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.

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.

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.

More like the most foolproof way.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

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.

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/

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.

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...

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).

Can you share the names of the textbooks?

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.

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.

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

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

> 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.

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 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.

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).

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.

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.

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.

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...

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!

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.

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.

• 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.

"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.

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.

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...

> 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

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

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

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.

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 :)

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.

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.

[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.

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).

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

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".

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

https://openstax.org/subjects/math

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

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

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.

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

https://m.youtube.com/user/professorleonard57/playlists

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