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?
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.
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.
> 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 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.
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.
* 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.
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 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.
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.
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!
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.
> 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.
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?
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.
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.
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.
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)
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.
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 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.
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?
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 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.
[+] [-] gavinray|6 years ago|reply
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)
[+] [-] sn9|6 years ago|reply
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
[+] [-] dhimes|6 years ago|reply
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.
[+] [-] pflats|6 years ago|reply
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.
[+] [-] bemmu|6 years ago|reply
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.
[+] [-] br_hue|6 years ago|reply
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.
[+] [-] jostylr|6 years ago|reply
* 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.
[+] [-] vector_spaces|6 years ago|reply
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/
[+] [-] ssivark|6 years ago|reply
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.
[+] [-] raidicy|6 years ago|reply
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...
[+] [-] injb|6 years ago|reply
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).
[+] [-] Proziam|6 years ago|reply
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.
[+] [-] daniel_reetz|6 years ago|reply
[+] [-] look_lookatme|6 years ago|reply
https://minireference.com/
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.
[+] [-] photon_lines|6 years ago|reply
https://github.com/photonlines/Intuitive-Overview-of-Linear-...
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!
[+] [-] dougabug|6 years ago|reply
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.
[+] [-] giornogiovanna|6 years ago|reply
• 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.
[+] [-] sriram_malhar|6 years ago|reply
"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.
[+] [-] rectang|6 years ago|reply
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.
[+] [-] nikofeyn|6 years ago|reply
https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...
https://www.amazon.com/Sequences-Combinations-Limits-Dover-M...
https://www.amazon.com/Geometry-Israel-M-Gelfand/dp/10716029...
https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/081763677...
https://www.amazon.com/Trigonometry-I-M-Gelfand/dp/081763914...
https://www.amazon.com/Method-Coordinates-Dover-Books-Mathem...
the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.
https://www.amazon.com/Street-Fighting-Mathematics-Educated-...
https://www.amazon.com/Art-Insight-Science-Engineering-Compl...
[+] [-] happy-go-lucky|6 years ago|reply
> 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.
https://github.com/srigalibe/NCERT_India_Grade_Mathematics
[+] [-] FigmentEngine|6 years ago|reply
i guess its idiomatic if you know "Greek".. ∑ is sigma, greek letter S, so Summation. And Π is pi, for Product..
[+] [-] laichzeit0|6 years ago|reply
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.
[+] [-] nsainsbury|6 years ago|reply
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.
[+] [-] p1esk|6 years ago|reply
[+] [-] melling|6 years ago|reply
https://news.ycombinator.com/item?id=19811715
https://news.ycombinator.com/item?id=20446796
https://news.ycombinator.com/item?id=11469494
[+] [-] deostroll|6 years ago|reply
[+] [-] hackernews7643|6 years ago|reply
[+] [-] daxfohl|6 years ago|reply
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.
[+] [-] integerclub|6 years ago|reply
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.
[+] [-] dorchadas|6 years ago|reply
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.
[+] [-] wyqydsyq|6 years ago|reply
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)
[+] [-] zerubeus|6 years ago|reply
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?
[+] [-] tildedave|6 years ago|reply
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.
[+] [-] kevstev|6 years ago|reply
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.
[+] [-] bo1024|6 years ago|reply
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.
[+] [-] strls|6 years ago|reply
[+] [-] proc0|6 years ago|reply
https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
[+] [-] angry_octet|6 years ago|reply
[+] [-] injb|6 years ago|reply
[+] [-] tprice7|6 years ago|reply
[+] [-] jcurbo|6 years ago|reply
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.
[+] [-] peatfreak|6 years ago|reply
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.
[+] [-] dwrodri|6 years ago|reply
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.
[+] [-] thorn|6 years ago|reply
[+] [-] emmanueloga_|6 years ago|reply
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.
[+] [-] leto_ii|6 years ago|reply
[+] [-] strls|6 years ago|reply
[+] [-] manu_ss|6 years ago|reply
[+] [-] mikorym|6 years ago|reply
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.