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Mathematics for the Adventurous Self-Learner (neilwithdata.com)
1232 points by nsainsbury on Feb 24, 2020 | hide | past | favorite | 210 comments

I know this is going to be the case for likely nobody, but I have browsed most of the self-study math threads that pop up here as a forever-on-my-todo-list thing and I have a remark to make:

I have yet to find a guide that does not start with the assumption that you graduated highschool.

That is a very reasonable assumption to make. We are in a community of technology and engineering, it would be a bit ridiculous to assume the people you are surrounded by did not have a fundamental base of mathematics.

But the times I have tried to go through these teach-yourself materials, it went from zero to draw-the-rest-of-the-fucking-owl real quick. [0]

I have been programming for 14 years, but stopped doing schoolwork around age 12, and never did any math beyond pre-algebra.

Does anyone know of materials for adults that cover pre-algebra -> algebra -> geometry -> trigonometry -> linear algebra -> statistics -> calculus? At a reasonably quick pace that someone with a family + overtime startup hours could still benefit from?

[0] https://i.imgur.com/RadSf.jpg

(Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier)

I would use Khan Academy. Start at a level that feels too easy, even if it's elementary school math. The key to learning anything is to start at a level that feels too easy and gradually increase difficulty.

As you finish a subject, see if there's a corresponding book in the Art of Problem Solving store [0]; you can revisit the subject at a deeper level that will strengthen your foundation. The AoPS books will also expose you to areas useful in programming like discrete mathematics.

Before any of the above, take Coursera's Learning How to Learn course. You'll learn lots of effective strategies to get the most out of your efforts. For example, you can use Anki [1] to remember definitions and concepts you've managed to understand and to schedule review of problems you've already solved.

[0] https://artofproblemsolving.com/store/recommendations.php

[1] http://augmentingcognition.com/ltm.html


Myself and my daughters use Anki every day. It has been the defining factor in turning my mediocre job into a career. And Ankidroid on Android is open source.

Can you tell me what type of cards did you use for your career? I'm thinking to start using Anki, but not sure how to structure the cards. Thanks!

Lots of cards for Linux commands, programming gotchas, and client-specific details like "Foo prefers H:i:s date formats".

Can you elaborate in how it's been that transformative? I get why it's a useful tool, but you seem particularly enthusiastic. How are you using it?

A big shout-out to both Anki and Ankidroid. I wish there was a way to donate to them.

Like other sibling comments, I would also like to know how you have used Anki to transform your career.

I like Anki but I have never been able to figure out how to capture complex and often long business related information in tiny cards.

> The key to learning anything is to start at a level that feels too easy and gradually increase difficulty.

Is it though?

In this particular case, for this particular person, may be. But personally, I always found myself bored and ADHDly switching to something else in 5 minutes if I wouldn't drop myself right in the deep end. And from my interactions with different software engineers over the years, I doubt I'm the only one.

I didn't say it was the only way.

More like the most foolproof way.

I hope I'm not too late here, but if you are in the US I would highly, highly recommend signing up for developmental classes at your local community college. You are exactly whom those classes are for. If you've tried on your own before and struggled to stay motivated, doing it in a structured way, in 15 week "sprints", may be just the kickstart to your self-study program you need.

Disclaimer: I was a full-time community college professor for a decade. I had no idea what a resource they were. It's small money compared to either the alternative of a university or not succeeding. If you use them you will succeed. It's what they do, and they've been doing it for a very long time.

Seconding this. After high school, I worked in industry until I got bored, and went back to school starting with community college. I'd never thought I was good at math, and I placed into pre-college algebra. Programming had taught me to think methodically, so I crushed those early classes. Fast-forward a decade; I've got a PhD in math (nothing like what I set out to do; I just followed my passion).

That is extremely inspiring. Can you speak more to how you went from industry to a community college to getting into and completing a PhD program? Did you quit your job to return back to school? What area did you end up specializing in and what do you do now?

Long and short, I got sick of the tedium in web development, quit my job and went back to school. The dot-com bubble had just burst, and I had been taking occasional classes including a very inspirational data structures course which planted the seed with formal proofs.

After I went back to school, I tutored in the math study center to pay the bills, which really helped cement not just the learning but also the notion that I could survive academia. I'd gone in with a plan to study engineering, but after I transferred to university, I kept dawdling on the math prerequisites and not taking the engineering courses that needed them. So it kinda gradually dawned on me that math was what I loved, and away I went.

I never strayed too far from computers. I'm a graph theorist, specializing in computation; had I known better I'd have gone into computer science because that's where I see the most progress being made.

> had I known better I'd have gone into computer science because that's where I see the most progress being made.

Interestingly this is similar to what my two advisors (one from the math department and one from CS) suggested to me. It would be easier to do the math I like in a CS department than it would be to do the CS I like in a math department. Do you feel like math departments are more conservative when it comes to working outside the discipline?

You, seriously, just made my day!

I'd also like to chime in with support for community college math classes. I was pretty good at math to the point that it was becoming a problem for my JR High, so they sent me to the community college to take the math series starting from the bottom up. Went all the way through Differential Equations at community college.

Maybe an even better way to start would be with trying to take a math class during the summer semester. 8 weeks of intense study to kick things off. I bet after doing some programming algebra would be a breeze.

I'd like to further recommend this course of action. Community colleges are a wonderful resource and at the one I attended the Math Department stood out as being particularly good. The student body tends to be extremely diverse too, from high school kids to retirees and everything in between. Oh and it's extremely affordable.

Might want to ask current students about those classes first.

My experience with community college math developmental classes was: "Go to this AV room, watch these videos, do these workbooks. If you have questions, I have office hours on these days.". Which was terrible. I did have someone I could come to with things I didn't understand, but it was obviously not something they liked doing and you had little choice about the material used.

This was decades ago, but I'm betting that now it's: "Watch these Youtube videos, do these workbooks. If you have questions, ask them in our online forums." Which is likely worse than just doing your own thing.

Holy crap! I wonder if it's a rural vs. metro area thing. Our CC was definitely classroom work- again, decades ago.

Same. Was a normal college thing. Classroom, teacher, chalkboard, occasional glazed eyes... but also the expectation you'll ask questions, and the ability to get feedback or clarification.

I worked in the math lab of my community college while in high school tutoring returning education students. 16 year old me teaching algebra and calculus to 55+ folks. They were some of the most engaged people I have ever taught and they were so happy, like tears of joy happy to finally get material that plagued them for most of their lives.

Anyway, my recommendation is to check out "the vibe" of the mathlab or whoever are the folks doing the tutoring. If they engaged and love answering the same questions all day long, then definitely sign up. If the tutors are like watch this video and do this quiz, if you have a problem sign up on the sheet kinda attitude, then find another place.

> Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier

One thing I tell my high-school students: mathematics always looks harder than it actually is. One of the essential skills in succeeding in math is looking at a page of arcane "stuff" and having your reaction be, "Whoa! Can't wait to learn what this means," rather than, "Whoa! This looks so complicated!"

Mathematical is its own language that has developed across continents and millennia. It has its quirks and foibles, but overall, community consensus has guided its notation. Mathematicians want things to be simple and "make sense", especially the notation they use. It's never as terrible as it looks.

Sigma specifically is a Greek letter, but the notation is not Greek. Like a large amount of modern mathematical notation, the convention came from Leonhard Euler in the 18th century. It was a disambiguation choice because the letter S was overloaded.

Single-symbol identifiers are enormously popular in mathematics because mathematics is not computing. Because math is (even now) essentially a handwritten subject, its design plays to the strengths of handwriting. Line size, height, and character layout are essentially freeform. Character accents and modifiers are easy. discrete_sum would never fly in a handwritten world, just like ∑ wouldn't pass code review.

I don't think it's because of handwriting. Math notation (for math) is just as useful when reading and writing it on computers or printed pages. I think it's just optimized for its subject matter, e.g. a small set of names and well known operations.

Important to note though that Euler probably didn't think of students trying to follow their professor at 8 in the morning while he scribbles proofs on the blackboard using ∑, ∈, E, e and S.

I solved this by just buying all the high school math textbooks and going through those on my own. I preferred this way, because it lets you have the same background as everyone else.

In attempt #1 I was jumping ahead to read the interesting stuff (calculus), and while I could make some progress, it was needlessly difficult because I didn't start from the basics.

It attempt #2 I started from the very beginning (course 1 out of 10 mandatory high school courses), and focused on doing exercises. However progress was slow, because I would just continue forward when I felt like it.

Finally attempt #3 was successful. I committed to doing exercises in order consistently every day after waking up. This felt great, as every week I was making noticeable progress, and having all the prerequisite knowledge for each next step made progress much easier than I had imagined it could be.

With the slow start but gaining pace towards the later courses, I finished this self-study project in 2 years (could have been close to half that, had I gotten into the groove from the beginning), and found it quite enjoyable. It didn't feel like a chore at all, more like the highlight of each day.

While I'm not sure there actually are a lot of high school prerequisites for higher maths, I think the experience of attempts 1-3 reflect pretty much what might be the key to studying maths successfully.

So in my first year of studying maths, I had 8 hours of maths lectures (and 4 hours of a minor which was of negligible effort). The exercises that came with the maths lectures made this a full-time program (and I estimate that while I didn't usually study all weekend, I typically only had 1-2 weekends per year in which I didn't look at anything at all). So one thing that can easily go wrong is underestimating that for every minute spent reading / listening to a class, one would want to spend 4 minutes working the problems.

The other comment I would have is that, yes, university level mathematics is (at least it was for me) incredibly hard. The reward is also astonishing: All these hard exercises I struggled with one year, are easy to do on a napkin during breakfast in the next year.

Can you clarify how step 3 was different for you from step 2? (I’ve had similar unsuccessful steps 1 and 2, wondering what they keys were for you to turn the corner)

In step 2 I was just making progress when I felt like studying, so many days might pass between reading about some new concept and then needing to apply it. By that time I might have already forgotten some, and progress felt so slow that it wasn't very motivating to come back to it.

Doing it every day on the other hand, I would quickly need the thing I just learned, both reinforcing the learning and making it easier to apply it. Then as progress was much faster, it was more motivating as I could see myself making gradual progress each day, such that completing each course seemed like a doable undertaking.

It's nice to see that I'm not alone in this situation.

I completely ignored math during high school due to a number of reasons (bad influences, even worse teachers...). I then went to college and managed to pass through calculus classes, mostly thanks to pure mechanical memorization and professors turning a blind eye to my lack of understanding.

Since my graduation (~5 years ago) I've been trying to fill this gap, but like you perfectly described, all materials expect you to have a solid basis. I think the problem is that math is huge and people spend a good chunk of their lives learning it (4-17 for the fundamentals alone!), so we fail to see how much it involves and how hard is for somebody that didn't have a proper education to learn it.

I have been making solid (although slow) progress with https://www.khanacademy.org/. I tried to learn from the top a bunch of times, but always hit a wall and dropped it. I only started moving forward when I decided to go through the basics, algebra and trigonometry 101. It has been a hard and slow journey, but each step comes faster and becomes more rewarding.

A few ideas:

* Guesstimation:Solving the World's Problems on the Back of a Cocktail Napkin. Math is a tool. Start using it with some simple arithmetic and scientific notation. Once it becomes something you can use and play in that context, everything else becomes a lot easier. This is water cooler talk and is something actually usable immediately.

* Speed Mathematics Simplified. From the 1960s. Wonderful book about doing arithmetic from left to right. Also has some good stuff about decimals/fractions/percents as well as checksums. Being quick with arithmetic and getting that number sense makes everything else easier.

* Burn Math Class. Gives an appropriate viewpoint for a lot of math. Gets a little whacky as it goes on, but the core ideas should help you take ownership of math.

* ... gap not sure what to put in ... Maybe Precalculus in a Nutshell... But play around with GeoGebra. Exploring geometry, trigonometry, and precalculus visually is key to getting an intuition about. Get to know the behaviors of the functions, but don't get lost in trig identities or solving random algebraic equations. Things like Newton's method (or the Secant Method) are more important for learning about than lots of arbitrary algebraic simplifications (they can be important too)

* Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. At some point, if you mastered K-12 math and want to get a good mix of theory and application and efficiency, this book by John and Barbara Hubbard is really quite nice. It puts linear algebra in one of its primary contexts of being the main foundation for solving nonlinear systems.

I like books by Ron Larson, particularly his Trigonometry and Applied Calculus books -- the applied calculus title (intended for social science and business majors) vs his "Calculus" book (intended for math, physics, and engineering majors) is much gentler for people seeing this material for the first time. Although I do highly recommend his Calculus book once you have the other book down.

Gelfand also has some nice texts on Algebra, Trig, and Geometry that are reasonably cheap, especially if used.

I'm older and went back to school later in life to study math, and these are the books I learned that material (for the first time -- I flunked math thru high school) from.

Here are the exact titles and ISBN-10s:

Ron Larson, Calculus: An Applied Approach, ISBN: 0618218696

Ron Larson, Trigonometry, ISBN: 1133954332

Israel Gelfand, Trigonometry, ISBN: 0817639144

Israel Gelfand, Geometry, ISBN:1071602977

Israel Gelfand, Algebra, ISBN: 0817636773

And as others have mentioned, Khan Academy is pretty good, although I tend to prefer patrickJMT's explanations a bit more: http://patrickjmt.com/

IIRC, the Schaum series books have been very good for this level. Very concise, no nonsense, gets to the point simply, and has a bunch of exercises.

One thing I’ve realized from experience: most books with lots of pictures and thick stacks are faking it (i.e. most college and school math/physics textbooks are not even worth the paper they’re printed on). The whole conceptual basis of these subjects is to distill everything down to a few simple ideas, which can then be applied in different contexts. The concise books typically tend to be much better at conveying the essence without bullshit. You just need to read a couple of hundred pages without getting stressed, rather than getting lost in an 800 page book and losing the big picture.

I'm in a similar situation. I'm having to learn linear algebra,calc,basic prob, and brush up on all of my holes in between. I haven't found a direct path, however. What has helped me is this deep learning book[0] which has spelled out very plainly what I need to learn. From there I use a combination of math is fun[1], better explained[2] and 3blue1brown. math is fun really helps by just giving you examples and definitions strictly based on the subject instead of assuming knowledge in another category. better explained helps with intuition. And, 3b1b was second to none for really painting a picture of linear algebra and calc. Even though I've had to watch the videos 3-4 times each to get it, I'm extremely happy with what I've grokked. One last resource is Eddie Woo[4]. Super clear and enthusiastic intuitive lessons from him teaching highschool and slightly beyond math

Good luck in your journey. I know how frustrating it is to not be able to find the math resources you need at an awkward level. If you happen across even better resources please share.

Also a tip that really has been helping me: when you "read" a math equation, don't simply recite the variable names and numbers. Try to say out loud what they represent. I've found that if I can't then I don't really understand the concept I'm working with.

[0]https://d2l.ai/chapter_preliminaries/index.html [1]https://www.mathsisfun.com/ [2]https://betterexplained.com/ [3]https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw [4]https://www.youtube.com/user/misterwootube/playlists?view=50...

I won't repeat the good suggestions already made, but I'll add this: find a set of the Open University MST124 text books. The OU publishes their own maths books, which are specially written for self-study (since that's your only option with the OU) and their courses generally assume very little or no previous knowledge. There's an even more basic course (MST123 I think) if that one is too advanced. These are serious courses that form part of their Mathematics degree course so they are very thorough and, I hate this word, but rigorous.

Of course you get the books when you sign up for the course, but it's way cheaper to get the books and study on your own, and you'll get 90% of the information that way.

I don't know if you'll find the pace quick or not; personally I would say that learning maths is very hard, and the materials you use are unlikely to prove to be the botteneck (spoiler: it's you).

I'm studying for a Math BSc with the OU and I think their books are great. They are designed for 100% self-study and are polished over the years. The range of topics in level 1 math courses such as MST124 or MST125 is huge, while keeping the appropriate level of difficulty.

Are you in the current presentation of either 124 or 125 by any chance? I'm in both, planning to do the BSc too.

I'm actually finishing M208 Pure Math this week. It's been great.

Which university, if you don't mind me asking?

Can you share the names of the textbooks?

OU = Open University. It's a UK distance learning university. Website is at https://www.open.ac.uk/

The courses the comments refer to are the level 1 courses listed here: http://www.open.ac.uk/courses/maths/all-modules

The text books are the texts for these courses. I don't know if you can get the books from OU without registering for the courses.

The OU; the Open University. It's a British university founded sixty or so years ago with the intent of widening access to higher education. Their campus is centred in Milton Keynes, a bit north of London.

I took a Masters in Maths with them and it was brutal. Serious exams are a youngster's game. I got a glimpse into why so many Cambridge wranglers were also serious atheletes.

The books are called "Essential Mathematics" 1 and 2. They correspond to these modules. http://www.openuniversity.edu/courses/modules/mst124 http://www.openuniversity.edu/courses/modules/mst125

Second-hand OU books are usually bought from https://www.universitybooksearch.co.uk/

Could you provide the direct links for the books used in: MST123, MST124, MST125? I could not find them in your links.

If you're not in the UK, I see that Thriftbooks appears to get at least some copies[1].

1: https://www.thriftbooks.com/browse/?b.search=MST124#b.s=most...

Search for "Open University MST124" or similar on www.ebay.co.uk or www.amazon.co.uk. There seem to be plenty of copies of 124 available on both at the time of writing.

It seems that the more basic course is Discovering mathematics (MU123). http://mathschoices.open.ac.uk/mu123


> Question 1: Sara wants to buy a desk before she starts her Open University course. She has chosen a suitable place for it but needs to measure the space before going to buy it.

> Which of these units is the most suitable for measuring the width of the desk? (a) millimetres, (b) metres, (c) centimetres, (d) kilometres.

Seems pretty arbitrary to me. Okay (d) is arguably out. (b) is maybe out? (c) vs (d) seems a toss up to me. Who cares if it's 1234mm or 123.4cm? How is this even a good question? Not a good sign

Note: the answer they want is (a). I don't think I have a tape measure I could get accurate millimeters on especially given measuring a space where I need to bend the tape at 90 degrees like next to a wall.

I might hazard that you, by virtue of being here and by virtue of having dug so far into the question, are exactly not the kind of person this question is targeted at.

Imagine being someone who cannot produce an answer to this question. Who cannot discuss it.

The vast majority of people for whom this entry-level course is aimed at do not think like you. The purpose of the question is not to elicit a correct answer, although I note the effect intended actually worked perfectly on you and the question has been a success in your case. Maybe you are exactly the kind of person this question is aimed at!

The specs for furniture and home appliances do use mm to specify dimensions. Also any 5m measuring tape in Europe has mm.

The reason for the question, IMO, is not to get the given answer.

The reason is to learn why there is an answer, at which point you can give an opinion as to whether cm/mm are better. You've missed the primary step, and almost certainly are - as others point out - over qualified for this question (which is a precursor to discussions of decimal precision).

Often when helping the kids with homework I find myself answering "well this is the answer they want ... but the real answer is ...".

tl;dr the question is there to make you think, not because there's only one answer.

I have no direct experience of the OU, but I did have a mathematics teacher (head or deputy head I think of the department at a large school, and taught the more advanced courses) who used it.

Not taught courses, took its courses himself, in mathematics.

It went up in my estimations then for sure, because he was already a great teacher, intelligent and knowledgeable in the subject (certainly more than enough for up to 'Further Additional' at A level) and studying for his own interest (or I suppose a career change or further study for all I know).

I know a person who tried Khan Academy to get caught up on math after a pretty poor childhood education (they didn't finish middle school). Their feedback was that they were able to get their fundamentals down enough to actually pursue more advanced topics fairly easily (about a year of self-study to go from zero to 'high-school graduate' level.)

From my own personal experience, I would recommend getting to the high-school graduate level and then taking some classes at your local university or community college for topics beyond trigonometry. You'll likely be able to handle everything up to that point on your own without much support, but many people struggle at that point and benefit from having people to ask for help when they need it.

I would wish you the best of luck, but I have no doubt you'll get to where you want to be without it.

I had the same problem. There is a series of books for folks like us -"Pre-Algebra Demystified", "Algebra Demystified" etc. I used them to catch up for a graduate level statistics course. One hour every morning for three months took me from pre-algebra through trig, and I did very well in class. I loved these simple books. They contain exactly zero owl-drawing.

The Demystified series is amazing. You can learn anything - from Pre-Algebra to Supersymmetry!

The No Bullshit Guide To Math And Physics


I bought the Math and Physics copy because it has an ebook option and the first chapter is the Math guide. I’m going through a few pages a day and it’s crisp and straightforward. There is a sample of the first chapter on the site, I suggest you check it out to see if this is what you are looking for.

Found via an HN thread from last year on this topic.

Ha, I came here to shamelessly self-promote, but I see my book has already been mentioned!

Here is a direct link to the preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...

I won't claim that Chapter 1 is complete and detailed review of all of high school math, but I did my best to cover all the essential topics needed for mechanics and calculus so that the book will be self-contained.

Is it still in progress, or does the sample omit some information on purpose? E.g the logarithmic operations. Good book, good idea!

The preview is a "special build" of the book that removes most of the content and leaves only the section formulas and headings --- this way to ensure the internal navigation works, but it's confusing since it looks like chunks are missing. Previously I would cut only selected pages but then the links didn't work. Sorry for the confusion. You can check the preview on amazon for the real page numbers: https://www.amazon.com/dp/0992001005/noBSmathphys

Great work. I read the sample and bought a copy myself.

The book looks lovely. Did you write it using latex? What is your tooling to author a book like this?

Yep, standard LaTeX with the book class.

The preview linked is a little outdated so it shows the old figures, but the latest version has all the figures done in TikZ (a vector drawing package using LaTeX syntax, see https://www.overleaf.com/learn/latex/TikZ_package for examples).

There is no "tooling" per se for the book production (just run pdflatex to get the PDF, then upload to lulu.com and amazon.com, and they take care of the printing). I do have some scripts to enforce naming and notation conventions though, and there is some advanced git-rebase kung fu going on that allows me to reuse the high school prerequisites from Chapter 1 of the MATH & PHYS book for the LINEAR ALGEBRA book as well (basically when I fix typo in the master branch, I have to rebase the LA branch).

One thing that has been tremendously useful and I highly recommend for any authoring task, is using text-to-speech for proofreading: https://docs.google.com/document/d/1mApa60zJA8rgEm6T6GF0yIem...

I'm actually self-studying as well, and I try to compile everything I learn and the notes that I take into 'Intuitive Guides' which I'm going to make available on my github repository. I actually have a guide on Linear Algebra which you can find here:


I'm going to release one on Maxwell's equations next week, and I started working on a Calculus and General Relativity guides as well, so hopefully it helps!

Precalculus in a Nutshell is a beautiful little book by George F Simmons, which pretty much captures everything you need to know to undertake the study of calculus. https://www.maa.org/press/maa-reviews/precalculus-mathematic...

Linear algebra is quite a beautiful, approachable subject; and a certain amount of it is necessary to make the leap from single variable to multi-variable calculus. Without a good grip on calculus, you can’t really what’s going on under the covers with linear algebra. What you need to do is precalculus (Simmons) -> single variable calculus -> very introductory / elementary linear algebra -> multi variable calculus (Apostol) -> less introductory linear algebra but still fairly basic (Gilbert Strang Intro to Linear Algebra) -> mathematical analysis (Apostol) -> linear algebra done right (Axler). You have to apply a spiral method where you return to subjects as you gain the tools you need to understand them better. You’ll never be done understanding geometry, algebra, or analysis.

Also, math is a problem solving art, and you can’t solve problems by reading, you solve them by thinking. Seek out problems that challenge and consolidate your understanding. You should be able to prove everything in Simmons and it should seem totally natural and intuitive. Then you’re ready to struggle with calculus, which is a subject humanity struggled with for centuries before getting a rigorous handle on. You probably want to get a handle on the mechanics and intuition, first, and for that I’ve heard that “Calculus Made Easy” by Silvanus Thompson is good.

Don’t try to eat too much all at once, you’ll make yourself sick. Don’t try to cheat yourself of the patient struggle to understand, confusion is completely natural when striving to really know something.

Simmons? WOW! He's a serious mathematician, e.g., as in his

George F. Simmons, Introduction to Topology and Modern Analysis.

I studied it one summer in an NSF program at Vanderbilt. Then I concluded that he is one heck of a good math writer.

He has elementary textbooks on calculus and differential equations, as well, so you get an entire development from precalculus all the way to topology and analysis from the coherent viewpoint of a legit (but sympathetic) mathematician.

I know these are extremely common suggestions, but…

• 3Blue1Brown has great introductory series on linear algebra and calculus.

• Khan Academy covers pretty much all of US high school mathematics, and you can go through it at whatever pace you want.

• I can send you a few Australian high school textbooks if you want.

Just the book for you:

"Who is Fourier: a mathematical adventure" https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

It is a simply brilliant book that takes you from basic trigonometry, logarithms and so on through calculus and finally fourier series.

> At a reasonably quick pace

Consider discarding that requirement.

> that someone with a family + overtime startup hours could still benefit from?

I suggest drilling fundamentals with easy exercises in moments of low quality time. (I often do a few Khan Academy skills during bouts of insomnia. For others it might be the commute, or just before bed, or...) Periodic repetition over the long term is more powerful than cramming.

Save your best quality time for your family and your job. Accept that you will progress in math at a slow pace. Before too long you will nevertheless end up ahead of many successful (!) software engineers who do not have strong math foundations.

israel gelfand, a top russian mathematician, has books aimed at high school students that served as education materials for distance learning in russia. the series covers algebra, functions, trigonometry, graphs, geometry, and more.







the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.



> I find these books very helpful whenever I need to brush up on math

> Each grade folder has a number of chapters, each chapter with a number of exercises, and answers to these in a single file.

> Exemplar Problems (for in-depth learning) with Answers

> Try solving the exercises and problems using pen and paper.


> Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier)

i guess its idiomatic if you know "Greek".. ∑ is sigma, greek letter S, so Summation. And Π is pi, for Product..

Our site (brilliant.org) is designed for your use-case, among others.

I use Brilliant for the same reason.

I found Khan to have too many endless lists of equasions, while Brilliant is much better at building intuition.

Yep, brilliant.org + the first book from minireference.com are what OP needs.

I found that the community on brilliant.org can be quite toxic. I was a member for awhile, but quit due to the people on the site.

It gives me great pain to hear this.

Please do email me silas [at] brilliant.org, so I can figure out which failure mode you hit and evaluate if it has improved at all since you quit.

In addition, as others have noted, our courses have no community aspect to them at all.

I’m using brilliant, but just the courses, not any of the community parts. What was the failure mode?

I think the final thing was on the daily challenges; I always tried to solve them before looking at the answers even If I don't know the "proper" math to do so. I had posted my (correct) solution that I got to by writing a program to work it out; and I got some replies saying I was an idiot for not doing in the proper (math) way.

Ah, that type. Thanks for the heads-up.

https://schoolyourself.org/learn/algebra is great. You start from addition subtraction and go through your list for the most part. Khan academy is great too, but with schoolyourself you can walk through it a bit faster.

SAT prep books such as those from Kaplan or Princeton Review tend to be extremely cheap, if not free, and cover almost all of those topics pretty succinctly :)

Speaking as someone who was kicked out of precalc and then never touched math again despite working as a dev, your point resonates with me :)

I'm in a similar situation - been programming for 25 years (off and on) since I started in middle school on my own. But in school we had 'Algebra I' and 'Geometry I' and I never went beyond that. Struggled with quadratic equations and factoring, never heard anything about trigonometry (which I now realize starts in basic geometry), calculus, and never heard the words 'function' or 'intuition' in regards to maths in school. Tried to do the classic Andrew Ng course on data science and I was completely lost every step of the way because of new language and the fact that his class was nothing like the way I had been taught so long ago.

I'm in the same shoes, and as ridiculous as it sounds but I've been in programming (self-taught) for 10 years now without an especially good knowledge of math. I've been slacking in HS on math classes, despite the fact that I loved math and was really good once I sat to listen, practice and understand the material. Also, I'm painfully aware of the fact that my math is really bad and that I'm missing on utilizing it in my job, so I've decided to practice at least one hour a day on Khan Academy, and eventually enroll to comp-sci university this year.

What is an example where you missed mathematical knowledge to do your job? Honestly curious, I never needed any math during programming for my job.

I guess the thing is that for any bit of reasoning, “there’s math for that”. So whether it feels like you’re using maths or not, you are.

Restructuring code, for example, often needs a good grasp of negation in logic.

Do you need formal training to do it? Not really. It’s advantageous to have a good math grounding though. My colleagues that have a good math education can often reason and communicate using graph theory, especially when it comes to architecture. Set theory is also super useful - I’ll often see people writing crappy algorithms because they don’t know about using sets (again, understanding computability and complexity would have helped here).

Maths is all around us, it is really just the study of patterns after all. That applies more so in software, even if it’s not immediately apparent.

> What is an example where you missed mathematical knowledge to do your job? Honestly curious, I never needed any math during programming for my job.

Basic arithmetic comes up all the time: pro-rating a monthly plan, figuring out how much to scale up or down a system in response to changes in data, figuring out when your system will run out of memory/disk.

Statistics and probability are also pretty common. I'm often calculating standard deviations and finding expected values of non uniform random variables. For example, how fast does a queue have to be to handle 1 second tasks 90% of the time, but 30 second tasks 10% of the time?

Derivatives comes with many graphics tasks, such as 3D graphics or animations.

Yeah, I always think I "should" brush back up on my now-rusted-solid math skills—I doubt I could pass the final for any math class I took past maybe 9th grade without studying, let alone anything I took in college—but they've rusted for a reason. I never fucking use them. If I do it's some narrow little thing that I look up, do, then never look at again.

[EDIT] and then there's "what is math?". The memorization from early grades that everyone shits on, with some simple algebra, is what I actually, ever, use in my life, plus some very basic geometry when doing stuff around the house. If it's for work it's some practical application thing. "Real" math like proofs? Never, ever.

> [EDIT] and then there's "what is math?"

Good remark. I wouldn't call arithmetic math, nor would I call using a Boolean expression math. I am currently working on a compiler bug that has to do with liveness analysis. That is an algorithm, which kind of is math, but the actual bug is just 'oh, for some reason the registers that the function arguments are passed in are not marked as live', and I wouldn't say that I had to use any math.

In my job, I'm usually either fixing bugs, parsing formats, making different API's work together by converting stuff or writing wrappers around API's. I wouldn't call any of this 'math', and if you'd ask me I have never used math at work (which is a shame really, because I really love math).

In my case, having to write WebGL shaders required me to brush up on mathematical concepts, but I very much enjoy being granted the opportunity to learn about those things.

Now I'm curious. IANAProgrammer, what programs have you made that don't have any maths in them?

Can you share example code? I can't see how that works.

It would be easier to show example code that actually used math! I might use math once a year, or every few years. Someone can write standard enterprise REST/SQL endpoints for ages and never use a bit of the stuff.

I'm also in a similar situation. I graduated highschool and passed mathematics, barely. I didn't absorb any knowledge and switched classes halfway through my last year.

I'm studying CS in University now and will have to do a math-related subject and I'm quite nervous about it, because my mathematics skill is extremely low, and has been my entire life.

I've also been bookmarking guides like this but haven't gotten to looking into them (pure haziness) other than reading the introduction, which usually says "this guide assumes highschool level mathematics".

I first learned about calculus from, I think, this book, "Calculus the Easy Way" [1] (or, at least the first chapter or two). It does make use of some algebra, but you're not necessarily limited to the strict progression of pre-algebra -> algebra -> calculus.

[1] (amazon: https://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/dp/...)

Consider getting a private maths teacher. At this level it's probably not very expensive but you'll be able to learn so much more efficiently.

I went to the local library and borrowed high-school level textbooks. Just start from the level where you absolutely understand everything, however low you need to go. Solve all the exercises. You will blaze through the easy stuff, but the exercises will make it clear to you when you need to begin paying attention.

I have seen the Chrystal's "Algebra: An Elementary Textbook" as a recommendation for a guide to most of Algebra a working CS person would need. It is about 100 years old, and appears to be lacking in modern pedagogic cruft and is to the point and densely packed with useful information.

I'd highly recommend OpenStax math textbooks, they are completely free, openly licensed, peer reviewed, and by Rice University. They also start from Pre-algebra.


It’s not unidentifiable once you study more. Sigma “is” “roughly” the letter S, and stands for sum. Squiggly S stands for integral (roughly a sum). Intuition for why these things roughly “are” each other takes perhaps more study.

Simmons - Precalculus Mathematics in a Nutshell [0] (128 page booklet)

[0] https://books.google.com/books?id=dN1KAwAAQBAJ

> (Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier)

S for Sum, T for Total, N for Number, I for Index, etc. might though.

At least in stats, the use of greek letters is helpful. Usually, the upper cases are kept for function like T(x) or to denote random variables. The lower cases for observed values. So when we see a greek letter, we know right away it's constant or a parameter.

I agree completely, and would emphasise 'at least', or change it to 'for example' even.

I only meant that criticising anyone for using them in the first place is unfair/hypocritical, because we'd quite readily do the same with the Latin alphabet.

Not everyone speaks English. It's nice to keep that in mind when naming things in the universal language that is maths. S, T, N might make sense in some languages but not others. More greek would at least level the playing field or something.

I didn't for a second mean that they should be used instead of Greek; just that 'we do it too, so don't be surprised that somebody else abbreviated things with their script'.

although it’s not exactly the resource you need to be able to work algebra problems, I highly recommend that you read https://www.feynmanlectures.caltech.edu/I_22.html - Feynman starts with “let’s assume we know how to count”, and in just a few pages takes you through a lot of math. Even if you don’t follow all of the details, it’s a nice overview.

Math from scratch https://learnaifromscratch.github.io/math.html

There's also Prof Leonard YouTube channel, get the book(s) he uses, covers exactly what you want:


khanacademy has content for those school work level.

Khan Academy is great as they start from the very beginning. If you’re a good problem solver you can skip most of the videos. If you learn by videos, they’re very helpful.


tl;dr - don't be afraid to go back to math concepts from elementary school to help you along the way to learning more math.

You have a gaggle of responses to go through, but I want to put this out there anyways.

Algebra is talked about as a 'breaking point' for many Americans; however the solutions rarely look at what transpired (or didn't) all those years before a student reached algebra.

Math standards in the United States are set so that ideally:

Kindergarten: learn to count

1st and 2nd grade: learn to think additively (+ and -)

3rd and 4th grade: learn to think multiplicatively (x and division); learn fractions

5th and 6th grade: learn to think in ratios and proportions; learn to think algebraically

Throughout all of those grades you are also supposed to be learning the properties of operations.

By the time you reach an algebra course in 8th grade or 9th grade, it requires you to call upon all of that previous knowledge.

Common problems:

- learning the properties of operations by rote and thus not understanding how to use them to manipulate algebraic equations

- not making the leap from additive to multiplicative reasoning, which hurts a students ability to understand fractions, which hurts a students ability to understand ratios and proportions, which hurts a students ability to reason with algebraic equations

- I forgot exponents. Most students only know those by rote or a bit about them before suddenly seeing huge exponents and negative exponents attached to variables in algebra.

Algebra itself may not be a problem. It is however a strong indicator of knowledge of the above. It's also where the house of cards falls down for students like you and me.

Source: was student who math fell apart for in school. I learned all about this when I left the business world to teach 4th grade, eventually created and piloted an "Arithmetic to Algebra" course for students to put all of this into practice, students learned, we rejoiced

Check out OpenStax: https://openstax.org/

Sure ∑ would pass code review. And in Maple and Mathematica it is code.

artofproblemsolving.com is excellent. Many of their books, while intended for middle-schoolers, are replete with problems that many adults would find challenging.

Khan Academy. Start with Khan Academy.

the art of problem solving series

khan academy. lessons and practice problems. go for it.


I pretty much followed the same route as OP re-studying mathematics seriously after 10 years in industry after initially doing a CS degree and doing mostly software engineering but transitioning into Data Science the last 3 years. When I saw Book of Proof then Spivak then Apostol on his list I chuckled because that’s exactly the route I ended up following as well. Studying from 04:30 to 06:30 in the week and about 8 hours split up over the weekend, Spivak took 8 months to complete (excluding some of the appendix chapters) but if you can force yourself to truly master the exercises - and Spivak’s value is the exercises - then you’re close to having that weird state called “mathematical maturity” or at least an intuition as to what that means. You can forget about doing the starred exercises, unless you’re gifted. Spend a lot of time on the first few chapters (again, the exercises), it will pay off later in the book. It was a very frustrating experience and I had so much self doubt working through it, it’s an absolutely brutal book. Some exercises will take you literally hours to try and figure out.

If you do Book of Proof first you will find Spivak much easier, since Spivak is very light on using set theoretic definitions of things. Even the way he defines a function pretty much avoids using set terminology. Book of Proof on the other hand slowly builds up everything through set theory. It was like learning assembly language, then going to a high level language (Spivak) and I could reason about what’s going on “under the hood”. Book of Proof is such a beautiful book, I wish I had something like it in high school, mathematics would have just made sense if I had that one book.

I read a quote somewhere, think it was Von Neumann that said, you never really understand mathematics, you just get used to it. Keep that in mind.

Heh, nice to find someone who walked a similar path! :-)

Ah Spivak...yes, I absolutely agree it's one of the best books to build up that mathematical maturity everyone talks about.

For me Spivak took about 6 months and I managed to do almost all of the starred exercises - Gifted? No. Brutally determined: yes. And I was quite fortunate to be in a place in life where I could put serious hours in to it at the time.

After that, I learned to relax a bit more as I realised I had pushed myself way too hard and was close to burning out. I still love looking back at that damn book though. There's just something that's so special about it...the way the exercises build upon each other and connect together. It's really unique.

So, you've made all that effort, how does it help you in your new role as a data scientist? Is there anything you do now that requires "mathematical maturity"? Or is it something that can be learned much quicker on as needed basis?

There are a lot of charlatans in the Data Science space who lack the necessary mathematical background for their roles. For me it was necessary to get a rigorous understanding of probability theory, and applied probability theory is basically what mathematical statistics is about. My background was CS and software so R, Python, data visualisation and ML operationalization is by and large the easy part of Data Science to me. If you pick up any book like Bishop's or ESL you will be extremely frustrated if your mathematical background is not there. I didn't feel comfortable creating predictive models for production use that I didn't completely understand what was going on "under the hood", the assumptions being made and how they could fail. It's the only ethical thing for any engineer to do.

I do deep learning research for a living. I've taken graduate classes in probability, stochastic processes, optimization algorithms, and signal analysis (ECE PhD). I almost never completely understand what's going on under the hood of my models as soon as they get larger than a single neuron XOR mapper. That does not prevent me from finding ways to improve the performance of very large models (millions of parameters and dozens of layers). I agree that there are some papers (or the two books you mentioned) that can be quite dense and heavy on math, but I can't say I've ever felt like I needed any math other than basic calculus, linear algebra, and prob/stats 101 to understand almost all ML methods that people actually use in real world. Obviously if you want to make breakthroughs in theoretical ML, then sure, you do need the mathematical maturity (mostly because you will need to be formally proving things), but if you're a regular data scientist? Can you give some example what kind of math is involved in your predictive models?

Initially I heard about Euler's famous Basel problem. Years later I got to solving it for my self (for curiosity and fun). I guess what intrigued me was to think of trigonometric sine as an infinite polynomial...After I worked it out, I had indeed seen the fire in Euler's own eyes...I could see how excited he was at having discovered something amazing...But this got me into hooked into math history. What I really wanted was how people came about discovering the Taylor's series...the intuition behind it. So that is how I came across John Stillwell's book. I have to warn people it is rather academic. But if, you, as a self-learner, is excited about mathematics, I would suggest Norman J Wildberger's youtube lectures on mathematics history. I find the buildup to calculus quite fascinating. J. Stillwell's book was the recommended reference in those lectures...

One thing I don’t think is discussed enough is the process of how self-learners in math get critical feedback. Most advanced level math textbooks do not have solutions to check their work against nor do they have a way to get feedback by an expert and this is essential for learning. Least with programming, you can get immediate feedback and know whether what you did is correct or not.

Usually, some people will post some or almost all of the solutions online if the textbook is really famous, for instance, baby rudin.

Although rarely, some authors do provide solutions, like Knuth's books, Stephen Abbott's Understanding Anaylsis, etc.

For immediate feedback, maybe you can checkout [0] to learn some formal proof by doing interactive proving.

[0] http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game...

BTW, you can always ask questions on https://math.stackexchange.com

It's not immediate feedback, but you do learn when you right and when you are wrong. You learn when you are bullshitting yourself. You learn that you need to be able to justify every step in a proof, and if you can't do, then you are wrong. Trying to bullshit the right answer is the most common way to end up with a faulty proof. Of course, you can also end up with a faulty proof because you can't differentiate your own bullshit from truth, but this is less common.

Learn proofs well and you get pretty good and knowing when you’re right. Enough for almost any problem you’ll be likely to encounter in a math textbook anyway.

I strongly disagree.

It's a little like saying "learn programming well enough and you'll know if some piece of code works as expected without running it."

> "learn programming well enough and you'll know if some piece of code works as expected without running it."

That is 100% true.

Obviously the code needs to be self-contained (not calling into other unknown code) but so do mathematical proofs.

That's what seperates them both, it's not about immediate formal checking from a compiler here, but rather if our proof is being justified adequately according with the rules of deduction and reasoning, it's a more intuitive approach in math, although you do end up knowing whether something is right or wrong, akin to programs.

This is so difficult. I've been doing it off and on for twenty years and not made much of a dent in things.

The hardest part I think is understanding and measuring your progress. In school you've got exams and classmates to compare against, profs to talk to. Alone it's much harder. "Do I understand this well enough?" "Did I do the problems right?" (Especially with proof problems, how do you know you're right?). "I can work through some problems one by one, but it feels like something fundamental I'm missing. Am I, or is this chapter really just about some tools?"

Then it's way too easy to say well I'm never actually going to use any of this so why am I doing it ... and take a few months off and come back forgetting what you'd learned.

I've tried a few things recently that help with that:

1. Don't do exercises unless you want to. Completionism is a trap.

2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self.

3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful?

4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is.

Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback!

(edit: formatting)

this is excellent execellt advice. seriously anyone interested in learning math, chancing on this comment, should write it down. i wish i could upvote many more times. i have a bachelors in pure math and am 10 years out. i have time and again revisited things and didn't make good substantive progress until i came to these same exact conclusions.

especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them. a lot of exercises are a hazing ritual or imagined by the author to be a dose of bitter medicine (i'm looking at you electrodynamics by jd jackson) since they mistakenly believe all readers are formal students.

the most important exercise is to mull over and consider what you're reading/learning. naturally dovetails in to asking question: what happens if i remove a hypothesis from a theorem, what happens if i add one, is there an analogy to another object/group/measure/etc, etc.

also read multiple books (http://libgen.is/ is your very very good friend and generous friend). a lot of math authors (no matter how esteemed they are) are terrible writers or make mistakes (look up errata for previous editions of your favorite book).

the only thing i'd add is to learn to use LaTeX to take notes - it is much easier and faster and neater.

> this is excellent execellt advice ... especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them

I think this is deeply mistaken. In a well-chosen book, such as the ones in the submitted article, doing the exercises is not to test your memorisation, it's to develop your understanding.

Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that.

Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.

> In a well-chosen book, such as the ones in the submitted article, doing the exercises is not to test your memorisation, it's to develop your understanding.

This is a great point and example of the problem with a one-size-fits-all strategy. For some books, exercises are an essential part of comprehension. For others, not so much.

> Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that.

My experience is that by taking excellent notes and asking why, you engage with the material to a similar degree, if not a greater degree, than by doing exercises. (Once again, depending on the book, as you mentioned.)

> Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.

I would argue that's the point. Usually self-taught math is about self-growth. Getting new ideas, being exposed to new concepts, recognizing patterns. Being able to actually "do it" on-the-spot is beside the point (and is the quickest level of skill to evaporate once you stop focusing on that material, anyway.)

This, 100 times. Mathematical understanding can only be obtained by doing, fighting with the concepts, causing that pain that you get behind the eyes. Just reading the text will give you a surface knowledge, maybe enough to impress at interviews or parties, but nothing more ...

>Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.

Isn't that literally exactly what I said?

> if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them

The submission and this entire thread is about learning math. That, to me, implies learning to do, not learning about. Yes, you said:

> if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them

There's ground in the middle, and this thread is about that. This thread is not about learning for tests and qualifications, nor is it about "being exposed", it's learning how to do the math.

And for that you need to do the exercises. You don't need to do all of them, you don't need to be completionist about it, but if you don't do the exercises, if you don't actually do the math then you won't actually be able to do the math.

Specifically, you said (quoting again):

> if you're ... just interested in learning ...

There's a difference between learning about and learning to do. If you meant just "learning about" then you are at odds with the entire thread. True, in that case you don't need to do the exercises, but I don't think that's what people are talking about here. I think people are talking about being able to do the math.

And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.

Otherwise it's fairy floss, and not steak.

My apologies if all this seems overkill, but there's a real danger of talking past each other and being in violent agreement, and I wanted to state explicitly and clearly what I mean, and why I thought you said something different.

> you won't actually be able to do the math

but i'm not a mathematician. i don't need to be able to do math anymore than i need to be able to do history (while reading serious history books).

>And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.

no i didn't. that's precisely why i used the word "exposed".

>violent agreement

we don't agree but i'm not being violent. but my responses are short and yours are long.

i do not see the exercises as essential for anyone other than practicing mathematicians. i have read a great many serious math books (i just recently finished Tu's Manifolds book and am now reading Oksendal's SDEs). i read them without doing absolutely any exercises but following the rest of the guidelines in the post i responded to. the experience is gratifying because i learn about new objects and new ways of thinking about objects i've already learned about. that's absolutely the only thing that matters to me.

but let me ask you something

>That, to me, implies learning to do, not learning about.

here's a fantastic explanation of the topological proof of Abel-Ruffini


would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?

We agree that if you want actually to be able to do the math then you need to do the exercises.

Do we agree that if you don't do the exercises then you probably won't actually be able to do the math?

You are discussing learning about the math, and not eventually being able to do it, because you say that you don't care about becoming a mathematician, therefore you don't need to do the math. Fair enough.

But my reading is that that's not what this thread is about. This thread, and the original submission, is about learning how to do the math.

> i do not see the exercises as essential for anyone other than practicing mathematicians.

I think you're wrong. Knowing how to actually do the math has proven useful to many people for whom it is a tool in their craft/job/employment. Learning Linear Algebra properly, being able to actually do it rather than just talk about it, can be enormously useful in Machine Learning.

>> That, to me, implies learning to do, not learning about.

> here's a fantastic explanation of the topological proof of Abel-Ruffini ... would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?

Understanding a single proof implies very little about one's ability to actually do the math. I've met many people who are math enthusiasts and who have watched hundreds of math videos. They say they understand all of what they've seen, and yet they are unable to do the simplest proofs, or the most elementary calculations.

My experience of people's abilities is that if they haven't done the exercises, they usually can't actually do the math.

But you complain about the length of my replies, so I'll stop. I think I've made my position clear, and I think I understand what you're saying, even if I don't agree with it.

>You are discussing learning about the math

You keep repeating this but you're evading the question about abel-ruffini and the question about whether reading a history book is "learning about history" as opposed to learning history.

You're making a weird distinction. People learn in different ways. Some by doing exercises and some by just playing with the objects. I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.

You edited your response.

>I've met many people who are math enthusiasts and who have watched hundreds of math videos

There's a difference between watching numberphile or whatever and essentially watching a lecture on a proof. Very few people are watching/consuming rigorous expositions. I think that's the difference not the lack of exercise.

Learning about history is not the same as then being able to do research in history, nor being able to apply the principles learned from it in context. So no, reading a history book is learning about history, not necessarily being able to "do history".

> You're making a weird distinction.

As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math, and headed a team doing research in technology, this is a distinction that I can clearly see. My inability to explain it to you is regrettable.

> People learn in different ways.

Yes they do.

> Some by doing exercises and some by just playing with the objects.

Doing the exercises is playing with the objects to try to answer specific questions. Good exercises are carefully constructed to help the reader learn how those objects work in an efficient manner.

> I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.

In my experience research mathematicians learn now math from papers by, in essence, constructing their own exercises based on what they're reading. In general it takes significant experience and training to be able to do that.

Clearly you don't think one needs to do the exercises subsequently to be able to do the math. Good for you.

I disagree.

>As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math

Me too so now what? I don't think your credentials give you any real authority but just make you look like you're gatekeeping.

>Doing the exercises is playing with the objects to try to answer specific questions.

Great so then we're in agreement: playing with the object is doing the exercise.

The funny thing is that at one time I actually did all of the exercises in volume 1 of apóstol's calculus. You know what effect on me it had? I was so bored I didn't read volume 2. And today I'd still need to look up the trig substitutions to do a vexing integral.

> I don't think your credentials give you any real authority ...

It wasn't intended to, it was to provide a context for my opinion.

So let me state my opinion as clearly as I can, and then I'll leave it.

* Math is a "contact sport" ... you have to engage with it;

* Reading books is not, of itself, engaging with the math;

* Watching math videos is not, of itself, engaging with the math;

* Well designed exercises are a valuable resource;

* If you can easily do an exercise, skip ahead;

* If you can't do an exercise, persist (for a time);

* Ignoring the exercises is ignoring a resource;

* For the vast majority of people, doing the exercises is an efficient way to engage with the material;

* To say "ignore the exercises" is, for the vast majority of people, an invitation to not bother engaging with the subject;

* Doing all the exercises is probably a waste. Doing none of them is an invitation to end up with a superficial overview of the subject, and no real understanding.

See? It's pretty hard. This is what I've been dealing with for the last 20 years of on and off trying to get through the bigger Rudin book and a couple others.

Just reading doesn't get much at all. Not even a superficial overview. I tried it. It's essentially a meaningless combination of words after a certain point.

Reading extremely thoroughly is actually marginally useful. Stopping to think, do all these assumptions matter, why, what if one of them changes, etc, pencil in hand, making notes, testing things out. I've managed to "understand" the topics when doing this, and so far it's been the highest ROI method. But it does still leave one feeling like something is missing. Just because you can sight read music doesn't mean you're an expert on the piano.

Doing exercises is a huge jump on investment, and the return on that investment is a bit questionable from my experience. A couple reasons: first you don't know if you did them right. If you did them wrong then that's negative ROI. Second you don't know what a "reasonable" workload is. It varies by author. Is it three problems per chapter, is it all of them, are some orders of magnitude more difficult than others? Without some guidance it's hard to know if your difficulties are due to not understanding basic material, or due to that problem being a challenge geared toward Putnam medalists. So they may cause you to question your understanding and thus mentally roadblock you unnecessarily. And finally with proofs (and this may be a me thing), it's pretty easy to say "I guess this is okay(?)" and move on, even if you're not sure. Since nobody is ever going to review it, and it's just a homework problem, it's very very hard to will oneself to make sure every assumption is correct and you're not missing anything, even if you feel like there's a good chance you are. Or perhaps I just don't have the constitution to do so.

So while I think doing exercises is necessary for a deeper understanding, I don't know whether the ROI is worth it outside of a classroom perspective. You need feedback for exercises to be beneficial. At least, I feel like I do.

Finally, is even taking a class that useful if the end state is that two years from then you'll have forgotten most of it and so what was the point. Can you claim knowledge of a subject that you've never actually used beyond some homework problems and exam questions, or is this still a superficial understanding? Having an ends where that knowledge gets used seems critical.

I feel like I have some knowledge but I don't feel like I'm there yet. But I don't know if I know where there is. Maybe that's the biggest challenge. Does completing a Ph.D. even get you to there? No idea. But, I guess it's up to the individual to decide what they want out of it. Nobody can determine that for you.

Shared a similar thought here: https://news.ycombinator.com/item?id=22401420

Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.

For all the adventurous self-learners out here, we would like to invite you to our self-study group named Integer Club.

IRC: https://webchat.freenode.net/#integerclub

Slack: https://bit.ly/integerclubslackinvite

Mailing list: https://groups.google.com/d/forum/integerclub

We pick up old concepts from popular textbooks and literature as well as new stuff from new literature in both mathematics and computer science. We plan to have online meetings periodically to share what we learn, work through popular literature, and have a few talks on interesting topics.

It is a tiny community right now that hangs out at Freenode IRC but the Slack channel is there too if you are more comfortable with that.

I think it's great that people are posting book links like this, however, what I've found most helpful is actually having someone to help guide you.

I realize how lucky I was that I found a Discord server ran by a math PhD graduate who is willing to help us guide our learning. From this, I've started learning Algebra and Analysis (just starting with the latter). It's nice to have someone to discuss problems with when you get stuck and to guide you. Likewise, he can suggest exactly which problems I should do for a give chapter, that way I don't spend my time doing ones that just repeat the same simple things over and over and can focus on nice, conceptual ones. So, if you can, please try to find someone to help guide you, or be that guide for someone else! Having it has made me seriously consider going back for a mathematics masters (and maybe PhD), switching from my physics background.

I would say that learning mathematics is virtually impossible without a teacher. Like, would you try learning Karate without a teacher? How about if you get with a bunch of friends and you all try to learn it together? No way.

Even the professionals try to find someone to learn a new concept from. There's something about mathematical writing that is too fragile/brittle for wetware.

One other strategy: I've noticed the really smart (arrogant?) people just don't bother reading new mathematics, they re-invent it themselves. It's actually worth trying if you can stomach it.

Could you share the discord server? Thank you.

I'm not the owner, and it's a small server, so I don't feel comfortable. Sorry. I think I found the owner through /r/math, so you might be able to find his old post there though.

As someone who dropped out of highscool after 10th grade and never went to university/college one great way I've found for learning mathematics without any foundational basis is trying to learn CG/3D programming.

I always felt like maths was too abstract to keep me engaged, but when the output of your work is immediately observable visually it becomes a lot more engaging. There's just something so much more satisfying being able to "see" the results.

Plus as a self-taught programmer, I find it much easier to learn front-to-back by deciding on a desired outcome and working towards it, rather than progressively building up abstract fundamental skills that can later be combined to achieve a desired outcome (which is essentially the traditional academia path for learning STEM fields)

This is why I love to do game development without using a game engine. It gives you a reason to learn math, optimize your code down to the metal, all while having fun playing your game.

Yeah I've been self-learning 3D "the hard way" and have been really enjoying it.

Keeping it as low-level as possible, I'm using CycleJS for dataflow management and Regl.js for drawing via a CycleJS-Regl.js driver.

All state is explicitly managed observables/streams in CycleJS, which maps out to Regl.js draw commands, which are basically raw frag/vert shaders with some bindings mapping my state from CycleJS to appropriate uniforms/attributes.

I probably would be able to produce some usable output much faster if I used an engine like Unity or a framework like Three.js, but I feel like I would have missed out on gaining so much knowledge by only working with high-level abstractions and never having to even touch GLSL code.

I came to the IT industry after a bachelor degree in math, 5 years in and all the math I know is gone I still remember some Fourier, signal processing and probability statistics that I never used in my day job, or anywhere else.

Time is valuable, it's the most valuable thing a human being has, I understand it's the hobby of OP to learn all this math, but unless you are going to use it why wasting all the time?

I've been pursuing mathematics as a hobby for the last 2 years or so. I got a mathematics major in undergrad so my motivating factor was mainly to explore some areas that I hadn't done coursework on, primarily algebra and number theory. (I focused more on logic in undergraduate/grad.)

I really enjoy how the subject is divorced from a lot of the modern attention demands and encourages more of a 'zen' thinking style.

As others have highlighted, it can be difficult. I work full-time as a software engineer and at the end of the day there's usually not much left in the tank in terms of "creative work". The morning is usually more productive for me - generally I'll spend 10-15 minutes on the commute in reading over the proof of some lemma or working through some computational exercise.

Things that have helped me:

- Focusing on a particular problem area rather than just "mathematics". The classical problems of Gauss and Euler tend to be more my speed than the modern mathematical problems of Hilbert or beyond. What started my journey was looking into the insolubility of the general quintic polynomial equation, something you learn in high school as a random factoid but has a lot of depth.

- Studying from small textbooks that I can fit in a backpack, so I can "make progress" during my commute. Dummit + Foote might be a great algebra reference but it's just too bulky to transport.

- Limiting the scope of how I think about the activity - my goal isn't to master these concepts on the level of a mathematics graduate student, it's more on the order of Sudoku. If I don't get something, that's okay. People spend their whole lifetimes learning this material and I'm just trying to fit this into whatever creative time I have left after the full-time job is done.

Do any of you all have some tips for understanding mathematical notation? I feel this is often poorly explained, and it feels like a language all its own that just does not speak to me. I did pretty well in calculus, but I still don't really understand what the dx was supposed to represent and in reality I was just really good at pattern matching when it wasn't supposed to be there anymore.

I try to read papers now and again with a math orientation, and I quickly get lost when trying to translate the concepts into cryptic formulas, and often when they make the "obvious" transition from step 3 to step 4 I just have no idea how they got there.

I feel this is by far my biggest barrier to understanding most mathematics, and I have thus far found no way to overcome it.

I think usually the problem is "almost getting it" and trying to move forward, which means small uncertainties add up and all the sudden one is totally lost without being sure exactly why. So it's important to go back and make sure each piece of notation is crystal clear before moving forward.

Any statement in math is meant to be directly translatable to human language. You should be able to read it out loud in English and know exactly what you mean when you say it.

Unfortunately, sometimes math uses awful notation. For example, df/dx. This is a case where df doesn't mean anything (or at least it's not normally well-defined), and dx doesn't mean anything either (same comment). But the notation as a whole means something. If we write g = df/dx, then we can understand that g is a function whose input is x and output is the slope of f at x.

Sounds like you might simply not understand the definitions for these operators and symbols. In other words, it's not a notation problem. I find that it's helpful to mentally replace the symbols like dy/dx, sum, lim, integral, and so on with the concepts they represent. That is, go from operators to definitions.

The most key piece of advice is to take walks. Walking is essentials for mathematics. Many times when walking with my father he would turn for home and start walking faster, and by that sign I knew that he wanted to get home and write down a lemma.

I'm glad you posted this, because I use walks this way too. And because it reminds me of William Rowan Hamilton and the quaterions!

I emphatically agree. There is something that walking does to your brain that really helps you see the big picture.

This is a solid read, with good book recommendations. After several years of tinkering with self-learning I bit the bullet and applied to a MSc in Applied Math program. (via ep.jhu.edu) I've had to take some pre-reqs to get started since it's been almost 20 years since I have been in a college math class, but it's been an enlightening journey re-learning calculus and now dipping my toes into differential equations. I don't think I could have gotten this far with self-learning, but I realize YMMV.

I will say I don't feel like single-variable real number calculus tells the whole story. I had taken that and linear algebra in undergrad but never any further, and now that I've taken single and multiple variable calculus, with real and complex numbers, plus integration of linear algebra ideas, the mathematical model feels a lot more like a cohesive whole to me, highlighting fundamental ideas that only barely peek through in a typical Calculus I class. I would encourage anyone talking to calculus to at least do the typical Calc II class, if not Calc III/multivariate. There is a beauty and structure to building up from calc I through III that I was missing before.

I'm pretty skeptical about these "best of" lists of books for self-directed mathematics education.

I have my own "best of" list that is very different to this list, although there are a couple of crossovers.

If you are fortunate enough to have access to a university library (or libraries) I would _highly_ recommend inquiring about access to their general collection. I was also fortunate enough to study mathematics to a university-level three-year degree at a research university. So I had an excellent head start.

A HUGE part of my journey of collecting my "perfect library" of mathematics self-tuition and reference books (and course books) was to do my own research on collecting the perfect titles. I started when I was in the early days of my mathematics degree and I used resources like Amazon, Usenet, libraries (already mentioned), and ... that was about it.

Another important question to ask yourself is the following:

"Why am I doing this?"

Life is short and by the time you hit middle age, if you have a family or bills to looks after, are you REALLY going to want to lock yourself away in your study room to learn Lebesgue integration instead of focusing on the rest of your life?

Consider that people fail to emphasise is that mathematics is a social activity much more than many people realize.

Exercise: Find the topics of mathematics that are important to your goals and are missing from the list and find your favorite books or two that cover/s these topics.

Exercise: Consider whether your interest in (self-directed) mathematics is so sincere such that you have a serious application in mind, that you might be better off enroling in a course? Even if it's a night course that last a couple of years, you will meet a LOT of people who can help in ways that are immensely more productive than trying to do this all by yourself.

I recently purchased volume 1 of my favorite calculus and analysis book. It's an incredible masterpiece. The coverage of topics is much broader and more interesting than Aposotol or Spivak. The latter books are both very good but they also have myopic, one-track pedagogical approaches and limited themes in their coverage.

Exercise: Find your own favorite introductory calculus book that is suitable for the motivated student.

> I recently purchased volume 1 of my favorite calculus and analysis book.

Which book would that be if I might ask? I'm wagering... Courant? ;)

Indeed :-)

Courant & John, to be exact.

I recently had the experience of taking my first graduate-level probability course. It assumed quite a strong familiarity with real/complex analysis, and I suffered quite heavily. Something of note was that once I finally managed to "peel back" the analysis, the underlying intuition made a lot of sense for the simplest cases in probability (e.g. hypothesis testing between two normal distributions is a matter of figuring out whose mean you are "closer" to).

I am of the opinion that notation is a very powerful tool for thought, but the terseness of mathematical notation often hides the intuition which is more effectively captured through good visualizations. I would really like to take self-driven "swing" at signal processing, this time approaching it through the lens of solving problem on time-series data, since as a programmer I believe that would be quite useful and relevant.

In my opinion, the issue here is notation and a bit more. I did about eight years of college in math, changed paths, changed careers, changed careers again to ML/DL research, and now will finish a CS undergrad degree this month.

I put it in context because it's not quite a direct comparison since I have been in greatly different situations and ages between studying math and CS, but putting that aside, I have to say I have greatly enjoyed the computer science means of teaching more than math, doubly when it comes to self-learning. Concepts in math are generally taught entwined with the means of proving those ideas. That's important if you're a grad student looking to be a math researcher, but (IMO) it is not so great if you're a newer student or learning on your own and trying to grasp the concept and big picture. A proof of a theorem can be (and too often is) a lot of detail that really doesn't help you grasp the concept the theorem provides or is used towards, often because it involves other ideas and techniques from higher levels or just different types of math, both of which are out of the scope of the student learning the topic. Worse yet, it is standard for a proof to be written almost backwards from how it would be thought out. Anyone from a math educational background has the experience in homework of solving a problem, then rewriting it almost in total reverse to be in the proper form to submit. This means not only is the proof of the theorem not useful towards conceptual understanding, reading the proof doesn't show you chronologically how you would discover it yourself. That is a lot of overhead cost to break through to get to real understanding, real learning. As you mention, notation as well is another thing you need to break through.

I have found computer science and related classes to be taught more constructively. Concept is given first, and then your job as student learner is to construct it. Coming from the ML field, I love comparing math and CS proofs of topics here. Explanations from CS people of back propagation, for example, are always visual, and books/courses will have you construct a class and methods to do the calculations. Someone with a bit of programming knowledge can follow along in their language of choice. Math explanations get into a ton of notation from Calc 3+, and it's going to take a lot of playing around and frustration to get a working system out of the explanation. Even the derivation section on Wikipedia is not something most people will understand and be able to turn into useful output.

The more I see other ways concepts are taught, the more I wish math had been taught a different way. There is a lot to break through in order to get to real understanding, just by the way it's formed and taught.

I am always astonished to learn that there are such self-learners in the world. I wonder how it is even possible to have a family and spend whole day building a startup - I cannot imagine that startup work is less than 8 hours a day - and then at evening they learn math or other complicated branch of science. What time and more especially how much energy they have for the family? Are these guys superhumans? I never was able to achieve such level of daily energy spent without trapping in burn out. I am not critiquing or being jealous here, just having genuine interest. How is it possible to be sustainable across so many years?

I have my doubts that the people who write these kinds of posts truly did everything they say. As you say, it just does not seem possible to thoroughly work through all those math books + raising multiple kids + maintain friendships + working out daily (as he claimed he did) + work full time + things like cleaning and shopping and other chores.

He talks about spending ~10/week on math, with more dedicated when he was between gigs. If I replaced reading for pleasure and all television (not that I watch a ton, but maybe an hour a day or so), I could carve out 10 hours a week without making other changes.

Not everyone is at a startup. I’m a bartender with a physics degree who’s learning more math in my spare time for fun, and to eventually switch to some sort of data science career if I tire of bartending.

Wow that's a brutal list of books... I'm impressed the author could work through all of that in just six years! I feel like math is a subject you need to get back again and again to refresh in order to retain. I got some pretty good grades in linear algebra back in the day... but I don't really remember much about it right now, sigh.

My strategy to get back to study math these days is getting to learn Wolfram Mathematica and Sage. Once I can move around those two, I feel like I will be able to create a tighter feedback loop on whatever Math subject I'm happen to be studying at the time.

Does anybody have any experience with How to Prove It? by Velleman? Recently I was thinking of starting on it, but I'm not sure about the level of commitment necessary.

I worked through this book to learn how to do proofs. It turned out way more fun than I expected. The book really did demystify proofs for me. It took several months of studying - there are many exercises. But completely worth it. I'm glad I have read this book before studying Group theory and Real analysis.


Seems to have some intro to logic and math language, if you have never read an math books as the ones referred in this post, that book should be a nice way to ease into it.

I see, thanks. I do have some experience with proofs and maths in general, but I never went through a text dedicated explicitly to developing proving techniques.

I might give it a go anyway, but do you know of a more advanced version of this book as well?

Not really. I'm a math major, what you usually do is have an intro to logic course and then just jump into those books.


I would recommend Conceptual Mathematics: A First Introduction to Categories by Lawvere.

It is written by a true pioneer. And also, you will impress your friends by your hipster foray into category theory.

However, this book is far from being hipster. Also, I would not be surprised if a high school student would be able to follow this book over the course of a year or two.

If you titled the book: Sarcastic introduction to how simple set theory is then I would actually be fooled that it were the correct title.

There is 3 books listed that essentially cover the freshman (first year) courses in Calculus. And 4 for Linear Algebra. If you work your way through even 2 different books for one topic, you are going to have a broader foundation in the topic than a normal math student in a normal university after completing the corresponding course. And you will have spent much more time, too. University courses don't usually cover everything that is in a textbook. And students don't usually read books through. In fact, students usually try to skim the course notes just enough so that they can solve the weekly problem sets.

There is maybe nothing wrong with being thorough with the elementary topics if you're studying for fun. But if you're studying for applications, I think you should cover the basics only adequately, and then quickly move on to more advanced topics. Basic Calculus is only the foundation, stuff that is actually useful in applications comes later. Basic Linear Algebra can be useful in its own right, but the advanced stuff is even more useful.

I suggest building an adequate foundation, not a comprehensively thorough foundation, and then moving on to the more powerful stuff. Which varies depending on what you actually want to use math for.

To be clear, I don't at all advocate that people work through all the Calculus books. Likewise for the Linear Algebra books. My aim was to provide alternative options (which are easier, cheaper, etc.)

Another book similar to Morris Kline's _Mathematics for the Nonmathematician_, which the OP mentioned, is Lancelot Hogben's _Mathematics for the Million_. Originally published in 1937, it has been in print ever since, through several revisions. This also takes a historical approach, beginning with numbers and counting, measure and Greek geometry; and eventually covering calculus, matrices, probability, and statistics.

You can take a look at it, at the Internet Archive,


I found the Real Analysis course to the really really hard back in university. I thought it was like the CS information analysis when looking at the course title. It was nothing like that at all. It didn't help that the professor teaching it was pretty bad. I remember he used to jog into the class carrying a tennis racket and in tennis sporty dress with headband, seemingly just coming back from a tennis practice and acting to be cool. People just rolled their eyes. The teaching was just reading off the book. Darn, it was one of the top schools. How did this clown get in?

The perenial self-learning mathematics curriculum is hbpms.blogspot.com.

This is fantastic. I recently been studying set theory and discrete mathematics as a self-directed learner and it is incredibly helpful to see others hewing the same path.

I love stuff like this. I graduated with an MS and never understood Calc II+. It was always memorization and repeating various theorems etc on the test. But, I didn't truly grok the fundamentals. I was just a good test taker and it bothers me to this day. So, learning math has been a continuing project of mine and things like this are beautiful.

So grateful. The world is wide open to the self learner in this day and age.

We are very lucky.

Completely unrelated question but: how do you go about finding out that there is an opportunity to build a small business selling Microsoft teams and slack integration apps? I’m stuck in the mindset that software companies make billions of dollars or no money at all. I’ve not seen the right kind of indie hacker post that talks about how exactly people size their ideas and how much money is possible to make.

It was a great read. I bookmark this post hoping one day to buy a couple of the recommended books. It's just too hard for me right now to find the time. Life is too busy with a family, a full time job, and all the distractions around. Self learning is one of the most enjoyable things.

Practice makes a master. More importantly, the true understanding only comes through practice. Also, more often than not to "understand" something comes down to simply getting used to it (which is how we learn things in the elementary school). Practice is the key.

Fyi, for those who like working with lecture notes, much of the material for the math course at Oxford is online as well (no solutions, though):


Not sure if mentioned, another good book: Analysis by its History

http://www.goodtheorist.science/ this is a step by step for theoretical physics.

Does anyone know a good historical approach to maths? Like, start with Pythagoras?

Even something purely in the modern era, learning about Fourier and Weiner, harmonic analysis, etc.

The best historical treatment I've yet found is the Time-Life book on mathematics. https://www.amazon.com/Mathematics-David-Bergamini/dp/B0007G...

I know that Freeman Dyson attributes his proficiency in math to his love of math, which he claims was kindled as a teenager by reading Bell's "Men of Mathematics"

Buy a chalk board! There is no more enjoyable way to work through problems. This is especially true as the problems become more complex.

Is there a list that inclines towards abstract math that can be helpful to solve programming problems?

Thank you for such an inspiring piece

Aren’t applications like Brilliant actually really good to do this?

Not quite sure why everyone needs to learn so much math.

I am going to suggest something that might go against this idea of self-studying math.

Do not do it alone. I mean, it is okay to self-learn mathematics as much as possible but don't let that be the only way to learn. Find a self-study group where you can discuss what you are learning with others.

I think the social-effect can be profound in learning. I realized this when I used to learn calculus on my own. My progress was slow. But when I found a few other people who were also studying calculus, my knowledge and retention grew remarkably. I think the constant discussion and feedback-loop helps.

With round the clock internet connectivity, it is easier to find a self-study group now than ever.

It's not super clear to me how this actually works in practice. I've seen there is one public math meetup in SF, but the topic is usually different from the one I want to study.

I'm glad to see there are online options for groups like Stack Exchange or tighter group's like the one integerclub mentions, but I still seem to run into the same problem. For example, I'm not sure how to get a group of people that are interested in reading book X when I want to start it. If anyone has advice on that, please share.

Yep, this has been the story of my learning experience. I've studied mathematics pretty much entirely on my own, but it's not because I wouldn't love to have company!

Having said that, I think it probably would be sufficient to find _just one_ other person who is at the same level of mathematical maturity and has the same degree of commitment to change the entire learning experience for the better. You don't need a big group.

Agree. And if one is a well paid software engineer, one can definitely afford to pay a maths grad student for an hour week, preferably a bit more than whatever pittance the local univeristy pays them for being a tutor. You will progress far quicker and with fewer wrong turns. It is also far cheaper than enrolling at a university. A personal trainer for the brain.

I actually did this as an undergrad, despite barely being able to afford school. I left school for a bit, so I could figure out how to actually pay for it. After getting that worked out, I came back and realized I'd forgotten way more math than I had anticipated. Between my CS courses and math I was getting overwhelmed with the sheer breadth of information I needed to be have mastered to comfortably follow along. I took a one semester long remedial class that served as a refresher to all high school level math.

After that I worked through my classes with a tutors help. Everything up to and including linear algebra and numerical analysis with the help of a extremely kind PhD student named Adnan. He had the patience of a saint and ended up becoming a very good friend.

The most valuable part of having someone like this available for an hour or two every week is that it increases your knowledge or understanding/minute rate dramatically. It's like having Google or Khan academy on steroids. Someone that did everything already and knows exactly what page of a text book to look lat to help you understand, but they don't even need the textbook, because they know how to explain the concept you're having trouble with.

To this day I work as one of many data lscientists on a team where we all have fairly diverse backgrounds. In fact I'm the only person that is only CS and does not have a graduate degree. I have a teammate that did her undergrad and Masters in mathematics and if I'm having a hard time with something math heavy after some googling, the first thing I do is ask her for a quick explainer. She does the same with me for CS or programming issues as well and I help her with informal code reviews.

I know this will seem like basic teamwork to a lot of folks, but far too often I see people in our industry exert huge amounts of effort to understand a difficult concept that likely someone they're sitting a few feet away from has a very good understanding of and would be happy to help them with, so they're not banging their head against the wall for hours. I had to have a similar conversation with my intern a couple years ago. She was spending hours doing pen on paper math to understand Kalman filters. Things went much more quickly after I talked to her about my process of working with my colleagues and asking for help when I didn't understand something.

TL;DR Ask for help sooner rather than later. We all stand on the shoulders of giants.

Ha - I actually suggested the exact same thing before seeing your post. It's definitely much better to have a group. Since I've found this group I'm currently in, I've also been much more motivated, but also able to get feedback from more advanced people, and pare the problem numbers down to only the ones that are useful and will help me build concepts, limiting how many "calculation" problems I have to repeat.

Made a group announcement on the same topic on this thread: https://news.ycombinator.com/item?id=22401750

See the user profile of my account for more details.

In mathematical history, there is a lot of discredit to Indian mathematics and their contributions.

Number System, Algebra, Geometry, Trigonometry, Calculus, ...


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