Teaching myself an upper level pure math course (we almost died) 

Get the LED Solar System and other science toys over at STEMerch! stemerch.com/collections/science-toys Then something I forgot to mention for this textbook/class specifically, the pre-req's would be single variable calculus, proofs (induction, direct, indirect, contrapositive, etc), basic set theory, logic, and functions/relations. Last time I made one of these videos it was about going through the book 'how to prove it' and that covers everything you would need to take go through this real analysis book (besides the calculus 1 and 2 concepts).

A mathematician calls a problem "hard" as long as he hasn't fully understood it yet. When he finally does, he calls it "trivial".

As a freshly minted PhD in math, I assure you the word "obviously" in a math textbook actually just means "there was something about this list of expressions that was annoying to LaTeX."

"...staring at a problem and having no clue how to start it." Welcome to the life of a math major

"They use the word 'obviously' way too loosely in upper level math courses." Yes. In upper level math textbooks, the word obviously gains an additional meaning. It can either have its original meaning, or it can mean "proving this fact is too inconvenient for me to bother with right now, so I'm not gonna."

Real Analysis is where you talk about all the times your Calculus professor told you not to worry about it.

Flammable maths: "Today we are going to do proofs" Zach Star: "what was that noise?" Flammable maths: You mean proof? Zach Star: There it is again.

I learned my favorite proof strategy from my Linear Algebra professor, “Proof by Force.” Where you state “clearly this is true” and then no one will have the confidence to call you out for being wrong.

"The proof is left to the student as an exercise."

"Well of course it's obvious.....ONCE YOU REALIZE IT!" Truer words have never been spoken. It sounds like a contradiction, but so many times in my Math Major as an undergrad, I would go through exactly that. I'd read something that claimed to be obvious 15 times over for an hour straight before giving up. Then the next day I woke up and I just understood exactly what it was saying. Like, of course it's true. Why wouldn't it be true?

The "it is obvious, once you realise it" part reminds me of a funny story: There's a professor in a math class, during the lecture he declares a theorem and says that the proof is trivial, then moves on. After class a student comes up to him and asks him about the proof that the professor claimed was trivial. The student says he doesn't see how you would do it, and it doesn't seem trivial to him. The professor then looks at the problem and thinks about it. He realises that he doesn't actually immediately know how to prove it. He tells the student to talk to him the next day. That night the professor looks at the problem again and spends all night figuring out how to prove it. By the morning he's figured it out, and is able to prove it. The next day the same student comes up to him and asks about the problem. The professor says: ah yes, I thought about that problem some more, and I can confirm that yes, it is indeed trivial.

More like teaching yourself how to be such a g-dang beautiful person

Undergraduate Analysis: "Ugh... I can't believe I stared at it for an entire hour not knowing what to do" Graduate Analysis: "Holy shit dude I only spent 2 days staring at it not knowing what to do."

Imagine not dealing with phase - This post was brought to you by Complex Analysis

A physics student walks up to his Prof and asks him whether a certain statement can be considered trivial. The Prof, deeply reflects on the statement, but cannot answer the student right away. The next day, the Prof happily concludes to the student, "This statement is indeed trivial."

As someone who is interested in Pure math, Physics, and Engineering, to be able to appreciate them all together is what makes them engaging in any shape or form.

I loved this class. Partly because the teacher was great, and partly because if you are teaching people this class, they are going to be a bunch of fast-witted, snarky, sarcastic dicks. Because these classes teach you to find loopholes better than a lawyer. And we had a lot of fun because of that. But the best part of the class was the 2nd test. I was given the question "prove or disprove the validity of this statement: there exists a rational number R and an irrational number I such that R•I equals a rational number." And I looked at that question for about 10 seconds, and wrote down "R=0" and moved on. When he graded it, he wrote "I technically have to give you credit" One of my friends got the question wrong, and when I told him R=0, he turned bright red and about died in embarrassment because, " I forgot zero existed". But the best part is that on the next test, he asked the same question, but with the added "for R=/=0" in it. It was great.

Mathematicians: Let E>0 Engineers: So E=0

"It was the longest build up ever" Clearly you've never read bertrand Russell's principia mathematica, in which they took several hundred pages to prove that 1+1=2.

My current Real Analysis Part 2 professor had Rudin as an advisor and I feel extremely honored.

"Sometimes I could draw a picture. Which is a good strategy." *laughs in complex analysis*

Hahaha man I've never felt so dumb as when I went through a serious maths textbook. That thing about "just give me one I can solve" really hit home.

Me: thinking about studying math at an advanced level. My brain: Softly 'dont'

“You can spend an hour reading a page.” I thought this was excessive until I realize I’ve sunk 300+ hours into JRPG and competitive games before, which is the pessimist estimate for 1 hr per page on a 300+ page textbook. This is encouraging me to self teach in preparation for my future courses.

Title : *we almost died* Me, a college math student : *I felt that*

Dude this is seriously so validating. I've tried to read through upper level texts a few times but I always get stuck on something or other and I get really discouraged. I feel like I'm going too slowly. Then I just kinda give up. Hearing that you go through the same thing really helps. It's encouraging. Thanks!

Those black and red circle around his eyes... proved 😅😂 Why ( We almost died)

Mathematician here. In undergrad, the first Real Analysis course is generally considered to be the hardest course in a math degree. In graduate school, the Real Analysis course is also generally considered the hardest course in the program. But there are some simple tricks to it. The year after I finished the graduate analysis course, a friend was taking it and was stuck on a problem. She explained the problem to me, and then I said "I didn't follow any of what you just said, but it's analysis, so just approximate it and you'll be fine." She wasn't too enthused with my answer, but 3am on the morning the assignment was due, she remembered what I said and just gave it a try. She approximated the function and then the problem just worked itself out. Complex analysis though, that's different. Way nicer to deal with, and some of the most beautiful mathematics you'll see.

"In mathematics the art of proposing a question must be held of higher value than solving it." Georg Cantor

It's always soothing to hear someone who went through the pain I went through doing Real Analysis

I have no degree and I taught myself programming to a pretty advanced level. Now I’m teaching myself linear algebra, with the goal to learn multi-variable calculus (I know you can learn one without the other, but this is just the path I’m going on). One thing that helps me stay consistent and motivated is learning everything with its applications. Thanks for another awesome upload. You’re definitely inspiring!

Wow, this video is very encouraging, I've been reading Rudin's book and sometimes I get stuck at one concept or exercise for hours, that can be a little discouraging most of the time because you feel that you can't continue.. but you made realize is pretty normal and it's part of the learning process, so let's continue study and never give up!

Mannnn, your videos are gold! I was planning on self-studying throughout the holiday and this particular video miraculously popped up. Thank you.

“10 real world applications I can use in my life, tonight. Go!” -Zach

Even though Im not interested in learning mathematics, I found your video to be very uplifting for my psyche. Hearing that you struggled this much yet perservered and grew your mind past your initial limits...thats encouraging to me! I want to self teach simpler tasks and concepts, hopefully apply the same down to Earth discipline you had summarized.

The problem I have at school is that you have to continue even if you dont understand what are you doing

Man, I absolutely hated Real Analysis. As someone who was pursuing a degree in Applied Math but sadly had to complete Real Analysis as a mandatory course, it took me 4 ATTEMPTS to actually pass that class. Just think about it, I failed 3 TIMES.

I am PhD student in math. Advice I would give to those teaching themselves from a book is to not try an read too much in one sitting. Too much new information all at once becomes overwhelming and discouraging. It depends on your level and how the book is written, but a good starting pace is one to two sections per sitting. You could even do more than one sitting a day. But pace yourself. It will not come as quick as you want. Math is hard. Also to Zach Star, if you liked the advanced calculus/real analysis that you learned, I recommend looking into a similar level book on complex analysis. It discusses similar concepts but with functions over the complex plane. You will see too that there are many "nice" things that work for functions on the complex plane that do not work for functions on the real line. Before going into that point set topology book, I hope your real analysis book discussed metric spaces.

You have no idea how much I needed this video. Self teaching math is something I’ve really been wanting to do once I have some time on my hands and Real Analysis is very early on my list (I’ve already done the calc series). I’m glad I have a first hand account of what its like and what to look out for.

Its cool to see your passage through math on your channel. Each new video you come across more intelligent and more motivated. Thats a great motivation for all of us.

Fantastic content Zach! I’m a math major and I was sort of independently following your path, Velleman and then an intro to analysis to prepare for those three courses I have to take. I’m glad I found your videos on self teaching, you address a lot of the issues (like not getting discouraged, when to move on etc) Thank you!

Very wise young person. A great use of your time, thank you for sharing,

It is great pleasure to see someone as passionate about mathematics as you are.

Dude, just loved this video. This is one of the best descriptions I've seen on how to learn. Because let be honest we have some people good at teaching/learning, but in reality we don't actually know how to do it. (From a rigorous stand point) The steps you've noticed and spoke about on the moments when you do stuff when you actually didn't learn it. Or the moments when you can actually realize and pinpoint that you found something you don't know. Because it's literally impossible to just think of something you don't know to learn. So being able to pinpoint these moments is SUPER important and no one talks about it. Also moving with the conclusion we also don't know how to teach because no one can actually pinpoint what someone is thinking that is wrong, why and what's the thing the person should think to realize they're wrong - and that would allow you to cause the "eureka moment" we call teaching. I mean at least outside of this video I've never seen anyone even speak about it.

Great job! I love self-learning and am on my own self-learning journey for many subjects including teaching myself upper level maths that I never took but for psychology it’s important to really understand these! I would agree with you that reviewing and tests are very important, I make myself take exams, there are a lot of exams or problems on the internet and workbooks, so yea, it’s great that any self-learner be sure to review and test themselves! So nice seeing other self-learners!! 😊

This is a cultural difference I guess, but here in Europe, Engineers take that course. It's usually mandatory, and a fourth one is either suggested or common enough (the third one usually is a 70 h real analysis + functional analysis + distributions + (more serious) serious complex analysis + pde, the fourth one is about unbounded operators, spectral theory, variational problems, etc) Also there is no single "proof" course, all math courses from the first one consist in proving theorems and then a (usually younger) assistant holds an "applications" class where you do exercises

I'm glad you liked that textbook. We use the same one at my university. I remember being so happy when I realised calculus was just a bunch of useless things I was told, since I thought there wasn't that much rigour behind it. And I'm the kind of person who is unable to enjoy a lecture if they're not able to properly understan d it. I hated calculus so much...

this is one of the best videos to watch when feeling confused with compsci/math! your methods are extremely helpful ty brotha

I love how passionate and down to earth this video is

I'm an engineering student that's graduating this month, and this video makes me excited about all the self-learning I'm going to be doing post-graduation!

The bit about "obviously" being used way too loosely in math texts was great. I have a bs in math and, yes, spot on.

Love some Zach Star! Your an amazing man and engineer!

Really like these kinds of videos where you go through textbooks on your own!

That’s the beauty of mathematics. You start from the bottom and step by step you build a structure that explains the world in a way you understand. I recommend V.I. Smirnov’s book called “A Course Of Higher Mathematics”. It goes from variables to limits to derivatives to integrals to complex numbers. This book is remarkable and teaches you mathematics the correct way it should be taught.

As a math bachelor student, what I sometimes do as well is just read the additional chapters in my books we don't cover. They usually have a fun selection of additional topics you can delve into. Topology and Measure Theory especially have tons of interesting branches after you've familiarized yourself with the main material.

Zach, I'm at Cal Poly too, and this video very well might help me make it through. These are methods that I haven't heard before and sound like they will help me a lot, especially since so many of the professors can't teach and so we have to rely on self teaching. Thanks!

I self-teach myself a lot of content that supposedly should be taught at school, and I can relate to your experience so much!! I used to get discouraged from getting stuck on a problem or having trouble grasping the logical flow of things. But after a while, I got used to it. I can understand concepts better as I teach myself from the ground up, with the fundamentals instead of taking pieces of knowledge prescribed by teachers. By making sure I fully understand before moving on, I am able to apply those concepts better in the long term. Highly recommend self-learning!

I love that book! It was the assigned book for my first real analysis course. I enjoyed it immensely. It is the one that I recommend to others learning real analysis for the first time.

Analysis is a beautiful subject indeed. In terms of applications though, one has to learn a bit more to see its full power. Well known engineering methods like finite element have their roots in functional analysis/measure theory; and so does quantum mechanics (it's certainly not necessary to know these subjects for QM but you'll definitely appreciate QM more if you do). Pure maths can seem very abstract and unmotivated but its real power is precisely in its ability to apply these abstractions to a whole range of different contexts. Innocent theorems about a set of linear equations are suddenly generalized to study differential operators on infinite dimensional spaces. My experience has been that I have come to deeply understand the physics only after I have rigorously studied the mathematical theory behind it.

I like your honesty Zach. Your appreciation of the details of Rolle's theorem and how it's generalised to get the Mean Value theorem reminds me of why I liked Real Analysis despite not connecting with it initially.

Thanks for doing this! I personally would like to have more of these recommendation-videos on advanced topics (maths and physics) that do not belong to the standard engineering curriculum.

I loved real analysis, that was such an intense, rewarding experience.

u are my inspiration bro

Thats beautiful. Like an art piece. Good content maye

Thank you so much for taking the time to share this video. I am doing a Real Analysis course this semester and the textbook we are using is 'Analysis with an Introduction to Proof' by Steven R Lay. I've heard quite a bit about the Rudin's text, which I intend to have a look at. I'll also be looking at the text you highlighted in this video. As a Mathematics Major, I can say with absolute certainty that Real Analysis is truly in a league of its own. It demands much more patience and practice than I have ever had to put out for Calculus and Statistics courses. Even Linear Algebra was much more manageable. Now I'm off to do some work in Metric Spaces. I liked and subscribed! Thanks again.

Congrats! That's a huge achievement and you should be really proud. Can't wait for the next video. P.S.: I really struggled with real analysis but I found that it became more intuitive once I learned the basic definitions and theorems from general topology (properties of continuous maps, compact domains, connected domains, etc.).

I used that book in my undergrad in my real analysis class that was a pre requisite for the Rudin based real analysis class! I too strongly recommend this book when you're trying to learn the material. One thing to note is that this text is that the approach was done using sequences whereas most real analysis texts use the concept of distances between objects in their proofs. Abbott does talk about the the latter a decent amount, but my personal takeaway was the using sequences in my proofs which can cause some misunderstanding if one were to go on to a topology or complex analysis text. Earlier this year, Springer, the publisher of the textbook, was allowing folks to download a pdf file of Understanding Analysis for free on their site, although I recently checked and it seems to me that the offer has ended.

This is amazing. Being able to learn textbooks on your own seems like an almost impossible feat for me, but now I am starting to understand how it can be done. Thanks a lot my man for making these videos! Also, I'm probably going to be taking this course in a few years so this video seems like it can help guide me through the darkness and hopelessness of terrifyingly difficult classes. I can't imagine being able to learn an entire book without a TA. I guess once you no longer have a paid staff to help you out, you really start to appreciate what their help.

Thank you for this video! I really needed it.

I feel your pain here - I'm taking real analysis as a pre-req for Econ PhD programs, and with the remote learning, I'm basically teaching myself. I'm not sure how anyone would take these classes before the internet age, so much is online that helps explain concepts just enough better than the textbook for the concept to click.

I mean at my university (Pisa) at the First year of physics we have a calculus exam like that. We Also prove the local uniquenes and existance of the solution of a particular cauchy problem class and everything that leads to It rigorously Like the dirichlet function that Zach described (1 on rational and 0 on irrational) Is the perfect example of non rieman integrable function. A rigorous course Is really Great because you know absolutely where everything comes from and calculus should be thaught like that even at eingeniering degrees

Thanks for this, as someone who also tends to spend free time learning (seemingly for fun) to make an effort to do only enough examples to get fundamental points of how a system works, then spending time learning sideways to patch up occasionally 'obvious' information then realizing that most of the holdup is due to some metaphorically applied definition that is commonly proposed to help understand said system; Having to take a break or switching gears to another subject or area of application can help solidify selflearned topics while you are the most stuck and motivated, walking away motivated and stuck usually gets the gears moving latently if you're not in any rush to cram the knowledge or fatigue one self mentally if you are trying to keep at it and feel the gears gring to a halt before leaving in a stressful state of unmotivated. Of course needing to apply the theory to any project makes for a easier time learning as mistakes being made can physically manifest themselves, but when learning about inapplicable or uninvolving topics can only really be explored if you have someone as interested in the field to discuss with about the theory even if they are not as learned or experienced with the topic as you may be. I struggle to find said persons and it's few and far between that discussion about self application is useful. You do well to propagate your struggles as to help others understand how to avoid and work with those assumed topic's flatness factors . (As in flat on it's face obvious) Sometimes it's the simplest things that we forget we know, when the vastness of a subject matter consumes one's mind and hearing the face-palming fact from someone aware of that you'll overlook it, is the reason I feel most people are not compelled to continue any topic and/or self teaching, Keep up the motivation, it's deep rooted and visceral, and like sunshine on a rainy mental day.

I’ve never really watched Zach Star before (although I watch papa flammy and Dotson) but I really, really enjoy this video. Thanks man, this is nice

I'm in Real Analysis 1 right now, and I like it and hate it at the same time. Its the hardest math class I've ever taken, but its also the easiest class to put work into. The feeling of catching some clever little thing in a problem and it all falling into place is just amazing. Better than building a circuit and having it work first try.

Great video! Thank you for describing the process of learning rigorous mathematics in such an accessible way. I can relate to so many of the experiences you had, and the feelings too. I want to suggest that all of this becomes even more important as you go deeper into mathematics. As a graduate student, I would often recall a theorem, maybe a "small" result, but realize that I didn't quite remember why it was true. I would sit down and prove it to myself again. Sometimes this would be inspired by a question or comment in teaching an undergraduate course. Sometimes it would be in the course of proving a deeper result. Regardless, I always had the "stranded on a desert island" approach. If I were shipwrecked on a desert island with no math textbooks-but perhaps plenty of paper, pencils, and good eraser (don't forget the eraser!)-*how much mathematics could I recreate?* Could I recreate integral formulas? How about Bézout's identity for the GCD? Fundamental Theorem of Algebra? Burnside's Lemma for counting orbits of group actions. Could I construct isomorphisms between finite groups presented in different ways? Bijections between sets of combinatorial objects? etc. This has given me a sense of ownership of and facility with the ideas, like they're mine. So I can use them and play around with them without fear. It's difficult to overestimate the satisfaction of not only knowing a result, so you can apply it, but also to be able to construct it from scratch if need be. Because, as you point out, the techniques in the argument can be just as useful, if not more useful, than the conclusion that you reached with them.

I really like how honest are you about your experiences with the book like saying I spent 2 hours staring at a problem having no clue how to start solving it

You are amazing Zach! Keep with you on Patreon

I have started to do something similar!* I realize that I only ever took Calc. 1 and that's it. I wanted to try my hand at Machine Learning (ML) stuff and when I was going through a course online I realized I had absolutely no clue what was going on. So now, I am learning Linear Algebra with 3blue1brown's help (Also yours! Thank you for that course), after which I'll try ML once again. I'm probably still missing some statistics, but that's this is my start.

An important side effect of struggling with difficult concepts and figuring them out on your own is a deep sense of satisfaction and fulfillment. How did you "feel" as you worked through this book and how did it affect your motivation?

Thank you sooo much. I share so many common points with you as a self taught advanced calculus and advanced algebra course. At first, i used to panic a lot and feel like I have a problem in my self or the way I study. 3 years after I passed this fear and panic and yes, i still get stuck sometimes but I take it simple. I do more research and review different texts of the topic because so many math topics can be explained so loosely. Thanks a lot

Really needed the advice for self-learning, thanks I always feel anxious when I get stuck on a problem with my engineering courses especially that I am self-learning it :( online classes really sucks

Good to hear that I am not the only one who attempts the "stare the problem down" technique fruitlessly, ad infinitum.

My experience is that anything called "obvious" (or similar) in a maths text book is a way to get you to work it out for yourself. Its never obvious. its always something worth knowing

Sir, I learned alot from you.thank you from the bottom of my heart❤️

This video is inspirational, I am planning to read Griffith's intro to electrodynamics, Taylor's classical mechanics and Shankar's Classical mechanics in the next two years. I'll come back to comment once I have completed this quest to update you.

One major problem I usually witness with people struggling in maths is that many people lack a lot heuristic skills, which are, IMO, very essential for math proving. Most would say: "Just do math proofs daily and you'll get better": Now while this does hold some truth, it doesn't really do much if you're just trying to prove something by let's say trial-and-error technique (which takes very long if the proof is more complex). Exercises only help if you develope some sort systemical approach to solve a problem, which requires several heurisitic skills (and I don't mean math proving techniques only; I am talking about more general approaches, such like psychological strategies (sounds crazy, but it's true)). I recommend the book: "The Art and Craft of Problem Solving". You will notice that it can also be applied to pretty much every area.

Me too smooth brain to understand this

awesome content bruh i love seeing when a guy has done something new but doesn't try to brag about it, instead tries to include the listener and give em some interesting stuff to investigate later, rly cool proof there

Zach you are so right, Understanding Analysis is my favorite analysis book for self-teaching. My prof actually mainly used this book but also took some problems from Carruthers. If you already know some analysis, then maybe you would be able to brave Rudin to learn some more, but honestly I was able to take further analysis courses pretty much just based on my background with Abbott. I love this book.

Andrew Dotson comments on all Zach’s and Tibees’ videos... how cute :)

I'm a computer science undergrad, this video feels like he's just summarised my engineering life, even though this is about applied mathematics 😂

My TA recommended this book and I am so glad that he did. I started with real analysis about 3 months back and I hope to have found this video back then, would've been really useful. It's still useful nontheless. Thank you, Zach.

That textbook is the exact one we are currently using in my Real Analysis class. My professor isn’t great IMO, but the textbook is really useful to me. Would definitely recommend it for the same reasons mentioned in the video. There’s also a lot of different resources on the internet of course, and they have been really helpful for certain parts I may not understand at times. I’ve really enjoyed Real Analysis and am looking forward to more in another class soon.

I have a B.A. in Mathematics. The "they use obviously WAY too loosely" hit me hard. Yes, I wholeheartedly agree. My professors told us to avoid using "obviously" unless it would be obvious someone in HS Algebra, at the highest.

I struggle with math horribly and this is pretty much every day of my life with a self-taught PreCal class 😅😂

Amazing video!

Wow real analysis, that was the beginning of my pain as a math major. That class tested my love for math. Self teaching had no effect for me, even though I read the book more than once. Test/quiz day would always show me why I was wrong. Thank goodness for my colleagues and professors, this was a class I just couldn't self teach myself. Damn tests........good times lol

Now try algebraic topology, It is one of the most beautiful things un math

If i have understood what you mean, in italy people in engineering math and physics go straight to real analysis instead of calculus from day one. And yes it's pretty hard. I remember than one question of the theory section was something like : write a continuous function with his derivative not continuous

This is helpful and entertaining! Thanks!