The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is _about_.

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284
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21answers
15k views

On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole". For example, suppose you come across the novel term vector space, and want to learn more ...
226
votes
7answers
16k views

Best Sets of Lecture Notes and Articles [closed]

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
195
votes
28answers
27k views

Too old to start math [closed]

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
114
votes
9answers
9k views

What's the point in being a “skeptical” learner [closed]

I have a big problem: When I read any mathematical text I'm very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: "is the map well ...
81
votes
16answers
11k views

A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory? Background: In my current situation, I'm at an undergraduate institution where I have to take a lot of ...
78
votes
6answers
17k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
69
votes
4answers
5k views

Why learning modern algebraic geometry is so complicated?

Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the ...
68
votes
6answers
5k views

Why can't you pick socks using coin flips?

I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', ...
57
votes
13answers
5k views

How to stop forgetting proofs - for a first course in Real Analysis?

I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read ...
56
votes
4answers
7k views

Do you prove all theorems whilst studying?

When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
55
votes
9answers
9k views

How to effectively study math?

Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble ...
49
votes
12answers
5k views

I almost quit self-studying mathematics, but should I continue?

Before I move on to the main idea of this post, I need to tell you some background information about myself. Hopefully, it proves useful for you in giving me advice. I'm a 16 year old high school ...
48
votes
2answers
15k views

What is the proper way to study (more advanced) math?

Here's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I think ...
47
votes
15answers
10k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
46
votes
14answers
5k views

Nobody told me that self teaching could be so damaging…

Even though I've been teaching myself math for a couple of years now I only just started (a month ago) at the university. My experience is rather mixed. For starters, I'd like to mention that I'm 21 ...
45
votes
2answers
884 views

Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, "What happened?" I was ...
44
votes
25answers
3k views

What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?

I am struggling to pick out books when it comes to self studying math beyond Calculus. My situation is as follows. I have taken all math courses at my school (up to Calc BC and AP Stats) and I have ...
43
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most of ...
41
votes
6answers
5k views

What should an amateur do with a proof of an open problem?

Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for ...
40
votes
5answers
2k views

Self-learning mathematics - help needed!

First, I apologise for the nebulous nature of my title but I can't adequately explain myself concisely. I am about to start an MSc in pure maths after a fairly shaky undergraduate degree. I am very ...
39
votes
3answers
8k views

A path to truly understanding probability and statistics

I'm embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social ...
37
votes
9answers
4k views

Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
37
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1answer
1k views

Are difficult exercises good for beginners?

I'm self-studying Rudin's Real and Complex Analysis. I've finished the first two chapters so far, and I don't have any major problems understanding the definitions and theorems. I can prove the ...
35
votes
5answers
3k views

What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician ...
34
votes
10answers
3k views

Becoming Better at Math

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up. EDIT: Could anyone suggest books for someone with a math education that just ...
34
votes
9answers
3k views

How to self learn mathematics if you can't buy the books? [closed]

I'm trying to learn more mathematics, especially number theory and abstract algebra, with my goal being category theory, however, whenever I look for book suggestions here, most guys will link to ...
32
votes
11answers
34k views

How to start with mathematics? [closed]

I fell in love with mathematics a bit too late when I've already taken decisions regarding my future, career-wise. Now I would like to learn math on my own but I'm a bit confused as where to start. My ...
32
votes
8answers
14k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
31
votes
8answers
3k views

How to use math textbooks

I'm a higher schooler who was recently gifted a book by my teacher (Schaum's outline of advanced calculus) which is really awesome and I've started working my way through it. I have run into a ...
30
votes
13answers
14k views

Formerly good at math, but after 12 years I've lost most of my skills. Now I need them once again. Any advice to grow them back?

I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the ...
28
votes
8answers
3k views

Active learning vs Passive learning in Math

I am trying to improve how I learn in general but specifically in math and a common suggestion I keep coming across is the difference between active learning and passive learning. The problem is, most ...
28
votes
4answers
18k views

I want to get good at math, any good book suggestions? [closed]

I have never been good at math. In my senior year I failed algebra 2 in the first semester because the teacher would refuse to show me how to solve problems saying that it's my fault for not paying ...
28
votes
8answers
35k views

Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however,...
28
votes
3answers
5k views

Math Major: How to read textbooks in better style or method ? And how to select best books?

I'm a bachelor student of Math major. The question is about the better way to select and read maths text- or non-text- books, without considering the course and exam. (Because we follow the lecture ...
28
votes
4answers
2k views

Understanding mathematics imprecisely

For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down ...
28
votes
3answers
914 views

Do free online collaborative solution manuals exist?

I'm not a mathematician by training and a rarely come in contact with mathematicians. For this reason I find solution manuals to be incredibly useful - reading them allows me to see how experienced ...
27
votes
9answers
60k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
26
votes
8answers
4k views

Math every mathematician should know [closed]

This question is meant as a companion to previously asked questions like Proofs every mathematician should know. More and more I'm beginning to see that there is just too much math to learn. ...
26
votes
4answers
4k views

Pure mathematics curriculum for self study with interests in foundational issues [closed]

I wonder if I want to make my own pure mathematics curriculum to study along the next 4 or 5 years. What topics should I include? I want it to be like one which an undergraduate student of pure ...
26
votes
2answers
1k views

Efficient ways to read and learn a new topic [closed]

I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
23
votes
3answers
2k views

Choice of $q$ in Baby Rudin's Example 1.1

First, my apologies if this has already been asked/answered. I wasn't able to find this question via search. My question comes from Rudin's "Princicples of Mathematical Analysis," or "Baby Rudin," ...
23
votes
6answers
7k views

Self-Studying Abstract Algebra; Aye or Nay? [closed]

I am a high schooler with a deep interest in mathematics, which is why I have self-studied Linear Algebra and have begun my self-study in Differential Equations. As I am a man who likes to plan ahead, ...
21
votes
4answers
2k views

Number of subgroups of prime order

I've been doing some exercises from my introductory algebra text and came across a problem which I reduced to proving that: The number of distinct subgroups of prime order $p$ of a finite group $G$...
21
votes
2answers
2k views

What is the prerequisite knowledge for learning Godel's incompleteness theorem

I am very interested in learning the incompleteness theorem and its proof. But first I must know what things I need to learn first. My current knowledge consists of basic high school education and the ...
21
votes
4answers
3k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
21
votes
1answer
2k views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
21
votes
2answers
403 views

How can a beginner researcher or Ph.D. student efficiently and effectively learn new concepts while staying motivated?

I hope this question is appropriate for MSE. The situation is that someone is reading a book (e.g. a monograph), possibly helpful in his/her research, and the content is sufficiently extensive or ...
21
votes
1answer
731 views

Learning Math Efficienctly and Succeeding in Grad School

I'm currently a second year Ph.D. student studying pure math. I've recently come to the conclusion that I must be studying wrong. Actually, more to the point, I must be thinking about mathematics ...
20
votes
2answers
6k views

Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
20
votes
8answers
12k views

Best Math books or apps for adults to learn math from the beginning

I lost a possible job because I didn't know how to multiply and subtract negative valued integers. I also don't know how fraction manipulation works. What reference books can I read that can help for ...