real Analysis is certainly hard. But MY discrete math prof could not manage to explain MANY really simple ideas. A lot of people changed majors because of him despite him being regarded as brilliant. Maybe academia needs to revisit allowing gifted researchers and gifted instructors to do what they excel at and discard the presumption that every academic should be competent at both instead of forcing people do demonstrate their particular weaknesses at the expense of everyone involved.

[@CanadaElon]

For now, I am struggling with matrix derivatives, and I know that's because of my weak linear algebra.

[@SenorGato237]

I haven't taken real analysis, but for me it was trig. What got me was all the rote memorization, there was very little understanding of why things were the way they were.

[@Thoth608]

the hardest course for me was logic and set theory it was hell hard like realllly abstract shit i took me months of work to be able to write some proofs

[@mitchellhopkins9331]

How about differential equations? I feel like my friends and I would say it is that class.

[@jnclouddragon9648]

Perhaps one of the easiest and most fun math classes!

[@kshitizmangalbajracharya9451]

If you are doing Partial Differential Equations with all the theory, probably. Differential Equations after Calculus-III isn't really that hard. But again, that's just me. I love Differential Equations.

[@Supercatzs]

It is fairly challenging if you aren't great at calculus to begin with, but, I would still say proof based courses are much more difficult.

[@slottibarfast5402]

I have noticed in some descriptions of KZbin math tutorials how so many students depend on these to get them through. Personally most of my math teachers were bad. No KZbin back then. So why is that. One of my problems was foreign profs that had terrible English skills. Then there were professors that just could not relate as everything seemed obvious to them. If you had a question they would just repeat what they had said. Why is that?

[@dinesh4april]

Wow , your videos inspire me to work hard in maths , could you guide how to master trigonometry from basics. -this is sanvie.

[@lousupino9236]

Abstract algebra

[@andia565]

I was struggling a bit in measure and integration theory but the hardest course was probably pde. Not sure why those courses are in undergrad classes already.

[@Apuryo]

I made the mistake of taking accelerated diff eq which meant doing ode and pde in one semester. it wasn't too complicated, just cumbersome. writing out proofs of heat and wave equations then solving examples took HOURS

[@argonwheatbelly637]

I took RA. It wasn't as hard as CA (had a bad TA trying to teach it), which I did take, but it was fun, still.

[@citizencj3389]

Algebraic topology is a nightmare....ugh

[@abubakarmohamedmohamud2926]

Topology is the hardest math class I've ever taken. Still, I don't have any deep understanding of it.

[@chrisroberts314]

Projective Geometry was "something"...

[@gustavoholo1007]

I think once you said it was Mathematical Statistics, I wonder what you think of it now

[@rohanthomas5434]

Learn a lot about inequalities

[@billdicklechips]

im so angry that im graduating with a math bachelor’s in may and my university never offered real analysis in my last few years, so i didn’t get to do it. The same is true for PDEs

[@TheMathSorcerer]

Oh wow that's really strange that they didn't offer analysis.

[@manofsteel9051]

What type of math degree was it? Pure math or applied math? You can always self-study analysis if you're interested in it. It's incredibly difficult but very rewarding

[@billdicklechips]

@@manofsteel9051 pure math

[@billdicklechips]

@@manofsteel9051 Pure math, I think I will self study if I get time. I also need to review the undergrad physics curriculum a lot in preparation of my up coming PhD qualifying exam.

[@burak3924]

Functional analysis is by far the hardest course.