Ask HN: How to self-study mathematics from the undergrad through graduate level? - <AUTHOR> Hey HN community, I've been looking to get deep and build my math skills from the foundation up. I have the time to dedicate to this endeavor and I'd love to hear if you have any specific resources/curriculums you recommend.
Something like https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics would be ideal, but more focused on applied math.
One idea I had was to complete the MIT open courseware courses for the Applied and Pure math fields <comment><USER7> Here's my personal opinion about how you should approach this.

It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses.

However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything.

And so I ask, what really is your goal here? You don't have to learn everything about mathematics, because that is in fact impossible.

My advice is to FIRST construct a bunch of projects, tasks or goals that require knowledge.

It could be something like (a) implement a machine learning algorithm to do X from scratch (b) implement a simple physics engine (c) try to verify a number theory conjecture (d) be able to solve all the exercises in a book (e) be able to write up a compelling description/theorem/problem in math (d) numerically solve the quantum mechanics equations of a certain system

Spend some time on material that will inspire you first to help get these goals. Numberphile on YouTube, or any of Brady Haran's videos, is a good place to start. But make the goals your own and make them personal.

Math is not a spectator's sport. Make sure to DO mathematics, not just LEARN mathematics.

<comment><USER8> Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.

<comment><USER9> Not necessarily. I left undergrad after four terms and self-studied to the point of having published research, giving invited talks, and even being a visiting researcher for three months with my expenses paid.

Links:

[http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf](http://content.algebraicgeometry.nl/2017-2/2017-2-007.pdf)

[https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508...](https://projecteuclid.org/download/pdfview_1/euclid.ecp/1508292095)

Look for my name (Thomas / Tom Price) on these pages:

[https://web.archive.org/web/20170121000748/https:/wwwmath.un...](https://web.archive.org/web/20170121000748/https:/wwwmath.uni- muenster.de/u/deninger/about/)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_spring...](https://www2.math.binghamton.edu/p/seminars/arit/arit_spring2016)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...](https://www2.math.binghamton.edu/p/seminars/arit/arit_fall2016)

[https://www2.math.binghamton.edu/p/seminars/arit/arit_fall20...](https://www2.math.binghamton.edu/p/seminars/arit/arit_fall2017)

<comment><USER10> Wow, that's remarkable.

As an aside, your work on numerical cohomology appears to have been useful for a new result pertaining to lattices. Given the authors of the followup work it's likely helpful for the study of lattices in post-quantum cryptography.

<comment><USER9> Are you referring to the "An Inequality for Gaussians on Lattices" paper? They cite my paper, but it's to give an example of an application of their result (which I use), not because they built on it. But anyway, I think it's very fascinating that the people who discovered a key result that I needed for that paper, which could probably be best classified as arithmetic geometry, are mainly computer scientists!

<comment><USER10> Ah, thanks for the clarification. The computer scientists who work on quantum computational complexity and post-quantum cryptography tend to be much more mathematical than the norm :)

</comment></comment></comment></comment></comment></comment><comment><USER11> There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices.

Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).

Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.

Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.

Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.

After that, where you go is going to vary enormously. Based on what you're aiming to do.

<comment><USER12> Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.

<comment><USER13> From what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it.

Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.

<comment><USER14> When you refer to engineers, do you mean actual engineers (BSc in Engineering)? or your web designer with jquery skills who calls himself engineer?

<comment><USER15> In my experience (as a civil then software engineer with a MS in CS in ML/Big Data) "real" engineering is much more prescriptive than software.

When you're designing a real-world engineering project, the entire specifications are defined legally (through national, state and local laws) and technically in manuals/books. Many engineering specifications will describe the work done to a T before you even need to think about it i.e. "water main shall be constructed of 12'' coated DIP at depth no less than 2 ft". A lot of the challenges are managerial and logistical.

All of this is on purpose. "Traditional" engineering disciplines are more mature and have the constraint of being safe for the general public. There isn't much room at all to creatively deviate from what's already specified.

I've found software design to be a lot more technically demanding in regards to designing and building things. There's a lot less precedent, more moving parts and many different ways to do one thing.

</comment></comment></comment></comment></comment><comment><USER16> Here is a proven approach for at least the first part, building foundations and being ready for graduate work. Many Berkeley Ph.D. students passed through this route. Get the book "Berkeley Problems in Mathematics." It contains historical problems from the Berkeley math prelim exam, and solutions. Now don't look at any solutions yet.

This is the exam all Berkeley math Ph.D. students must pass within three semesters of arriving to stay in the program, and the fail rate is about 50%.

You will also need reference books, advanced undergraduate and beginning graduate textbooks. Buy, download, or borrow as appropriate.

Pick a problem (start with the older ones, they are easier). Set aside 30-60 mins and try to solve it. No devices, no references at all, go to a library or a coffee shop without your devices. Dont' give up till time is over. If you cannot (usually the case), still don't look at the answer. Hit the reference books (don't look up the problem online either, it will go right to the answer and you won't learn much). Read and try to understand enough so that you can solve the problem. It is ok if you solve it this way (in the course of reading about it).

For bonus points, students studying for the exam will typically take entire old exams (available from the Berkeley website), take that to the library and just sit down for three to six hours and try to solve all the problems correctly. Then self-grade harshly. When you can do that for a recent exam (and get a good score), you will have more or less mastered undergrad math to the point that you could teach it.

Most important: you have to struggle to solve problems. Reading a solution is about as useful as watching someone else lift weights: you get minor tips on form but not any stronger.

<comment><USER17> Thanks for the excellent analogies!

</comment></comment><comment><USER18> The hard but maximally useful thing to do, in my opinion, is to regularly meet with at least two other people and a blackboard and beat your heads against it together for about 3 hours at a stretch. Do this at least weekly--and preferably more often. Self study in between meetings is obligatory.

One of those people should be at about your level. The other should be farther along.

All three of you should trust each other enough that nobody gets caught in a shame/guilt/ego loop.

<comment><USER19> Just to add, it's also worth finding someone who might be a bit below you in terms of maths ability. If you can explain the concepts to that person, then you know the concepts.

<comment><USER18> Yep. You can do this, if you're lucky, with the triad. You and the other person at about your level can trade places depending on subject matter, each explaining to the other and reinforcing their own knowledge.

</comment></comment></comment><comment><USER20> As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards.

There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning:

 _At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat's Last Theorem, which says that the equation xn + yn = zn has no nonzero whole-number solutions if n is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem's simplicity, its solution uses some of the most cutting-edge mathematics around. "I understood nothing, but it was really fascinating," he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. "To this day, that's to a large extent how I learn," he said. "I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff."_

[1][https://www.quantamagazine.org/peter-scholze-and-the- future-...](https://www.quantamagazine.org/peter-scholze-and-the-future-of- arithmetic-geometry-20160628)

<comment><USER21> An approach might work well for a Fields medallist but less so for normal people.

<comment><USER22> I love generative art, so I quite often run into things like Lorenz attractors. I looked up how they are made and that lead me to differential equations which lead me to better understand calculus. So my love of pretty graphics lead me to learning math with a purpose.

I haven't won any awards yet.

</comment></comment></comment><comment><USER23> Look at any undergraduate course curriculum at your university of choice and just follow that. Some of them might even have lecture notes and exercises published.

As a maths undergraduate, I attended around 10 classes per year - you go see who the instructor is and then just read the lecture notes at your own time at home. Then go take the exam at the end and that's it. That sums up my 3 years of undergraduate studies.

Mathematics is well established, essentially has been frozen at undergraduate level for 50 or more years, so there is plenty of material. Also, you don't need any equipment, just your own mind.

In my opinion, it is the easiest major of all if you can follow the logic. No essays to write and no projects to do - just read the material.

<comment><AUTHOR> I was thinking about completing the MIT applied and pure math major course requirements via open courseware

</comment></comment><comment><USER24> I did this a few years ago, now doing an MS in math, after last formally studying math in high school, more than 20 years previously.

There is lots of advice in this thread, including long lists of books - <USER2> your level, I think that's unhelpful. To get started with the foundations, I think it's best to have one or two books at most, that you work through _completely_ , _then_ decide what to do next. Otherwise you can waste a lot of time time figuring out "what to do next" instead of actually doing math!

I would personally recommend "Engineering math" and "advanced engineering math" by Stroud. The first covers "foundation" topics like algebra, trigonometry etc, then covers the first year of math for a English engineering degree. The second book covers the second year.

Note that English engineering degrees are 3 years vs 4 in the US, but have no general education requirements, so the math covered in the two books is roughly equivalent to the 1st-3rd year of an engineering degree in the US. Both books are very much focused on calculations, not proofs, but will give you a fluency in handling mathematical calculation that is _assumed_ when doing more proof-based courses later. And they are both designed for self- study, which is important.

The other book that I would get, and do in parallel once you have done the foundation part of the first stroud book, Chartrand, Mathematical proofs. This will teach you how to do proofs, using mainly (high school) algebra and basic number theory to start, but going on to cover some proof techniques in analysis, advanced algebra and others areas.

Once you have done those 3 books, you will have a solid basis for further study. I also agree with the comments about interacting with others studying the same material if possible.

Happy to chat sometime if you want to message me (contact info in profile)

</comment><comment><USER25> The University of Oxford has all of their notes and exercises available online [https://courses.maths.ox.ac.uk/overview/undergraduate](https://courses.maths.ox.ac.uk/overview/undergraduate)

If you get stuck you could try asking on [https://math.stackexchange.com/](https://math.stackexchange.com/)

Although, if you can, finding someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.

<comment><USER10> Many of the courses listed there have no course materials available or only very incomplete materials.

<comment><USER25> Oh really? The ones I cliked through to did!

</comment></comment></comment><comment><USER26> Hello, I have actually done this. I learned Algebra up to a good amount of Vector Calculus over the course of four years mostly through self-study.

I would leave for work an hour early and either sit in my car or go into a Starbucks and do math. Doing time before work is important. That's when you are at your best. Then after work I would sit in my car and do math for an hour. Then on the weekends, in the morning, I would do three hours of math straight, sitting in my car. Then sometimes in the evening I would do a first pass over a section that I knew I would have to really think hard about the next day that way I had sort of taken the first layer of difficulty off.

Why in my car? Because I simply cannot sit at home and focus enough to study math. I've tried the library but it's too restrictive on what you're allowed to do "oh f __*! that 's how you do that" doesn't generally go over well in public places. Plus I feel very safe in my car. I can be relax which makes learning a lot easier.

I always take one day off a week (for me that's Tuesday) but aside from that I don't skip.

Get your books at half price books. Math book reviews on Amazon are almost always wrong. I usually have to try two or four books before I find one I can understand. I would just get a book and read it in your car. You don't need the pressure of following along with a schedule where you fall behind and miss out on a topic. You also don't want to miss a topic just because the teacher chose not to include it. Some of the coolest topics/examples don't get covered when you take a course because they don't have the time.

Also, when you get done with a book go through it again and take good notes on note cards that you can review on your drive to work or while you are waiting around. You'll find yourself going back to them over and over again.

<comment><USER27> Bet you weren't married at the time. I'm trying to re-learn calculus myself, and I have to hide it from her because she gets mad at me when I try to do calculus problems: "why are you doing this? Are you doing this for work? You don't have to do this. There's no reason for you to be doing this."

<comment><USER28> I hope you are kidding to some degree. Having to hide things you want to do from your spouse doesn't sound like a good relationship.

</comment></comment></comment><comment><USER29> SOME ADVICE BEFORE YOU DO A DEEP DIVE INTO WHATEVER YOU END UP STUDYING MATH- WISE:

My Background: Current Undergraduate in CS and I recently added Mathematics

The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. You will encounter countless difficult proofs in any mathematical topic you try to study. Read a few books on mathematical thinking and proof techniques before/during/after reading any other dense math book.

Like you, I realize the value of having a mathematical mindset and want to have a deep understanding. When I added math as a major, I had a very hard time jumping from computational courses (typical math courses, geared towards any major) to theoretical and conceptual courses (proof-based courses that use all the fun and interesting math books everyone has linked here). These books helped:

[https://www.amazon.com/How-Read-Proofs-Introduction- Mathemat...](https://www.amazon.com/How-Read-Proofs-Introduction- Mathematical/dp/1118164024)

[https://www.amazon.com/How-Think-Like-Mathematician- Undergra...](https://www.amazon.com/How-Think-Like-Mathematician- Undergraduate/dp/052171978X)

[https://press.princeton.edu/titles/669.html](https://press.princeton.edu/titles/669.html)

 __< 3 this is a great book, obvi since its george polya __

<comment><USER30> > The most difficult part for a person who hasn't done a lot of math to become > a person who does a lot of math is to read and understand rigorous proofs.

I'm afraid you haven't delved into any advanced topics. That's actually about the easiest part, and could be mastered by ten year olds (certainly myself when I was ten).

<comment><USER31> alnar is likely referring to the US system, or something like it: unless you are tutored externally (rich), an autodidact outlier (gifted), or selected for honors courses, you basically take computational courses for 14 years (with one cursory stop for euclidean geometry) and are then thrown into proofs at the age of 19-20, if at all. it's widely recognized as a problem in the math pipeline, which is why many US universities have "transition" courses for non- honors students.

so it's not that the basics are intellectually difficult as much as practically difficult (unfamiliar, disorienting) for many students. many "transition" books talk about the difficulty in adjusting from talent being redefined from perfectionist "plug and chug" (APs, SATs) to reasoning and creativity.

btw, i'm impressed that you could master college-level proofs at 10. i have a kid about that age who is pretty good at logical reasoning, but i'm not sure what topic (at that level) he could _do_ a rigorous proof about; maybe numbers, as in landau? can you say more about the materials you used?

<comment><USER30> This has little to do with education systems, I was talking about objective difficulty. Try to tell any mathematician that the hardest part of math is rigorous proofs; if they don't laugh in your face, they are just being polite. One may think the transition is "hard", until one actually gets into more advanced topics. As you said, it's just unfamiliar to the uninitiated, at best.

Reading and presenting rigorous proofs in elementary number theory, Euclidean geometry, etc. is easy for gifted ten year olds and definitely manageable for a lot of fifteen year olds. You asked about material — it doesn't actually matter, and I don't recall specifics; any entry level treatment of elementary number theory should do (really beautiful subject with a very low barrier of entry). For kids who have eyes on IMO, it's very common to be throwing around perfectly rigorous proofs at young ages.

</comment></comment></comment></comment><comment><USER32> IMHO the books being advised here are too high level. Its like trying to read about Computer networking basics from RFCs. Some advise are equivalent to searching for a reason to learn "C" by appreciating what can be done with "C" by reading books on Linux Kernel programming.

I will mention a few books which by no means are undergraduate level, but are per-requisite for undergraduate level study. Are you thorough with books given below? These books are available through [http://gen.lib.rus.ec](http://gen.lib.rus.ec).

1\. Higher algebra - Hall and Knight 2\. Trigonometry I - Loney 3\. Coordinate geometry I - Loney 4\. Calculus of one variable - <USER3>

Download the books and check if you already know all those. Once done, then go for the higher level books most the people advising here.

</comment><comment><USER33> If we are talking about content you can find plenty of advice here or elsewhere. But do yourself a favor and pay a tutor and/or find a study group. As in writing, dancing, etc. you cannot evaluate your own work good enough.

You will not improve your math watching youtube videos and reading books. You need to produce stuff that pass the "sniff test" to your colleagues.

<comment><USER34> _You will not become good at math watching youtube videos and reading books._

That depends on exactly what you mean by "become good at math". If you're talking about "becoming a mathematician" and doing original research in pure math, then you're probably right. But if one means "learning existing math well enough to apply it to a problem", I would argue that one can learn this stuff just using books, videos, etc. At least up through a certain level.

That said, I do encourage the idea of finding peers to work with. I used to coordinate a "math study night" at the local hackerspace for that exact purpose. It fell off because I got busy and couldn't keep committing to it, but generally speaking, it is a good idea to have other people to work with. I may well try to find a math major from UNC to hire as a tutor at some point as I keep working on this stuff.

And if one can't find somebody to work with in person though, and they need, say, a proof evaluated, there is the option of using math.stackexchange.com or the like.

<comment><USER35> I'm sure padthai meant that you need to do math to learn math, not just read or watch videos.

<comment><USER34> _I 'm sure padthai meant that you need to do math to learn math, not just read or watch videos._

Sure, I'm just saying that it kinda depends on what part of math one is referring to. I think sometimes in these discussions on HN, we overload the term "math" to mean both "calculation" or "applied math", and "pure math" or "math research" and it can be unclear which is being referred to in a given statement.

I believe you can learn the former - "applied math" \- (at least up to a certain level) just by reading books, and watching videos (and doing exercises, of course). But for the latter - "pure math" \- I agree that you need other people, since you can't easily verify your own proofs.

</comment></comment></comment></comment><comment><USER36> I've been following a breadth-first-based-on-my-interests approach for the last several years. It's not been very systematic, but it's been a lot of fun. I think math is essentially clear thinking, and it's best when it's explored from curiosity.

A couple resources I would recommend would be:

* A book called "Who Is Fourier? A Mathematical Adventure", which touches on a pretty good variety of math topics. It's aimed at children, but it's probably the best math book I've ever seen. It's the only math book I've ever read cover-to-cover.

* 3Blue1Brown's youtube channel: [https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw) (as well as other math videos)

<comment><USER37> 3B1B is amazing at explaining concepts. Lots of commenters in his videos studies math/cs/physics and tell stories about _finally_ understanding a topic they misunderstood in university.

His videos, I think, are best consumed twice: once before you explore a topic, so that you go in with a solid overview and know the key things to look out for; and again after you've studied in depth, when you are able to predict what he will say next, which reinforces you learning.

</comment></comment><comment><USER38> It is really hard to truly self-study in mathematics. Going through the opencourseware and reading textbooks (working as many exercises as you can, of course) will get you only so far. It is important to have someone (ideally with a PhD-level education in mathematics) who you can meet with to guide your study, correct mistakes and answer questions.

<comment><USER39> I strongly disagree actually. I think mathematics is the best field to self- study. There is nothing in mathematics that cannot be explained on the paper. There are many textbooks that are very good that in most universities professors won't be that good anyway. I went to UC Berkeley to study mathematics (ended up studying CS though) which is supposed to be a top department, but most of my textbooks were better teachers than my professors. I still prefer reading textbook to people explaining me math. I don't even think I understand math when people explain me. I need to first teach it to myself. Then occasionally people offering different perspectives is very beneficial, which, again, can be done on paper.

I don't think you need a PhD level educator. You need mathematical maturity. Mathematics follows a very specific logical structure that needs you to shift the way you think. Human brain simply doesn't work the way mathematics needs it to work. But this is a constant time overhead. Once you understand how to approach mathematical problems, I can't see why you cannot learn everything from a textbook.

<comment><USER40> As someone who did study both math and CS at university I disagree. I think there are numerous courses that can more easily be self-taught[1], mostly what one would encounter in their first ~2 years in a math degree. After that things get conceptually a lot more difficult.

For me personally I didn't really need an instructor for most of my calculus courses, or ordinary differential equations, or most of the linear algebra stuff. It was a bit more difficult around real/complex analysis, non-linear dynamics, and courses of that nature. The classes that taught me the value of having an instructor were abstract algebra and topology. Those were such a massive shift away from what I had perceived math to be that an instructor being able to impart intuition, correct my own incomplete or incorrect assumptions, and generally just help guide me to a different mode of thinking was invaluable.

The problem with books/texts in this instance is they are not reactive, they have no idea what you're thinking and can't steer you in the right direction. Worse is that as the person trying to learn the subject matter you don't know where to look to get on the right track and correct your own assumptions because you don't know enough yet.

Now I'm not saying you need an instructor per se, but having some place to ask questions where someone far more knowledgeable than you can help might be a good substitute. I'm sure there are some websites like this, although I don't know of any since I graduated a long time ago.

[1]This does somewhat depend on a person's skillset going into this.

<comment><USER41> The reality is that having a great mentor is a privilege, not a given. I don't think anyone is arguing that having a genius and brilliant teacher wouldn't help, rather that it is possible to reach an "advanced" (grad/undergrad level) understanding of mathematics without the luxury of having someone who is far more knowledgeable to turn to for help.

<comment><USER39> Precisely. GP here. I didn't argue against having a great mentor. If you go to a great university you clearly have an advantage. I think it's perfectly doable to teach yourself mathematics if you study intense enough eith correct tools.

</comment></comment></comment></comment></comment><comment><USER42> A nice foundational textbook to start with is Michael Spivak's text "Calculus." This text is fairly conversational, it motivates its concepts well with many examples, and it will help you build a strong foundation in writing proofs and reasoning mathematically.

As others have said, it might be helpful to have a friend or two read with you so you can answer each other's questions. Also, make sure you do a ton of exercises!

</comment><comment><USER43> Check out Open University ([https://www.open.ac.uk](https://www.open.ac.uk)), they have a undergraduate (BSc) up to graduate (MSc and PhD) level programmes in Mathematics, and they take distance-learning seriously.

That's what I would do if I were you. Learning this stuff on your own will be very difficult. I can't even imagine the amount of discipline needed...

</comment><comment><USER44> I was a homeschooling parent. I also have a stronger math background than average, though it isn't anything impressive for this crowd.

I will suggest you try to find a written curriculum from a respectable source as your first step so you have some idea what you need to cover to meet the stated goal.

Then you will want a variety of study materials for the subjects in question. You will want to verify that these are respected materials that are not full of errors.

When I brushed up on math to CLEP Alegebra so I could take college statistics, the computer program I used for self study had errors in it. I knew enough math that I knew what was wrong. I was just rusty and in need of a bit of practice. But it was a program I did not use with my sons to homeschool them because of the errors.

You want to know which materials math geeks like. You want them vetted, basically.

Then use whatever quality materials most appeal to you. Different strokes for different folks and self study let's you pursue whichever materials you like the best.

</comment><comment><USER45> You need to do lot's of exercises.

Shaum's Outlines has lots of cheap math books with exercises and solutions. You can use them to complement other books

[https://www.mhprofessional.com/9780071623667-usa-schaums- out...](https://www.mhprofessional.com/9780071623667-usa-schaums-outline-of- advanced-calculus-third-edition-group)

[https://www.mhprofessional.com/9780071635349-usa- schaums-300...](https://www.mhprofessional.com/9780071635349-usa- schaums-3000-solved-problems-in-calculus-group)

[https://www.mhprofessional.com/9780071383929-usa-schaums- out...](https://www.mhprofessional.com/9780071383929-usa-schaums-outline-of- calculus)

[https://www.mhprofessional.com/science- math/math/topology](https://www.mhprofessional.com/science-math/math/topology)

</comment><comment><USER46> If you want to start with a single self-contained book, then a really good choice is "Modern Mathematical Physics" by Peter Szekeres:

[https://www.cambridge.org/core/books/course-in-modern- mathem...](https://www.cambridge.org/core/books/course-in-modern-mathematical- physics/E899DB30C574E2F4D7C861B3097F9813)

It covers most of the mathematics needed for quantum field theory (which is a big chunk of applied mathematics) starting right from set theory.

</comment><comment><USER47> Lots of good points here. I can't speak to the graduate level, but for the undergrad level I think the trick is to find textbooks that don't suck ( _most_ good textbooks are free, or are old enough no one cares about copyright, and a million pdfs are online, or you could get it used for $15. Most bad textbooks are price gouging undergrads).

Then just put the time in with deliberate practice. I tried self-studying math for a year, and often skipped the harder practice problems, thinking just reading the textbook and examples is sufficient. That's actually entirely useless. You are't learning math (or anything worth learning) unless you spend 6 hours on a Saturday working through some hard problem that makes you so frustrated that you ask God why he made you such an idiot. If you spend enough time doing that you can actually learn something.

</comment><comment><USER48> The obvious route for self-study is to go through Halmos or Axler and Rudin, plus all additional materials and study aids such as Gelbaum & Olmsted, emulating the most popular Harvard Math 55 incarnation. Another nice Math 55 incarnation covered Hubbard & Hubbard, which is a wonderful book.

Although Math 55 is tough, if you are self-paced and have a bit of mathematical maturity I think it is doable. It's also an excellent pure math bootcamp that gives you a solid foundation to branch into any other pure or advanced math topic.

I have gone through Halmos & Rudin myself, and it is a great experience. However, if your end goals are more geared towards pure CS, an alternative route might be much more appropriate. Very interesting and promising parts of CS, such as formal methods, and the foundations of mathematics themselves depend on abstract algebra and logic: [https://ncatlab.org/nlab/show/computational+trinitarianism](https://ncatlab.org/nlab/show/computational+trinitarianism)

A minor problem is that beginner literature is not so polished as it is relatively young. But there are some excellent textbooks nonetheless. Some below. Other suggestions welcome:

* [http://www.cs.man.ac.uk/~pt/Practical_Foundations/](http://www.cs.man.ac.uk/~pt/Practical_Foundations/)

* [https://www.mta.ca/~rrosebru/setsformath/](https://www.mta.ca/~rrosebru/setsformath/)

* [https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen...](https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen%2C%20Urzyczyn.%20Lectures%20on%20the%20Curry-Howard%20Isomorphism.pdf)

* [http://www21.in.tum.de/~nipkow/Concrete-Semantics/](http://www21.in.tum.de/~nipkow/Concrete-Semantics/)

* [http://adam.chlipala.net/frap/](http://adam.chlipala.net/frap/)

* [https://softwarefoundations.cis.upenn.edu/](https://softwarefoundations.cis.upenn.edu/)

</comment><comment><USER49> I had math through some introductory calculus when I was an undergrad 20 years ago, but I let my skills lapse. But I want to understand enough math so that I can do some more complex electronics projects and some statistics / ML / intelligence analysis. This level of math, as I recall, is more or less the math core of the undergrad engineering program I dropped out of in favor of a philosophy degree (cause I am dumb as dirt).

In March of '18 I started doing lessons on Khan and just played around until I couldn't do the problems easily, and for me that was, like literally adding fractions and using exponents. So I had pretty basic skills at that point.

At thus point, I'm finishing the unit on using derivatives to optimize functions around min/max. Not a big deal, but a long way from where I started 10 months ago.

I've had a lot of luck with:

a) khan academy

the lessons are very simple and well broken up, the teaching is interesting, and the site is gameified in a way that is rewarding

b) doing it as close to literally every single day as I can manage

Math fluidity feels (at least to me) very much like my fluidity with music theory or programming. As such I need to do it regularly. Even if I don't get all into a flow state about it (which, I think is necessary on the scale of any given week), I do need do it a little bit every day... that both keeps me doing it and keeps it in the forefront of my mind.

c) keeping a notebook

I do all my work in a single notebook and use it to both track my progress and as a reference. It's also been neat to see how far I've come.

-=-=-=

I dunno if any of that is useful to anyone else. But I feel like I've had a lot of luck educating myself in math. At this rate, I should be through integral calculus by the end of the spring and through the linear algebra class by fall, and then I will move my deep, long-term learning projects over to something else, hopefully a deeper dive into electronics design.

<comment><USER50> One great thing about Khan Academy is the gameification aspect makes sure that you go over older material in a spaced reptition sort of way which helps really drill in the knowledge so it isn't forgotten

</comment></comment><comment><USER51> This book is fantastic and pretty much takes you through an entire undergrad mathematics course: [https://www.amazon.com/Mathematics-Content-Methods- Meaning-V...](https://www.amazon.com/Mathematics-Content-Methods-Meaning- Volumes/dp/0486409163)

<comment><USER10> The topics covered in that book are undergrad level, but that book is not suitable for learning the topics. It's more like a high level discussion of the topics looping them together with historical background. It's more appropriate for people who are already familiar with the material.

</comment></comment><comment><USER52> I discovered this amazing YouTube channel recently that I wish I had access to back in college. It explains some very difficult math concepts in a very simple and easy to understand way: [https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)

<comment><USER53> It's amazing how seeing something visually or having someone else solve something can aid in your understanding of a topic. I remember having the most difficult time understanding mergesort from my book's pseudocode and my professor's lecture. But seeing mergesort in action on youtube revealed what it truly was and made what seemed impossibly complex, so simple and elegant.

</comment></comment><comment><USER54> Terry Tao's post - [https://terrytao.wordpress.com/career-advice/theres-more- to-...](https://terrytao.wordpress.com/career-advice/theres-more-to- mathematics-than-rigour-and-proofs/)

[https://www.theatlantic.com/education/archive/2014/03/5-year...](https://www.theatlantic.com/education/archive/2014/03/5-year- olds-can-learn-calculus/284124/)

Math is not linear, So why do we teach math in hierarchical steps? - [https://prezi.com/aww2hjfyil0u/math-is-not- linear/](https://prezi.com/aww2hjfyil0u/math-is-not-linear/)

[http://ocw.mit.edu/resources/res-18-001-calculus-online- text...](http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook- spring-2005/textbook/) highly recommended for Calculus

</comment><comment><USER55> Hey, I am also on the same boat. I have been trying different things lately. I have gone through a lot of courses and a lot of books, but most of these experts have the curse of knowledge. They don't know how it is like to not know. They skip things that seems obvious to them. But, for begineers not having such information can be frustruating and feels like the author/lecturer is moving fast.

I recently stumbled upon this book "The Language of Mathematics: Utilizing Math in Practice" I have been reading this book for a while and the author takes a different approach in teaching maths. I am slowly beginning to learn how to forumulate a real world problem into mathematical model and then solve it. More importantly, I am able to read and understand mathematical expressions better through this book. This book has raised my hopes in learning and apply math. I would recommend you to give it a shot on this before you go through any other courses.

</comment><comment><USER8> I suggest you get the syllabi from a state college, and then follow that. "work through" the books : copy each page of exposition by hand, try to do the examples and proofs before looking, do all or most of the problems with answers, and get ready for a long journey.

I find a page of math textbook takes about 30 minutes (at least) to really work it over until you understand it.

Personally, I don't much like working in groups or watching video lectures. To learn math, you have to be able work problems by yourself, so you might as well just do that.

David Morin has some great self study books:

[http://www.people.fas.harvard.edu/%7Edjmorin/book.html](http://www.people.fas.harvard.edu/%7Edjmorin/book.html)

I have a math BA from a regional state college, usually scored in the top 5 on the tests, and I am self teaching physics / me /ee about 5 hours per week after work.

</comment><comment><USER56> Don't study "math." Find a topic or a problem you'd like to understand, look up the prerequisite knowledge for it, and start there.

You need to know what you want to do before you go looking for resources. There is no set agreement on what should constitute an undergraduate applied mathematics curriculum, and you are likely to get lost in the deluge of conflicting information. On the other hand, the undergraduate pure mathematics curriculum has been more or less stable for half a century. Any college curriculum will do at this point, and many are freely accessible online.

Either way, there is no shortage of information and resources available. Any topic you'd choose as a layperson likely already has a course or a seminar covering it, and the corresponding syllabus should give you what you need.

</comment><comment><USER57> I'm actually building an app specifically for this use case:

[https://getpolarized.io/](https://getpolarized.io/)

Polar is basically a personal knowledge repository.

The idea is that you use Polar for all of your education. Either official education or continued learning.

Every textbook or technical paper should go into Polar.

You can then annotate the textbooks directly including text highlights, comments, etc.

It supports spaced repetition systems like Anki so you can continually review what you've learned and NEVER forget anything.

It doesn't make sense to pursue a PhD or spend 6-12 months researching a topic to just forget 80% of it due to bit rot.

The reason I named it Polar is that it's meant to 'freeze' all of your knowledge so that you never forget it - ever.

<comment><USER58> Learning math isn't really about memorizing things.

<comment><USER31> very true, in the sense that rote memorization is not the point.

but false in the sense that doing math requires fluency - in applying a small amount of technique up to lower division math, and in applying a large number of definitions/results after that. in areas like abstract algebra, failing to memorize will kill you. it is as disfluent as writing text in a foreign language without having memorized the working vocabulary and grammar; even if you can eventually recall, derive or look up what you need, it's just not practical to work like that. (and i say this as someone who prefers to mentally index information rather than memorizing...)

<comment><USER30> I have to disagree, especially about abstract algebra. Most concepts and theorems feel like abstract nonsense (not specifically talking about category theory here) when you don't understand them, but should become pretty natural once you do. For true mastery you need to work with the concepts and results on a day to day basis for a while, by applying them; continually reading the text of definitions and theorems hardly helps if at all.

<comment><USER31> i agree that memorization by internalization (concepts) is different from (and superior to) memorization by rote (text); i was _agreeing_ with the other fellow that rote memorization is not the end goal. however, to me they're both "memorization" because they both represent work to achieve fluency in application.

however, i made my comment because i think i disagree that spaced repetition has no place in learning. i think that if you dig around in the Polar guy's earlier comments, you'll find threads where folks like michael nielsen are talking about using spaced repetition tools for much more than purely textual memorization of theorems - basically, cycles of repetition and (re)synthesis. so i don't feel it's right to completely shut him down about card decks. you may disagree, of course.

<comment><USER30> When your definition of memorization encompasses all forms of learning, saying memorization is crucial to learning is pretty much a tautology. No one claimed that amnesia sufferers are perfectly good for mathematics. Same goes for repetition. No one expects you to understand and never forget the theory of cohomology by writing down a long exact sequence once.

The thing is, the root comment of this thread specifically talks about "continually review what you've learned and NEVER forget anything" through looking at highlighted notes in the software mentioned (a somewhat out-of- place plug, I'd say). That's not how math works. You refresh your memory by tackling preferably new problems. Reciting proofs is largely pointless (except for certain very elegant proofs, in which case you probably won't need to recite them anyway); reciting definitions and theorems is even less useful.

> it's meant to 'freeze' all of your knowledge so that you never forget it - > ever.

Yeah, no, you don't "freeze" your mathematical knowledge.

</comment></comment></comment></comment></comment></comment><comment><USER59> Khan Academy has multivariable calculus, differential equations and linear algebra --- with many exercises, solutions, and worked solutions (under the "hints" link after a problem), and gamification as a motivation aid.

The videos are above average, but seem about 5x too slow on the steps, then rushing through the interesting stuff almost too quick to catch: boredom punctuated by rewinding. Probably because it targets highschool kids.

BTW You may find you need to cover even earlier material anyway (I did), to refresh and fill gaps. You can cope with some gaps and hazy recall, until there are too many... and some gaps cannot be filled piecemeal (because you don't know you have a gap).

</comment><comment><USER60> I don't follow curricula online. But you should be able to get student and teacher copies of textbooks. Hold yourself firm to attempting the problems before looking at solutions to simulate homework. To simulate lecture, pick 1-2 of the tougher problems and work through the solution for them. Then try your homework out.

What I love about math is that this is completely do-able. Good luck! Also, definitely get a dry erase board. There's something about having a large space that can be easily erased that helps working out problems.

Edit: deleted an m

</comment><comment><USER61> I have a Masters degree in mathematics and struggle so much to continue learning in my (relatively ample) free time. I was never good at self evaluating my proofs so often I solve a problem and have to scrutinize it for nontrivial steps made "intuitively".

It's like when you have Calculus and over generalize when entering Analysis or Differential Geometry -- but now I have more structural patterns to extrapolate from :/

</comment><comment><USER62> First I think you need to think about a direction. "Learn Math" is a very broad and vague statement. You will never learn all of math. So you really need to think about why you're wanting to learn it, and pick a direction that suits that.

If you're wanting to learn math just because you like the subject, then it's wise to get a sampling of a wide array of different areas. Then you have a better chance of finding an area that you like best.

If you're goal is to learn it so you can apply it in some other area, you should focus mostly on learning what math applies in that area. Then find the prerequisites and start studying.

Studying alone in your spare time will be nothing at all like attending a university. The full immersion you get to focusing primarily on one thing, and the ability to engage others at will also focusing on the same thing is a great help in learning. So set your expectations right. You won't be able to reach the level of understanding a 60 year old math professor has. Since he/she has been fully immersed in the topic for probably 40-50 years. So set your expectations accordingly.

</comment><comment><USER63> I actually started from the very beginning of mathematics on Khan Academy, thats starting from pre-school going through every video and exercise, taking the quizzes and each end test for each subject. I am currently almost done with Algebra 1 and has taken longer than expected. I highly recommend! The exercises are key and do everyone until you get 100% then move on to the next topic/subject.

</comment><comment><USER64> There are some awesome resources on github on many topics including math. Just search for "github awesome-X" like [https://github.com/rossant/awesome- math](https://github.com/rossant/awesome-math)

And here is a root project for all resources [https://github.com/sindresorhus/awesome](https://github.com/sindresorhus/awesome)

Personally I'd like to have something like a learning path. Not only a list of resources/topics but also some guide how to approach the learning process. O'Reilly has something like this but it's not mature yet [https://www.safaribooksonline.com/learning- paths/](https://www.safaribooksonline.com/learning-paths/) in my opinion.

MIT OCW has some guide for prequesities [https://ocw.mit.edu/courses/mit- curriculum-guide/#map](https://ocw.mit.edu/courses/mit-curriculum-guide/#map)

I hope that helps you

</comment><comment><USER65> This is really great.

One specific point:

> One idea I had was to complete the MIT open courseware courses for the > Applied and Pure math fields

A note about Open Courseware: they tend to be much of or all of the material handed out in a course (lecture notes, problem sets, etc). They aren't a "course" in the sense of Coursera, Khan Academy and the like.

A few years ago I needed to brush up on my thermodynamics and was able to read the material for the same course I'd taken as an undergraduate (well, same course number; I took thermo in 1983, and the prof and some of the material had changed). This was merely an undergraduate thermo class _that I had already taken 30 years before_ and it was still quite hard.

I don't mean in any way to discourage you!!! This is an excellent idea. But OPenCourseware itself is more like a box of legos for someone who wants to teach a class in the subject. You may find a different source better, especially for topics that are new to you.

<comment><AUTHOR> Ah very good points, thanks for the heads up! I'll think about this a bit more

</comment></comment><comment><USER66> One thing that worked for me was to have a _community_ of fellow mathematicians to study with. There's a lot of cultural stuff that is kind of hard to osmose from books, such as for example, how to _pronounce_ things that you read.

Mathematical notation is meant to be read out loud. It's shorthand for English, or whatever your natural language is. You should be comfortable seeing a sigma sign and thinking in your head, "the sum as ehn goes from one to infinity of eff of ecks squared times ..."

That's just the basic part of it. There's lots of cultural shorthands that are just kind of osmosed, seldom overtly acknowledged (for example, the precise meaning of "without loss of generality"). Maybe you can acquire them by watching videos, but if you can find people that you can hang out with that can teach you the cultural aspects of mathematics, that would help a lot. Picking up a book and reading and working through it and understanding it gets a lot easier once you have the culture down.

<comment><USER10> That is also part of the loosely defined term, "mathematical maturity." When you've reached it, you can generally grasp the basics of unfamiliar math quickly. Failing that, you can find out what you need to study to understand it and do so on your own.

You don't typically obtain that kind of maturity until graduate school, maybe upper undergrad if you're quite good.

<comment><USER66> Yeah! There's that whole thing where you understand how a work is structured, you know what part to pay attention to, what part to skim or read later, where you're required to do your own calculation to fill in something...

People complain a lot that books and presentations don't do everything and that mathematics has a bad "user experience". They may have a point, but the complaining alone won't fix it and will leave most people still feeling like frustrated outsiders.

</comment></comment></comment><comment><USER67> I disagree with the common advice that before you study math you should try to develop love for it or find an application you care about. If you don't know any math, how can you know what you like or want?

What I think one should do, initially, is try to be better than other people in some way. That's a motivation everyone has accessible. One way is what you're doing already — trying to learn math on your own. That's pretty cool.

From there I'd use Reddit and Quora to build yourself a curriculum. Most people start with calculus, linear algebra, differential equations, and real analysis. Look for textbooks that are known to be pedagogically well-written for self-study (rather than intended to be used as a manual in conjunction with lectures). Skim the table of contents and the first chapter of several books to get a feel for which ones you'll be able to stand going through. I like Spivak, Arnol'd, Axler, and Pugh.

In each subject, find a book full of problems, and do the hardest ones in each section.

</comment><comment><USER68> I started doing this, and the first stumbling block I hit was simply assessing at what level my math education falls apart and figuring out what courses I needed to refresh with, what materials I needed etc.

There's a market opportunity for building a math placement test product, or even better a fully automated math education platform with examinations at each stage, that takes you from K-12 refreshers all the way up to grad school. Something a bit more advanced than the existing online learning programs that are still very much based on the college lecture model.

There's a lot of products out there that do this for software development, but not for other subjects. I have noticed there are bits and pieces I can download as mobile apps, but they are designed to be supplements to certain college curriculum rather than an end-to-end education.

Obviously math is a subject where software is an optimal instructional tool, so much more could be done here

</comment><comment><USER69> These concept maps might be helpful as a general overview of the basics: [https://minireference.com/static/tutorials/conceptmap.pdf](https://minireference.com/static/tutorials/conceptmap.pdf)

They are extracted from my concise books for adult learners on MATH & PHYSICS [https://minireference.com/static/excerpts/noBSguide_v5_previ...](https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf) and LINEAR ALGEBRA [https://minireference.com/static/excerpts/noBSguide2LA_previ...](https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf)

I don't want to self-promote too much here, but maybe HN users who have read the books can add a comment to say what they thought.

</comment><comment><USER70> I think it is pretty much impossible to learn mathematics without a teacher beyond a certain point, multiple teachers are ideal. The reason for that is in order to do mathematics you need to solve problems and develop intuition. It will be very hard for you to solve problems (write proofs) without someone that evaluates the quality of your solutions, and it is very easy to deceive yourself that your reasoning is without gaps without experience. The same is true for intuition, without a master at whatever field you want to study it will be hard to work through most books (Hartshornes, Algebraic Geometry comes to mind. His seminars and lectures contained plenty of simple examples. The book itself is famously an uneasy compromise, which no one was really happy with but also with few better alternatives)

</comment><comment><USER71> When you first start out in math you won't even know the basics like e.g. "proof by contrapositive" but a basic "proof course" perhaps accompanied by "Linear Algebra by Hoffman and Kunze" will get you pretty far.

You can never ever ever ever learn too much linear algebra. It is pivotal in the development of e.g. "field extensions" and "differential geometry" to name two random items.

That said, partway through Hoffman you should start an algebra book like Dummit and Foote so you can see groups and rings.

*After you know how to prove things you could read an analysis text instead of Linear Algebra

Lastly, any serious math requires some topology, but it's less about what a topology is and more about knowing how to quickly apply the basics so certain statements are easy to state/prove/think about

<comment><USER71> Lastly, the further you go the more you'll see that there are no islands in math. Everything uses everything else. (For example, you can go back and forth between groups and topological spaces via e.g. the fundamental group)

</comment></comment><comment><USER72> Just take community college courses. Work lots of problems. Ignore the grand theorems unless you're going to be a math major (in which case you should go to college).

Applied math; if you deal with matter it will generally be linear algebra and differential equations. Vector math important also, and Calc-3 is inadequate; junior level classical physics mechanics book is how I learned stuff like action angle and rotating frames of reference.

If you deal with electronics/signal processing you'll need some kind of Hilbert space course to get you through Greens functions, Laplace transforms and so on.

And for computer science/machine learning/OR: just be really good at linear algebra.

(Applied obv) linear algebra is the highest leverage thing you can do.

</comment><comment><USER73> I don't at all recommend limiting yourself to self-study when you start this endeavor, it's too daunting given all the unknown unknowns. To start with it helps a lot to take courses, even if it means sitting in/enrolling in night courses at the local community college. It provides a reference point to what others understand, allows real-time querying with an instructor, and is more motivating than self-study. Once you have completed and fully grasped a difficult proof-based course (real-analysis, algebra, etc), then you can possibly embark on a successful self-study. I'm not saying it can't be done, but you will get to mathematical maturity faster this way and there will be less risk of early abandonment.

</comment><comment><USER74> We need more info:

How much math do you currently have, which courses have you taken and how long ago did you take them?

</comment><comment><USER75> Be a social human and find a local math prof or someone in industry to mentor you.

That aside: Haskell Road to Math and Programming. Concrete Mathematics. Pick your favorite computer graphics book to learn working with vectors and physics equations. From there chase your passions.

</comment><comment><USER76> Just follow MIT Open Courseware for curriculum as for how to learn either the lectures or textbooks, lectures are good for initial understanding but then you'll need a textbook to go over the definitions,theorems and very important EXERCICES . you might want to take it easy don't try to put everything in some sort of timeline like (I am going to do all undergraduate in 1 year) that's really stupid as may be advertised on the internet. Your real goal ,as you're taking this endeavor ,is to understand it by making it your own, get a feel for it.I cannot describe it but you'll know . Best of luck in your quests .

</comment><comment><USER77> Leaning math is primarily a solitary endeavor that takes time to learn properly. If you are an adult starting from basic algebra be prepared to spend at least 2 years of your life in order to get to where you need to be in order to begin physics. (Ie Allegra, function, trig, calculus) You have lots of options but assuming you don't want to go back to college these are 3 tools that are 100% free and as good if not better than any of the paid options.

1 Every day on this site [https://www.khanacademy.org](https://www.khanacademy.org)

2 Use this for graphing desmos.com/calculator

3 Use this to supplement (ie free text) myopenmath.com

</comment><comment><USER78> I'd very strongly recommend [https://en.wikipedia.org/wiki/How_to_Solve_It](https://en.wikipedia.org/wiki/How_to_Solve_It) to familiarize yourself with some fundamental processes and [https://www.amazon.co.uk/Playing-Infinity-Mathematical- Explo...](https://www.amazon.co.uk/Playing-Infinity-Mathematical-Explorations- Mathematics/dp/0486232654) before you lose yourself in a highly symbolic and abstract world, appreciate the beauty of it first, it'll make the long journey much easier.

</comment><comment><USER79> I went back and learned a lot of math recently. I found the 3Blue1Brown youtube channel to be hands down the most useful for developing intuition.

I honestly don't think there is better mathematical content than his being made.

</comment><comment><USER80> I know this is not what you're asking for, but I'd recommend stopping at the end of the sophomore level classes in the open courseware material.

I finished an undergrad degree in math from a school that had a large PHD program, and once I got to the junior level it was a lot of proving obscure facts about complicated objects (say, Lie Algebras) that have zero applications outside of very speculative theoretical physics. Proving theorems was fun sometimes, so if you're interested in math as entertainment, it's worth trying the more advanced material.

</comment><comment><USER81> Here's a reading recommendation list I put together over the years: [http://math.mit.edu/~notzeb/rec.html](http://math.mit.edu/~notzeb/rec.html)

Two caveats: first, it is obviously biased to the sort of math I find enjoyable, and second, many of the suggestions require the reader to have that elusive quality called "mathematical maturity". (I have no idea how one goes about gaining mathematical maturity, despite having gone through this process myself.)

</comment><comment><USER82> Is there a particular reason why you want to learn upper level mathematics? I have a pretty decent math background and I personally found majority of math boring (particularly linear algebra). The only reason why I did so well in Math is because I know I needed a strong math foundation to do the fun stuff (like fluid dynamics). Without this purpose, I would of passed my math courses in college but definitely would not of excelled in it.

</comment><comment><USER83> For those like me who learned programming first and want to improve their math, but struggle with the crazy notation inconsistencies and the "pure math" approach to things in general, I highly recommend this book:

A Programmer's Introduction to Mathematics by Dr. Jeremy Kun

I've only just started it, but so far it does a great job of introducing math while assuming just the right amount of programming knowledge to make math approachable.

</comment><comment><USER84> I've been catching up with things I've missed, already forgot or never learned in school with Paul's Online Notes: [http://tutorial.math.lamar.edu/](http://tutorial.math.lamar.edu/)

It's an incredible resource with great introductions to each topic and examples that are easy to follow and try for yourself.

</comment><comment><USER85> Not sure if its great for your purposes, but I like the 'No Bullshit guide to Math and Physics' at least as a reference. It might be a good place to brush up on lower level stuff before advancing to harder topics.

[https://minireference.com/](https://minireference.com/)

<comment><USER86> Can anyone share some insight on if there has been any improvements on this since the last time it was posted here [1]? From that thread it appears that it has few misinformation there.

[1][https://news.ycombinator.com/item?id=4994367](https://news.ycombinator.com/item?id=4994367)

</comment></comment><comment><USER87> John Baez (physicist / category theorist / smort dude) has a list of books to learn math and physics. Is an interesting source to crib from. [http://math.ucr.edu/home/baez/books.html](http://math.ucr.edu/home/baez/books.html)

Gerhard t'Hooft has a page on how to become of a good physicist. [http://www.goodtheorist.science/](http://www.goodtheorist.science/) Not math though.

I'm not necessarily suggesting that a mountain of books is the best way to go about it. I think I respond best to video lectures. I'm not sure where you're starting from or how applied/pure you want. Here's a quick mind dump some of my favorites and some that I haven't watched. Roughly in order of how much I liked them.

Gilbert Strang's Computational Methods for engineers was life changing for me. It is a two part MIT opencourse. [https://ocw.mit.edu/courses/mathematics/18-085-computational...](https://ocw.mit.edu/courses/mathematics/18-085-computational- science-and-engineering-i-fall-2008/)

A Stanford course on the Fourier transform [https://see.stanford.edu/Course/EE261](https://see.stanford.edu/Course/EE261)

Bartosz Milewski's Category theory for programmer's [https://www.youtube.com/watch?v=I8LbkfSSR58](https://www.youtube.com/watch?v=I8LbkfSSR58)

Stephen Boyd's courses are online. [http://web.stanford.edu/~boyd/](http://web.stanford.edu/~boyd/) Linear Systems, convex optimization. Useful stuff.

Francis Su's Real Analysis is very good [https://www.youtube.com/watch?v=sqEyWLGvvdw](https://www.youtube.com/watch?v=sqEyWLGvvdw)

Indian universities have an astounding collection of videos [https://nptel.ac.in/](https://nptel.ac.in/) I have a tough time with the accents, which is a bummer.

UCCS MathOnline has quite a haul [https://www.uccs.edu/math/vidarchive](https://www.uccs.edu/math/vidarchive)

I've been enjoying this Visual Group Theory course lately [https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTw...](https://www.youtube.com/playlist?list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv)

Math Doctor Bob [https://www.youtube.com/user/MathDoctorBob/playlists](https://www.youtube.com/user/MathDoctorBob/playlists)

Wildberger has some interesting takes on elementary and non elementary topics [https://www.youtube.com/user/njwildberger](https://www.youtube.com/user/njwildberger)

[https://www.perimeterinstitute.ca/training/perimeter- scholar...](https://www.perimeterinstitute.ca/training/perimeter-scholars- international/psi-lectures) Perimeter scholars lectures. Physics not math. Good stuff.

Federico Ardila has a number of combinatorics courses. [https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ](https://www.youtube.com/channel/UCWwECTsgjp_S-c73pO2c4gQ)

Nonlinear algebra course [https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrW...](https://www.youtube.com/playlist?list=PLRy_Pn1LtSpejKLClqbrWWBxGKxx333O1)

Also of course there is Coursera and edX stuff.

Godspeed.

</comment><comment><USER88> Good advice here.

If you choose the project/problem based approach I recommend [https://projecteuler.net](https://projecteuler.net)

You will learn a ton solving these problems (the programming aspect is the trivial piece in most cases).

<comment><USER89> This is absolutely the right approach!

</comment></comment><comment><USER90> This might also help a similar thread was posted a while back

"Ask HN: How to self-learn math?" [https://news.ycombinator.com/item?id=16562173](https://news.ycombinator.com/item?id=16562173)

</comment><comment><USER91> I'll add that, as a working faculty member, most people do not take naturally to writing proofs correctly. Even quite good students have to work with someone, at some length, before they get good instincts at it.

</comment><comment><USER92> I have found that talking to mathematians is the best way to learn mathematics.

</comment><comment><USER93> I think this is something like what you're asking for

How to Become a Pure Mathematician (or Statistician) [http://hbpms.blogspot.com/](http://hbpms.blogspot.com/)

<comment><USER94> This might be a good roadmap for pure math, but don't use this for guidance on how to be a statistician.

</comment></comment><comment><USER95> Start small. I would look at Stat 110 from Harvard, Strang's OCW lectures, and the intro calculus MIT course.

When you're done with that, it'll be easy to see where you want to go next.

</comment><comment><USER96> Use community college as a non-degree student to do the introductory undergrad classes. It's cheap, you are goal oriented and can get a lot of face time with your professor.

</comment><comment><USER97> My $0.02: Make a goal to fully understand General Relativity as Einstein published it. Then work through the evolution of some of its mathematics.

<comment><USER39> Why set an arbitrary goal like that? Why not understanding Curry-Howard correspondence or Godel's Incompleteness or Hilbert's 17 instead.

<comment><USER97> That was my $0.02 as someone who majored in Physics. You just gave $0.04. Good suggestions.

</comment></comment></comment><comment><USER98> Part I

I have some opinions on the question in the OP: I'm heavily self-taught from independent study in math; that study helped me with, and at times was part of my good career in, applied math and computing before my Ph.D. in pure/applied math and helped a lot for my Ph.D.

For a _curriculum_ :

(1) Do the standard high school math, Algebra I, Plane Geometry (based on proofs), Algebra II, Trigonometry, and Solid Geometry (based on proofs). In each of these, it is sufficient (A) to work most of the more challenging exercises and (B) try to think a little about what is going on. For (C), it would be good occasionally, say, 1-4 times a year, to chat about your progress with a good mathematician, Ph.D., likely a college prof. To find a Ph.D. who will give you an hour or so per visit, maybe do some _networking_. If you don't like the first choice, e.g., if they are not encouraging and helpful, then try a different Ph.D. mathematician.

(2) Do Analytic Geometry and Calculus. Use a good college textbook. Since highly polished texts have been widely available for decades, there's no need to pay big bucks for new copies of the latest. So get a good used text. Better yet, get 1-4 good texts, use your favorite one as your primary source and the others for more content, alternate descriptions, evidence of what is more/less important, etc.

Sets: Now in an important sense, essentially all of math is heavily about _sets_. For first calculus, you don't need to know much about sets; the following should be nearly sufficient:

By 1900 or so, a lot of math was known, but there was a nagging, somewhat philosophical, question of what math really _is_. The answer of _pure_ math was to construct a _foundation_ in, say, the deep basement, that would answer the question but not much change the rest. The answer was to start with _sets_ and then define everything else in terms of sets. In particular do some somewhat tricky and obscure work to define some sets that look, work, walk, talk, etc. like the numbers on the line, the _real_ numbers. We don't think of the real numbers this way, but the effort addressed the philosophical question -- or, if I can get you to believe in sets, then I've got you for the rest of math. Much of the early work on sets was from G. Cantor. Yes, soon there were some issues from Bertrand Russell and then Kurt Gödel and, later, Paul Cohen.

For a long time in math, you can regard a _set_ as a conceptual (imaginary) collection of some kind, e.g., all the peas in a little jar in the back of the refrigerator, all the people with an iPhone, all the real numbers on the line, all the water molecules in the earth's oceans, all the parabolas, all the trials you might do in a lab experiment, and much more -- really, from the philosophical approach to what math _is_ for all of math. Or intuitively _sets_ are the containers for what we are thinking about.

Functions: Much of math is about _functions_. E.g.,

         f(x) = 2x^2 + 3x - <USER4>

So, in _pure_ math, the function is f. The x is the _argument_ of the function, i.e., a _variable_ which is essentially always in math from,

         "Think of something; call it x."

where at least in calculus usually the thing are thinking about is a number. Or, to "solve for x", think

         "I suspect there is a number such
         that these conditions hold; I want to
         find if there is such a number and if
         so what its actual numerical value
         is."

Now you understand what math means when it talks about variable x and function f. Then, given variable (e.g., number) x and function f, f(x) is the value of f at x.

E.g., for this set stuff, a function gets defined in terms of sets, e.g., the set of all _ordered pairs_

         (x, f(x))

for x in some set called the _domain_ of the function. In calculus, the domain is usually the set of real numbers, that is, the points on the number line (yes, that's not the tricky set theory definition of the real numbers).

So, with the ordered pairs

         (x, f(x))

we have a very precise definition of a function in terms of sets (ordered pairs can also be defined in terms of sets).

Of course, in actual work, especially in applied math, nearly no one actually thinks of a function as a set of ordered pairs.

Completeness: There is an old joke, partly appropriate, that

         "Calculus is the elementary
         consequences of the completeness
         property of the real number system.".

Intuitively _completeness_ is, if you are converging to something by more and more accurate approximations, then there really is something there for you to converge to. This property does NOT hold for the rational numbers -- the _rational numbers_ are all the numbers of the form (that can be written as) p/q for p and q integers (whole numbers, positive, negative, and zero) with q not zero. Why? E.g., can use the rational numbers to approximate as closely as we please the square root of 2, but the square root of 2 can't be a rational number. Why not? Because if the square root of 2 were rational, then we would also have for integers p and q, q not zero

         (p/q)(p/q) = 2

or

         (p)(p) = 2(q)(q)

So the left side has some even number (possibly 0) factors of 2 while the right side has an odd number of factors of 2, and this violates the fundamental theorem of arithmetic that each integer can be factored in only one way as a product of prime numbers.

Well, a big deal about the real numbers is that they are _complete_ ; i.e., intuitively, if something converges, then there is something to converge to. Completeness is a big deal because it generalizes, especially to Hilbert space that you may come to.

So, in calculus often we make approximations that can become as accurate as we please, and we want the completeness property to know that what these approximations approach, _limits_ , can exist.

E.g., with the real numbers, if we make more and more accurate approximations to the square root of 2, then we know that our approximations will _converge_ to the actual square root of 2 which DOES exist as a real number.

Sets get to be quite important in math in the last years of college math and beyond. But for calculus, nearly all of that subject, especially what learn in a first calculus course, was quite solid well before G. Cantor. So, in practice, when studying first calculus, we don't see much about sets. So, for first calculus, sets remain mostly a topic in the deep basement that addresses an old philosophical issue. So, for first calculus, sets are no big issue.

Analytic Geometry: Here you study the _conic sections_. They are a bit amazing and important in physics, mechanical engineering, and more and something definite to work with while learning calculus. So, imagine a cone, say, an ice cream cone. Take two of them that are the same and put the points together so that they are both on the same _axis_ through the points (I'll let you get a precise definition of _axis_ ). Now imagine a sharp sword, say, as in the old John Belushi Samurai Tailor skit and use the sword to slice the cones and look at the cut edges. Right, we assume that the sword moves in a plane. Depending on where you slice, you will get (A) just a point (where the two cones touch together), (B) two lines crossed as in an X (the cross in the X is at the point where the two cones come together), (C) a circle (we are awash in circles), (D) a parabola (to a fairly good approximation, a grand slam baseball follows a parabola), (E) an ellipse (to a quite good approximation, each planet goes around the sun in an ellipse), (E) the two parts of a hyperbola (an electron shot at a negative charge will follow one of the halves of a hyperbola as it avoids the negative charge).

First Calculus in a Nutshell: Then for first calculus, there are two big topics, differentiation and integration. Differentiation is finding rate of change, intuitively the slope (as in high school algebra) of a tangent line to the graph of a curve (defined in terms of a function with domain some or all of the real numbers). E.g., for time t, let d(t) be the function that gives us our distance from home at time t. Then the derivative of function d at time t is the velocity, say, function v, at time t. So, in a car, the odometer gives d and the speedometer gives v. That is, at time t we have traveled distance d(t) and are moving a velocity v(t). Here to be simple, I do not make a careful distinction between what physics calls speed versus velocity.

What differentiation does, integration reverses, undoes. So, we can use integration on function v to recover function d. We have now introduced the fundamental theorem of calculus.

Right, v is the function, and v(t) is its value at time t, but in first calculus it is common to drop the distinction between v and v(t) and just say "the function v(t)" and mostly avoid mentioning v by itself.

<comment><USER98> Part II

Again, for _variable_ t, think

         "There is a number; call it t".

That approach to the meaning of a _variable_ works well enough for essentially all of math.

Calculus was invented mostly by I. Newton, mostly for physics, especially for explaining the motions of the planets. Since then calculus has become a pillar of civilization, especially Western Civilization.

Mostly in calculus, integration is finding the area (or more carefully _defining_ the area) under a curve, maybe a parabola.

But can also have a _line integral_ , say, the work do when carrying 100 pounds of hay to the top of the hay loft of the barn. Here we integrate from beginning to end of the work. If we let the hay fall, then neglecting friction, etc. get the work back as energy. So, have to pay attention to the direction of the integration, from the ground up to the hay loft or from the hay loft back to the ground. The two values have opposite signs.

You can teach yourself calculus: Just get 1-4, at least one good calculus texts and dig in -- read the chapters, follow the material, work nearly all the more difficult exercises, and maybe check 1-4 times a year with a good mathematician. I did that for freshman college calculus (but didn't check with anyone). I never took the course and, instead, started on sophomore calculus and made As. No problem.

(3) Linear Algebra. Likely the next course would be linear algebra. It would be good to do this subject 2-4 times from more elementary treatments to some of the more advanced material. Some of the more advanced material can cover linear programming optimization, classic two person game theory, the proof of the Nash result in game theory, some treatments of the fast Fourier transform in digital signal processing, group representations in the quantum mechanics of molecular spectroscopy, error correcting codes in algebraic coding theory, numerical methods, and more.

The start of linear algebra is just one or several _linear_ equations in one or several variables. E.g., for two equations in three variables, _unknowns:_

         2x + 3y - 6z = 0

         5x - 2y + 7z = 3

For positive integers (whole numbers) m and n, given m such _linear_ equations in n variables ( _unknowns_ ), the number of solutions, depending also on the numerical constants in the equations, is none, one, or infinitely many.

The main way to find all the solutions is just Gauss elimination.

That covers a good chunk of a first course in linear algebra.

The notation

         2x + 3y - 6z = 0

         5x - 2y + 7z = 3

gets to be clumsy so we write instead

         2 + 3 - <USER6>

         5 - 2 + 7

write that with big, square brackets, and call it a _matrix_ with two rows and three columns. The individual numbers are _components_. Of course, a matrix is conceptually close to an array with two subscripts common in programming languages. Maybe we call the matrix

         2 + 3 - <USER6>

         5 - 2 + 7

A. Since matrix A has 2 rows and 3 columns, we say that it is 2 x 3 ("2 by 3").

Then we write the x, y, z in a column

         x

         y

         z

apply the big, square brackets, and call it a matrix with 3 rows and 1 column and maybe call it v. We write the right side

         0

         3

also with brackets, and maybe call it b.

Then we define a product, a _matrix_ product Av so that

         Av = b

means just the same as

         2x + 3y - 6z = 0

         5x - 2y + 7z = 3

From then on we work with matrices and try to avoid notation like

         2x + 3y - 6z = 0

         5x - 2y + 7z = 3

In this way, a course in _linear algebra_ is commonly called a course, Linear Algebra and Matrix Theory.

A matrix with just one column is a _column vector_ ; with just one row, a _row vector_. In short, either is called a _vector_.

Linearity: Given m x n matrices A and B, m x 1 matrices u, v, and real numbers c, d, we have that

         A(cu + dv) = cAu + dAv

Here we regard Au as case of function A with argument u. That is, we could have said that matrix A defines function f so that

         f(u) = Au

With

         cA + dA

we form _products_ cA and dA and _sums_

         cA + dA

So, we have to define both of those: The definitions are close to just obvious.

The equation

         A(cu + dv) = cAu + dAv

means that matrix A acts like a _linear_ function. Big deal! The real world is awash in linear behavior, and linearity is an enormously powerful property mathematically. The rest of linear algebra and matrix theory is nearly all about the consequences of such linearity.

The fundamental theorem of algebra is that each (high school style) polynomial in variable x can be uniquely factored into a product of terms of the form (ax + b) where the a and b are possibly complex numbers. Essentially from this fact, the more advanced parts of linear algebra use the complex numbers and not just the real numbers.

At the end of a second course in linear algebra, will pay attention to the essentially geometrical notions of length and angle and, in particular, to orthogonality, that is, perpendicular. Big topics are the Gram-Schmidt process where find some orthogonal vectors and the polar decomposition.

The polar decomposition result says that a matrix acting on a circle will yield an ellipse, that is, for matrix A and vector u, if we let u take on the values of all the points on a circle, then the Au will generate all the values of all the points on an ellipse. The ellipse has two axes, and they are mutually perpendicular. So, if have the vectors of the two axes, then have enough to construct the whole ellipse.

In linear algebra, this situation generalizes to any finite dimension and yields the singular value decomposition, principle components, factor analysis, and more.

With more advanced work, this situation becomes the spectral theory of self- adjoint linear operatiors central to quantum mechanics.

(4) The Four Main Parts of Math

There are four main parts of math:

(A) Foundations, that is, deep in the basement with set theory.

(B) Algebra, as in high school, the integers, prime numbers, the fundamental theorem of arithmetic, greatest common divisor and least common multiple, the fundamental theorem of algebra, linear algebra and matrix theory, the generalizations of number systems such as groups, rings, fields (the rational, real, and complex numbers are all _fields_ but so is the set of integers modulo a prime number), etc. Fermat's last theorem, settled by A. Wiles, is part of algebra; so are the deep, difficult questions about prime numbers, etc. There are applications in error correcting codes and cryptography.

(C) Geometry, e.g., high school plane geometry, analytic geometry, and differential geometry, a deep subject important for relativity.

(D) Analysis, as in calculus, differential equations, functional analysis, e.g., Banach and Hilbert spaces, partial differential equations, e.g., as in Maxwell's equations, fluid flow, and relativity, probability, statistics, stochastic processes, optimization, and more.

In differential equations we are given an equation with the derivatives of some function and want to find the function.

E.g., for real valued function y(t) and constants k and b, we might have

         y'(t) = k y(t) (b - y(t))

where y'(t) = d/dt y(t), the first derivative of function y(t). At one time, that little differential equation saved FedEx from going out of business. The solution is a _lazy_ S curve and a first cut at _viral_ growth.

There is topology which is partly in geometry and analysis. The main idea of topology is _continuity_ , that is, changing without sudden jumps or some cases of wildly fast oscillations.

There is algebraic geometry partly in algebra and geometry.

Much of number theory has deep connections with analysis.

Much of linear algebra is an introduction to functional analysis in analysis.

The older applied math is mostly analysis.

(5) Analysis. Usually after linear algebra will study _advanced calculus_ and analysis. Broadly there are two approaches, (A) theory and (B) applications.

The theory is mostly to give fully careful proofs of the results and first generalizations of what you saw in calculus. For the theory maybe the most respected text is

R. Rudin, _Principles of Mathematical Analysis, Third Edition._

For this, get the third edition and not either of the two earlier editions.

So, will discover that the integral learned in calculus is called the Riemann integral because B. Riemann made the theorems solid. Rudin also does the easy generalization to the Riemann-Stieltjes integral.

<comment><USER98> Part III

Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for _overtones_ at frequencies w and where each frequency w is a whole number multiple of a _fundamental_ frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes.

The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers.

I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of

Tom M. Apostol, _Mathematical Analysis: A Modern Approach to Advanced Calculus_ , Addison-Wesley, Reading, Massachusetts, 1957.

Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly enough for what physics and engineering do with _vector analysis_ and Stokes theorem.

In H. Royden, _Real Analysis_ and the first half of W. Rudin, _Real and Complex Analysis_ , can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations.

As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes.

In W. Rudin, _Functional Analysis_ get a treatment of _distributions_ that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory.

From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant".

For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some.

One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as _significant_ based on what it does for the real problem and not just its contribution to pure math.

Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more.

In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year.

For a little on how to do the original parts of math research, there is the A. Wiles comment:

"Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...."

That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you.

Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.

</comment></comment></comment><comment><USER34> I'm working on this myself and while I'm not very far along, I can share my approach.

1\. Go to Youtube, find the Professor Leonard channel. He teaches math at Merced College and is a very good lecturer. He has recorded himself teaching everything from pre-algebra through differential equations, and a statistics class. The one thing he has not done yet is linear algebra, he he apparently plans to do it.

2\. Watch his stuff, and supplement that with the corresponding "Schaum's Outline" or similar book for the topic at hand. Also, if you desire, buy a few editions old used college textbook for the corresponding topic. This gives you more exercises to do and a reference to consult if anything is unclear.

3\. As desired, follow the Khan Academy lessons on the topic you're studying. KA has everything from arithmetic / pre-algebra up through at least Calculus and Linear Algebra. I don't remember offhand if they cover Differential Equations or not.

4\. For Linear Algebra in particular, the Gilbert Strang lectures on Youtube are very highly regarded, and he has a text that was written specifically to accompany those videos. So that's a good resource for Linear Algebra.

5\. For "higher" math (real analysis, complex analysis, topology, abstract algebra, etc.) you can almost always find complete lecture series on Youtube / OCW. Depending on the topic, there may also be a "Schaum's Outline" or similar study guide book you can supplement with. And you can always find a used textbook on Amazon, usually for not too much money if you go with an older edition.

You can also find a lot of freely available maths texts online. See, for example: [https://math.gatech.edu/~cain/textbooks/onlinebooks.html](https://math.gatech.edu/~cain/textbooks/onlinebooks.html)

If you don't have a background in doing proofs, which is kind of regarded as the dividing line between "simple" math and "higher" math, there are a number of books on that specific topic, including texts written for so-called "transition to higher math" classes. Some of those are freely available online as well. There's also a good class you can find on Youtube, "Math for Computer Scientists" which covers proofs and what-not pretty well. There's a freely available corresponding text as well.

Another thing to do is consult forums where you can ask for help if you get stuck. There is math.stackexchange.com, physicsforums.com, cheatatmathhomework.reddit.com, learnmath.reddit.com, mathhelp.reddit.com, etc.

Somewhere I have a Google doc that lists a lot of the resources I have been using, and have queued up to use in the future. If anybody is interested, I'll clean that up, and make it public and share the link.

One last note: I haven't done it myself, but I've heard that if you live near a University, it's not too hard to find maths students who will tutor you to pick up some extra cash. So that's an option as well.

Edit: somebody else mentioned 3blue1brown on Youtube, and there are a number of other really good Youtube channels, including: Prof RobBob, NancyPi, and Dr. Chris Tisdell.

</comment><comment><USER99> Get Thomas 13th or 14th edition and solve all the odd problems. That takes care of calculus.

</comment><comment><USER100> I need this too, thanks for asking what I was too afraid to ask

</comment><comment><USER101> Check brilliant.org. Got everything there, worth the money.

</comment><comment><USER94> What is the purpose of a proof? You may think that it's intended to solve a problem, but that's really only half the point. A good proof is one that communicates your solution to other people.

Mathematical writing is hard to learn well under the best of circumstances, but if you don't have someone else giving you feedback on whether they understand what you're writing, it's absolutely impossible. You need to have a mentor or at least an editor at some point. That's not impossible to find outside of the university system, but it's very difficult.

(This is the single biggest reason why MOOCs for higher math haven't taken off. There are a lot of people who'd love to communicate something about the field that they've dedicated their lives to, but the feedback system just doesn't scale. If anyone can figure out how to fix that, it'll be a game changer.)

So the first thing you have to do is to figure out what you can reasonably expect to get out of this process. You can learn the definitions and theorems of higher math, and that might be enough, but you're never going to develop an intuition for them without understanding how to produce proofs on your own. And don't fool yourself into thinking that you can evaluate your own proofs. It just doesn't work.

If all of that doesn't have you turned off, then here are some ideas on what to do.

A university level math curriculum is split into roughly three components: * Lower level classes that focus on basic definitions and calculations; * Mid level classes that teach some basic theorem-proving skills in subjects that are useful for people in other quantitative fields; * Upper level classes that offer serious practice in theorem-proving as well as the core ideas of mathematics. You don't generally have to do classes in any particular order, but you do have to master the skills of each level before you go on to the next one.

To begin, you must be very comfortable with the contents of a high school math curriculum. Serge Lang's book on basic mathematics is a great refresher if you're not, or you can use any of the various popular study guides (Schaum's, Barron's, etc.).

At the first level, you have calculus. This is generally split into three semesters, with the first dedicated to limits and derivatives of single- variable functions, one dedicated to integrals of single-variable functions as well as sequences and series, and the last dedicated to derivatives and integrals of multivariate functions. There are plenty of very expensive books with glossy page and many color pictures and few ideas, but if you want a serious introduction, look at Peter Lax's books on calculus.

At the second level you'll almost always find introductions to differential equations and linear algebra. Differential equations have historically been the workhorse of applied mathematics and you really need to have some familiarity with them, but I've never seen a book on the topic that I liked. I probably won't be satisfied by anything at this level, though, so look around and see if you can at least find something inexpensive.

Linear algebra is a more recent topic (with many of its key ideas actually originating in the 20th century), but it's probably actually more important now. Gilbert Strang's books are popular and are worth reading for a first look, but you really can't regard them as a serious introduction to the mathematical side of the topic. Axler is probably the best book in that regard, but it's best taken on a second pass.

I think that probability should be regarded as a core class at this level. I don't think that's a fringe view, but it's not as universal as I'd like. I learned from Pitman's book, and I think it's as good as any to start with.

You can also take classes on complex variables or "discrete math" here. I don't know what a good textbook for complex variables is--maybe Saff & Snider? --but I'm sure there are recommendations out there. Needham's "Visual Complex Analysis" is a fantastic book, but maybe not really suitable for a very first introduction. As for "discrete math" (a jumble of topics from logic, combinatorics and number theory), find the cheapest book you can get that has decent reviews on Amazon.

At the third level, there are three main topics: analysis, algebra and topology/geometry. You can think of these as the three main viewpoints in higher math, and other topics being populated by people who primarily look at things with the tools of one of those three topics.

Analysis starts out as the theory behind calculus. In a first course, you'll revisit a lot of what you saw in single-variable calculus, but you'll learn why it's true rather than just how to use it. For a single semester undergraduate course, Ken Binmore's book is probably the gentlest introduction.

Modern geometry is related to what you studied in high school, but with a few more centuries of development. It also doesn't get a lot of coverage at the undergraduate level, which is highly unfortunate. Stillwell's "The Four Pillars of Geometry" is a wonderful book and completely accessible.

Algebra is a bit difficult to explain without getting into the weeds. Pinter's "A Book of Abstract Algebra" is very good at motivating the topic and explaining the basics, which is the best you can hope for in an introductory textbook.

Beyond that but still at the undergraduate level, you can get electives in combinatorics (use Brualdi), number theory (?), logic (?) and some applied topics as well. Looking through the course offerings of various math departments will help you to fill in what the other possibilities are.

</comment><comment><USER102> Check out mathvault.ca.

</comment><comment><USER103> You first need to define your boundaries and limits. Don't get me wrong, but mathematics is grinding, and if you think you have an idea where you're getting yourself into because you've studied calculus and algebra as an undergrad CS, you will fail miserably. Don't make the same mistake I did :)

Undergrad math is tooling; I like to use the analogy that it's like entering a workshop where you have all the tools available, but you're blindfolded, and you have no idea what each stuff is used for. So you'll have to touch everything. Look down for the tooling advice.

When venturing in Mathematics, for your own sanity, please have an objective in mind. I'm down serious! Understand what you want to research and go in that direction.

One last piece, Find a mentor, share, talk to people. You won't advance from yourself.

As a piece of tooling advice, I have the following I've stolen from Reddit a few years ago (sorry I haven't found the source to it) __ TOOLING __ First of all, most important, GO LEARN ALGEBRA. Seriously, I know you think its bullshit but its the most basic skill in some ways that any mathematician should know. Second learn Calculus: Single and Multivariable. If you are still interested here are some things to go onto next:

Discrete mathematics: This includes equivalence relations (probably one of the most important things for you understand ever), propositional calculus (logic) proof techniques (induction) and some basic combinatorics (Pigeonhole principle). You can literally find any textbook and start reading. The theory is kinda a hodgepodge, but those are the major themes.

Linear Algebra: Again, one of the most important subjects you will ever study. Once you understand this, you are really on your way, and this stuff comes up everywhere. Many mathematicians have said many of the biggest proofs in the world come down to "just some linear algebra". The major point here is to understand that there is only one vector space for each dimension over a field and understand how a linear transformation becomes a matrix only after a choice of basis. Here equivalence relations come up again!

Differential equations: Unless you're focused on engineering math or serious applied stuff, don't worry too much about this. Seriously, it's not that integral (haha get it!).

Complex Analysis: Yes, mathematicians and Engr. Actually, do study "imaginary" numbers, but there is nothing imaginary here. This is serious stuff, do it.

Okay, so now you're about as a sophomore/junior level place in mathematics. How to finish it off? It's not that unclear:

Abstract Algebra -- Grab any book read about groups, rings, fields, vector spaces, and modules. Proofs will be difficult here but work through it. There are so many books here, avoid Lang (good book but not for starting out), Dummit/Foote is okay. As a undergrad I had a good time with Rotman's An introduction to abstract algebra.

Analysis -- Grab Baby Rudin. No seriously, Grab this book, sit in a room for a semester and just fuckn' read it. You will basically be "redoing" calculus. This is a trial by fire, go!

Topology -- Grab Introduction to Topology by J. Munkries. Its so well written it might as well be a coffee table book.

There now, you have done everything a math major would. Yes, there are lots of things that are missing, arguably the most important things depending on what your goals are. Typically one studies Number Theory along with Abstract Algebra, or studies Analysis and Differential Equations together or Analysis and Topology. Seeing the links across different topics is essential, but I'm just giving the overview here.

Not every mathematician studies logic, and there are LOT of fringe topics that I'm omitting (including some of my favs: Projective Geometry, Varieties, Lattice/Order theory, Combinatorics, Elliptic Curves, Coding theory, Harmonic Analysis, etc.). However, none of these are required courses at more than a 1% of programs

</comment><comment><USER104> I could give you a list of books that are used at the graduate and undergraduate level that are considered good. However, almost none of them really discuss the motivations behind topics. My suggestion is to start with this book by Arnold on Abel's Theorem. If it interests you then delve deeper into the topics once you are done with it.

[https://www.mathcamp.org/2015/abel/abel.pdf](https://www.mathcamp.org/2015/abel/abel.pdf)

</comment><comment><USER53> Math is a huge field and I'm not sure any can truly have both a deep and wide understanding of everything in math.

But if you want to give it a go, why not start with the tried and tested "Euclid's Elements" and a book on the history and philosophy of mathematics. If you are still interested, then tackle a course plan laid out by MIT or any other reputable university.

Also, I suggest getting an exercises/solutions book because math is doing. You can read the theorems, corollaries, etc and think you understand it but until you try to solve a problem yourself, you really won't know if you really understood the topic. And in your down time, you can check out math related channels/videos on youtube or elsewhere. It's going to take a lot grit and determination. Best of luck to you.

</comment><comment><USER105> Go to the library and read their math books. I don't know why the internet is filled with these questions that are always "how do I start doing x?" Just f* do it dude. Nobody is going to be able to or bother to answer anyway. The answers wouldn't make deep sense to you because they rely on having done it before. Once you've started and you're really stuck and everything you read doesn't seem to illuminate it, post a question. People will notice you've put some work and time in and will be eager to help you.

</comment><comment><USER106> youtube

</comment><comment><USER107> You want to learn more about both pure and applied mathematics. Pure math is very proof heavy which is excellent for self learning because you never have to take anything on faith. Proofs are meant to be very clear explanations.

One of the best resources for learning in this way is to find books that are structured to have you develop the theory via the problem sets. Of the top of my head, Tao's Analysis books and Atiyah & MacDonald's Commutative Algebra book are exemplars of this style. These books often show the basic definitions, some results, and then break up remaining results into step by step problems.

Proofs will be the hardest part to wrap your head around. Once you understand proofs, can read them, the remaining is going to be a question of time. Think of it like programming: you can't learn by reading, lectures, etc. You need to learn by doing. In this case, the doing is doing proofs.</comment>