How a layperson in mathematics learn mathematics from basics to at least intermediate level?

Question

From few weeks I have been watching videos on Mathematics and came across the mystery of prime numbers. I am no mathematician, Even I almost know nothing about this field but I don't know I find Prime number fascinating. I have watched several other videos too like the Pascal's triangle and few more math unproven Theories and mysteries. As I was curious but having no knowledge of mathematics, I picked up my tablet and started playing with Pascal's triangle of prime numbers see here and few more . I tried creating Pascal's triangle with weird combinations see here 😅 I don't even know if it make sense or not but I was interested and wanted to do something so I just did what came to my mind. Further I written prime numbers in different series by adding, subtracting and also tried finding pattern in twin primes by multiplying and dividing numbers between the twin prime by 2 and 3 See here and Here. I know what I was doing is all nonsense but it made me curious about things. I always wanted to learn mathematics and science but because of the way it was presented to me in school, I got less interested as schools focuses more on marks and treat every student as same level of intelligence and learning pace. Now I wanted to learn but confused from where to start and how to start 🥺. Recently I have completed my bachelor's in computer application ( considered as computer science ) and will be doing my masters soon but how can I study mathematics side by side effectively? Computer application is a 3 year degree program available in india and only contain discrete mathematics in 2 sem that also of medium level, nothing too hard.

My level of mathematics knowledge at the moment is negligible. Apart from basic math calculation operations I suck at everything. As I am computer student I was able to develop good level of logical ability used for making computer programs.

Edit: I live in india and here things are bit different. After our 10th standard one need to choose which stream he wants to go and the options are Humanities, commerce and science. You must have guessed it by their name what stream each represent and as i wasn't good at math I opted for humanities and did not studied maths for 2 more years. In my bachelor's I had mathematics of similar to what science students must have studied during those 2 years. During my bachelor's, students who came from science stream didn't faced any problem in the mathematics as they already studied those topics but I barely cleared the subject as it was too much for me at once since I has not studied maths for 2 years.

Another point I want to mention. The only option I have right now is self study ( because of some reasons ). I am not worried about the time it will take. I really wanted to learn mathematics atleast enough that I understand school level mathematics and all the basic concepts of mathematics well. Aiming for Undergrad or masters level at the moment is not approachable and also my aim is not to learn it for the sake of attaining knowledge but to learn it for myself.

Answer 1

I'd love to answer this question because I come from a very similar situation. I didn't like school and never really put the effort into it. I even failed maths in my final exams. I got interested in it 2 years after school, learned at my own pace, started a math bachelor and am just writing my bachelor thesis in mathematics. So you can certainly go from 0 to 100 if you're passionate about it.

But first things first: Welcome to MSE! This is actually a great place to start, because reading posts you're interested in can give you a good idea as of how things work. And you can ask questions yourself when you're stuck! If you do ask them well (see FAQ's to know what's expected) you will almost always receive an answer.

One of the comments suggested that you should get a math prof at your institution to take you under their wings. If you want to pursue mathematics seriously, and have some previous knowledge from you CS course, that's a great idea. But as CS programs can, in my experience, range from very mathy to not at all, you might not be there yet. This is a great tip anyways because maybe you find someone that's just willing to help and answer question. Doesn't have to be a prof, some doctoral students, postdocs or even undergrads will be fine. Getting to know people that do math as well is important anyways, more on this later below.

Ask yourself how confident you are with basic mathematics. Did you have trouble in your math courses at uni, or was it easy? If you find that university mathematics is hard for you, khanacademy.org (it's a non-profit and for free) is a great place to start. I brushed up all of my basic math skills there. You can find out what's missing by taking tests and then follow their programs to fill the gaps. Khanacademy was really motivating for me because I felt like I could go at my pace and it's slightly gamified (you get points and badges for solving exercises/ watching videos/ reading texts), which I personally find very motivating. The kind of maths you learn there is not strictly necessary for pure maths, but I find it helps if you're very confident in working with formulas and knowing the basic concepts on highschool level. But keep in mind: It's not strictly necessary, so if you don't want to, you don't have to learn this high-school math and what Americans call calculus.

Now, if you are interested in (and ready for) more university-like pure mathematics there are two things you can do:

Firstly, look at different textbooks. I recommend two pure math textbooks: one on (linear) algebra and one on real analysis. If that's your cup of tea, maybe there are nice books on discrete mathematics for beginners. You should certainly try different books before you settle for one - maybe you have access to some through your university library. Do as many exercises as you can! It is important that you use math books that are made for mathematicians because the hardest (and most rewarding by far) thing is learning how to proof stuff. You can only learn this by a) looking at a lot of proofs and b) doing a lot of proofs. Textbooks for mathematicians provide ample opportunities for both. Make sure you understand most of the proofs that you read. Get used that it might take you hours to read a single page! It's no failure if you have to put a problem, or a passage, aside for now and come back to it days, weeks or months later. There are book lists online, so I'm not recommending anything here. If you are interested in number theory specifically, though, you might want to ask another question which introductory textbooks are extra nice for that. For Algebra, Bosch's book is great for people with an interest in number theory, but maybe a little bit too advanced (depending on your background).

Secondly, your university probably offers a math program and you should certainly check their lectures out. This is important not only for the lectures (where you can ask questions), but also for the collaberation with others. Studying math together is really nice and actually (for most people) crucial to medium and long term success. Talking about the problems and concepts is invaluable.

One last option: There are a lot of great books that are written for a wider audience and treat mathematical topics. So, if you don't want to learn all the basics first (linear algebra, algebra, real analysis), then you might just enjoy reading a few of these.

Answer 2

To complement Guenterino's answer, in this answer I'll focus on enrichment/entertainment type books, and not on textbooks designed for learning specific areas of math.

For below calculus review and enrichment, see the Gelfand books and the Russian (English translated) Little Mathematics Library books:

Algebra and Functions and Graphs and The Method of Coordinates and Trigonometry and books in the Little Mathematics Library series.

At a similar school level, but especially focused on enrichment topics, see:

Mathematics for the Million and A Long Way from Euclid and The Borders of Mathematics and Mathematics for the General Reader and Mathematics and the Imagination and anything by W. W. Sawyer.

For books at a bit higher level but still strong on enrichment, see:

What is Mathematics? and Mathematics: Its Content, Methods and Meaning and Famous Problems of Mathematics and The Enjoyment of Mathematics and The Nature and Growth of Modern Mathematics and Mathematics. The Man-Made Universe and The Mathematical Sciences. A Collection of Essays and Kvant Selecta: Algebra and Analysis I and Kvant Selecta: Algebra and Analysis II and Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis and Elementary Mathematics from an Advanced Standpoint. Geometry.

Incidentally, I've mostly focused on older classics because I suspect almost all suggestions you’ll get will be relatively recent books.