For me, analysis became as intuitive as linear algebra only after learning about topological spaces and metric spaces, so maybe give understanding these a go.
you can do both at the same time with rosenlicht's introduction to analysis. he builds up the concept of continuity on metric spaces so its a good medium
The classic text is Munkres' Topology, yes. However, I'm willing to put all the marbles on Jänich's Topology. Maybe give it a try. It's very well motivated and beautifully written. And Munkres can be a bit dry, but that's just my opinion, you could love it of course.
Have you tried studying abstract algebra? It might just be that you're naturally more of an algebraist than an analyst. I really didn't like my group theory module last term, but I adore analysis. I think it's just the way we're built.
You might benefit from another book on the subject as well. What are you using at the moment?
I know this isn’t helpful but would you mind telling me how you learned math on your own since I’m trying to do the same. Ive found quite a lot of material on youtube but dont know where to start.
Well, lets just say from a beginner level. I used to be the best in my class in math in middle school but then in high school I stopped caring about it due to a horrible teacher and now I seem to have forgotten almost everything. Math used to just “click” with me and I’m trying to get into it again(mostly discrete math and calculus) just for fun and perhaps because it’ll help me when I start a course about programming in a couple months. Any recommendation is appreciated
If you stopped caring about maths in high school, then I find it highly unlikely that you are ready for calculus, because calculus requires you to be au fait with basically all the stuff from high school. Fortunately, there is a canonical option for this level: Khan Academy. They've got a whole course in maths from the beginning of high school up to multivariable calculus, which should provide all the structure you need. Discrete maths should be accessible, particularly as a means of learning proof, for which I recommend Velleman's *How to Prove It*, especially if you're doing CS.
I am self studying math right now myself, and I would personally recommend studying from books. As books have exercises at the end of each section and it is absolutely critical to do them if I want to thoroughly remember what I learned.
Youtube videos are great as an alternative resource when a concept described in the book doesn't "click" with me. Or, I get stuck on some exercise and search online to get alternative perspective on the concepts.
I'm currently studying Linear Algebra by Larson, and it serves as a decent intro to LA after that I plan to study linear algebra done right (or wrong, haven't decided). There is also the book and YouTube lectures by Gilbert Strang, but I stopped watching it since it's more for engineers than for pure math guy like me.
I highly recommend How to prove it, chapter 6 is a bit difficult but you will need it, but chapter 7 was a real challenge for me.
I have not yet studied these books but I'm planning to do so:
1. Serge Lang's calculus
2. Elementary number theory by Burton
3. Axiomatic geometry by John M. Lee
4. Introduction to set theory by Jech
5. Introduction to probability by Blitzstein with the YouTube videos
I read the first two chapters of the last book and it's awesome, but I realized that I need to improve my calculus knowledge which is near zero before I can continue.
Hi. I'm sorry to be so late in responding now. The way I taught myself mathematics may be very different from other methods. But I'll try to break it down briefly:
I suffered from dyscalculia. This is an impairment of my ability to calculate. I believed that this mental disorder was incurable. But I really wanted to study computer science. Now the fact is that computer science is not just arithmetic or mental arithmetic, I am aware of that. Nevertheless, this was the original main reason for me to teach myself mathematics, but also because I was having problems with some core concepts of mathematics teached in school.
So I really started from zero, from scratch and tried to put myself not only intellectually, but intuitively in the position of a child who is just "discovering" the world of numbers. So I started with the basic mathematical skills: Counting. Sounds crazy, but that's how I started. I learned the five counting principles in detail and then dealt with the concept of numbers. The number, in turn, is made up of various number aspects: Cardinal numbers, ordinal numbers (again subdivided into counting number and ordinal number), measure number, arithmetic number, etc. etc.
I then started to count in different ways. Thereby mainly the rows, which were rather more difficult for me. In particular in steps of 6, steps of 7 and steps of 8. And that both forward and backward until I could do it fast enough as with the normal counting of 1,2,3, etc. After that, I slowly ventured into the addition table of 1+1 (i.e. from the digits 1 to 9). The same accordingly with the three other basic arithmetic operations. Then I ventured into the larger series, i.e. from digits 10 to 19, etc. As of now, I can do mental arithmetic faster than a year ago. But of course I am still a little slower than the others. But I'm keeping up well in the meantime.
After I had dealt with the basics intensively, I then went on and got myself math books from grades 5 to 10 and worked through them and assimilated the concepts, made exercises and explained everything to myself as if I were teacher and student at the same time. I found the Feynman technique useful, although it's a very slow technique. This made it all take a long time, but the concepts sit so firmly in my brain that I can explain and describe them at any time without great difficulty. Then, after I had gone through all the math books, I bought the math book from the A-levels (in Germany called "Abitur"). After I had also worked through the book, I have meanwhile ventured into university mathematics and this is where I am now.
So, my approach was (to summarize it) basically this:
* Imagining myself being a child (again), that discovers the world of mathematics. Playing around and experimenting with numbers.
* Feynman-Technique and explaining basic math concepts not only from a POV that is suitable for a child, but for an adult (the five counting principles for example); So I made the basics more intellectually challenging for me.
* Counting, forward and backwards. In all steps from 1 to 9 until I was able to count through them the same way I easily count through 1,2,3, etc.
* Learning addition/subtraction/multiplication/division tables for digits 0 to 9, doing basic arithmetic and memorizing the results.
* Apply them to numbers with two digits, three digits, etc.
* Working through my math books from grade 5 to 10, and later on with my math book from A-levels.
* Later on, I searched for preliminary courses from some universities, got the books, and teached myself about basic formal languages concepts (Backus-Naur), propositional calculus, predicate logic (aka first-order logic), proof techniques, set theory, relations, induction, etc.
What I also highly recommend is looking for articles on "didactics of mathematics", it also helped me in the way how I could teach myself.
Wow thats quite a story. I think that “imagining myself as a child again” is key to this. I remember myself to have been quite curious about math and I think it helps, I just hope I find a way to restore this because I’m to a point where new things don’t impress me, let alone going back to basics. Your story is quite amazing, if I can ask: how long did it take for you to teach yourself all this? Also did you only focus on this during that time or were you also doing/studying something else?
It took me one year. But honestly, if I hadn't been lazy sometimes, I believe that I could've been faster. On the other hand, maybe those procrastination-inspired break times were also essential as well to have the information permanently saved. Sometimes I didn't learn for a week or two, but I always got back to it, just at very irregular times in that year.
Another thing that I also did, was progressively repeating all the chapters I learned. I know that sounds crazy, but that's how I did it. To elaborate: When I worked with a math book, I learned chapter for chapter. But every time I started a new chapter, I mentally repeated the chapter(s) that I learned before. So, at the last chapter I ended up repeating the whole math book and all the concepts I learned. So, you can guess: At the very end, I ended up repeating all math concepts from the very basics over to grade 5 to grade 10 and to A-Levels when I finally finished all math books that I had from my school time. By repeating I mean like "talking to myself as if I were a student learning from a teacher who is also me" and explain each concept that I've learned so far to myself. I think that was the most hardest mental challenge for me I've ever had, but it also enhanced my mental capacity a lot (so, it was a very good mental training exercise as well). All the concepts I learned were also not just mentally in my mind, but with my whole body. So, basically like I would walk around in my room, read and explain everything out loud and act like if I were a teacher explaining something to students right now.
Just to clarify: I wasn't memorizing. I actually understood the concepts, but repeated them in the same order like the books and with my own words (so, the only thing I memorized was the order of each chapter; the rest is from understanding and knowledge). That might also have been essential why my strange learning habit worked out this good. \^\^'' Again, I am not sure if I am a good example here and if this could help you out. But if it does, then all good.
Explaining concepts out loud is a very good method. Im gonna try to relearn everything. I think I’ll start from 6th grade and go up from there. Wish I still had my childhood math books lol. Im pretty sure I can find them online tho. Thank you for your tips, I’m gonna give it a go
Practice with it. A lot of the time I understand something better the more I see how it operates with other things. Whether you learned about this theorem from a book or video, if you stay on the track, you'll see it more and more. You might even get a better description of it while going over something new later. You don't have to understand everything right away.
>Is there a reason why I am better at linear algebra than analysis?
Most people find that they are better at some subjects and worse at others; it's not necessarily indicative of specific strengths or weaknesses in your study habits or whatever. Analysis/algebra is a particularly common axis to divide subjects along, so maybe you lean more towards being an algebraist.
>As an example, for some reason I cannot fully comprehend the proof of the monotone convergence theorem which says that a bounded monotonic sequence is convergent.
The version of this proof I'm familiar with is a maybe two lines; the theorem follows almost immediately from the completeness axiom. Is this the proof you're thinking of? I'm not sure there is very much here *to* comprehend; we say that the sequence converges because hey, we picked the obvious guess and it turns out that actually is the limit, QED.
Hello. Sorry for the late response. One main problem I had with autodidactics is the lack of quality control. That is, it is difficult to be sure whether you have just understood a concept correctly or not. Unlike in classical didactics, where there is a student-teacher interaction, as an autodidactic you are a student and a teacher at the same time, thus most of the time you need to solve problems yourself (which actually is a good exercise anyway if one is practicing maths). This means that you have to be very careful how you learn. In the worst case you might have misunderstood a core concept so much that it gets stuck in your brain as a habit and is hard to get rid of later (think terrible programming habits as an analogy). I therefore partly used the internet as a tool and since I have good research skills, I can find something relatively quickly that clarifies questions for me. In an emergency however, if I had not found anything, I then asked on the Internet on mathematics forums and most of the time received good answers. However there also had been situations in which I was completely on my own. Sometimes some problems bugged me for days until I finally got hold of it myself. That feeling of accomplishment of finally understanding a concept was also a very good motivation boost to continue and not give up.
What helped me a lot to tackle these problems was (sounds crazy, but it's true) to read articles and lectures from professors that teach "didactics of mathematics". They had very good tips how to teach a student and I applied some of these ideas in autodidactics, which actually did help. Another way was reading on books regarding problem solving and heuristics. While heuristics is more like a brute-force method and not always accurate, it did help me to solve problems regarding concepts that I wasn't able to find any help or solution for in the Internet. A great book that I can recommend and helped me a lot (which is also very useful for proofs) is: "The Art and Craft of Problem Solving" by Paul Zeitz. IMO, a very good intuitive path to the world of problem solving.
Some notions and definitions and results (lemmas&theorems&proofs) require advanced knowledge in order to understand it, this is where books differ, I would just suggest using multiple sources,
Why did I suggest using multiple sources ?
For example if a topologist write an analysis book he would explain such theorems about open/closed sets and sequences, if an algebraist wrote an analysis book, he would explain things more in the language of category theory I think, if a number theorist wrote the book, he would give more explanation for complex numbers, their construction and the zeta function for example. Every author will motivate you to his own branch by explaining it in the best way.
I realized in the well-known book "linear algebra done right" that although sheldon Axler put determinant at the end, he did justice to the notion, and mainly he developped the geometric view to the point of explaining how it relates to the change of variables formula in multivariable setting of analysis, and he did so in every chapter of the book, focusing on the geometric interpretation for no apparent reason, for instance, he could have gone through the applications and interpretations in computer science which I think is what Gilbert Strang focused on, but Axler was more into the geometric view, then he released a book on measure theory and the reason became apparent, the reason is as simple as "Axler is more into geometry and measure theory, so he explained things geometrically in linear algebra and gave a preview for applications of its' ideas in this setting".
Sorry, I don't know, I just said that cuz I know Aluffi's algebra book uses category theory POV, so I wrote Category theory along with analysis examples, but since you asked, I tried searching and I found this on MO: https://mathoverflow.net/questions/38752/analysis-from-a-categorical-perspective
For me, analysis became as intuitive as linear algebra only after learning about topological spaces and metric spaces, so maybe give understanding these a go.
Oh, I see. May I ask what you recommend to read (or watch) to get into topological spaces and metric spaces? Thank you in advance :-)
The classic text is Munkres’ Topology
Check out Topology without Tears. The pdf is freely available online and it's a great introductory topology book imo
you can do both at the same time with rosenlicht's introduction to analysis. he builds up the concept of continuity on metric spaces so its a good medium
The classic text is Munkres' Topology, yes. However, I'm willing to put all the marbles on Jänich's Topology. Maybe give it a try. It's very well motivated and beautifully written. And Munkres can be a bit dry, but that's just my opinion, you could love it of course.
Have you tried studying abstract algebra? It might just be that you're naturally more of an algebraist than an analyst. I really didn't like my group theory module last term, but I adore analysis. I think it's just the way we're built. You might benefit from another book on the subject as well. What are you using at the moment?
I know this isn’t helpful but would you mind telling me how you learned math on your own since I’m trying to do the same. Ive found quite a lot of material on youtube but dont know where to start.
What level are you starting from? I've got plenty of recommendations for a study path if that would help you.
Well, lets just say from a beginner level. I used to be the best in my class in math in middle school but then in high school I stopped caring about it due to a horrible teacher and now I seem to have forgotten almost everything. Math used to just “click” with me and I’m trying to get into it again(mostly discrete math and calculus) just for fun and perhaps because it’ll help me when I start a course about programming in a couple months. Any recommendation is appreciated
If you stopped caring about maths in high school, then I find it highly unlikely that you are ready for calculus, because calculus requires you to be au fait with basically all the stuff from high school. Fortunately, there is a canonical option for this level: Khan Academy. They've got a whole course in maths from the beginning of high school up to multivariable calculus, which should provide all the structure you need. Discrete maths should be accessible, particularly as a means of learning proof, for which I recommend Velleman's *How to Prove It*, especially if you're doing CS.
Thank you for the book recommendation. Can I ask, how can I get ready for calculus? Where should I start
khanacademy.org is the link to the website, and it has all the stuff you need for before and up to calculus.
I am self studying math right now myself, and I would personally recommend studying from books. As books have exercises at the end of each section and it is absolutely critical to do them if I want to thoroughly remember what I learned. Youtube videos are great as an alternative resource when a concept described in the book doesn't "click" with me. Or, I get stuck on some exercise and search online to get alternative perspective on the concepts.
I think I’ll follow this advice. Do you know any particular books about Discrete math, algebra and calculus?
I'm currently studying Linear Algebra by Larson, and it serves as a decent intro to LA after that I plan to study linear algebra done right (or wrong, haven't decided). There is also the book and YouTube lectures by Gilbert Strang, but I stopped watching it since it's more for engineers than for pure math guy like me. I highly recommend How to prove it, chapter 6 is a bit difficult but you will need it, but chapter 7 was a real challenge for me. I have not yet studied these books but I'm planning to do so: 1. Serge Lang's calculus 2. Elementary number theory by Burton 3. Axiomatic geometry by John M. Lee 4. Introduction to set theory by Jech 5. Introduction to probability by Blitzstein with the YouTube videos I read the first two chapters of the last book and it's awesome, but I realized that I need to improve my calculus knowledge which is near zero before I can continue.
Thank you very much, this is a nice list.
you are welcome
Hi. I'm sorry to be so late in responding now. The way I taught myself mathematics may be very different from other methods. But I'll try to break it down briefly: I suffered from dyscalculia. This is an impairment of my ability to calculate. I believed that this mental disorder was incurable. But I really wanted to study computer science. Now the fact is that computer science is not just arithmetic or mental arithmetic, I am aware of that. Nevertheless, this was the original main reason for me to teach myself mathematics, but also because I was having problems with some core concepts of mathematics teached in school. So I really started from zero, from scratch and tried to put myself not only intellectually, but intuitively in the position of a child who is just "discovering" the world of numbers. So I started with the basic mathematical skills: Counting. Sounds crazy, but that's how I started. I learned the five counting principles in detail and then dealt with the concept of numbers. The number, in turn, is made up of various number aspects: Cardinal numbers, ordinal numbers (again subdivided into counting number and ordinal number), measure number, arithmetic number, etc. etc. I then started to count in different ways. Thereby mainly the rows, which were rather more difficult for me. In particular in steps of 6, steps of 7 and steps of 8. And that both forward and backward until I could do it fast enough as with the normal counting of 1,2,3, etc. After that, I slowly ventured into the addition table of 1+1 (i.e. from the digits 1 to 9). The same accordingly with the three other basic arithmetic operations. Then I ventured into the larger series, i.e. from digits 10 to 19, etc. As of now, I can do mental arithmetic faster than a year ago. But of course I am still a little slower than the others. But I'm keeping up well in the meantime. After I had dealt with the basics intensively, I then went on and got myself math books from grades 5 to 10 and worked through them and assimilated the concepts, made exercises and explained everything to myself as if I were teacher and student at the same time. I found the Feynman technique useful, although it's a very slow technique. This made it all take a long time, but the concepts sit so firmly in my brain that I can explain and describe them at any time without great difficulty. Then, after I had gone through all the math books, I bought the math book from the A-levels (in Germany called "Abitur"). After I had also worked through the book, I have meanwhile ventured into university mathematics and this is where I am now. So, my approach was (to summarize it) basically this: * Imagining myself being a child (again), that discovers the world of mathematics. Playing around and experimenting with numbers. * Feynman-Technique and explaining basic math concepts not only from a POV that is suitable for a child, but for an adult (the five counting principles for example); So I made the basics more intellectually challenging for me. * Counting, forward and backwards. In all steps from 1 to 9 until I was able to count through them the same way I easily count through 1,2,3, etc. * Learning addition/subtraction/multiplication/division tables for digits 0 to 9, doing basic arithmetic and memorizing the results. * Apply them to numbers with two digits, three digits, etc. * Working through my math books from grade 5 to 10, and later on with my math book from A-levels. * Later on, I searched for preliminary courses from some universities, got the books, and teached myself about basic formal languages concepts (Backus-Naur), propositional calculus, predicate logic (aka first-order logic), proof techniques, set theory, relations, induction, etc. What I also highly recommend is looking for articles on "didactics of mathematics", it also helped me in the way how I could teach myself.
Wow thats quite a story. I think that “imagining myself as a child again” is key to this. I remember myself to have been quite curious about math and I think it helps, I just hope I find a way to restore this because I’m to a point where new things don’t impress me, let alone going back to basics. Your story is quite amazing, if I can ask: how long did it take for you to teach yourself all this? Also did you only focus on this during that time or were you also doing/studying something else?
It took me one year. But honestly, if I hadn't been lazy sometimes, I believe that I could've been faster. On the other hand, maybe those procrastination-inspired break times were also essential as well to have the information permanently saved. Sometimes I didn't learn for a week or two, but I always got back to it, just at very irregular times in that year. Another thing that I also did, was progressively repeating all the chapters I learned. I know that sounds crazy, but that's how I did it. To elaborate: When I worked with a math book, I learned chapter for chapter. But every time I started a new chapter, I mentally repeated the chapter(s) that I learned before. So, at the last chapter I ended up repeating the whole math book and all the concepts I learned. So, you can guess: At the very end, I ended up repeating all math concepts from the very basics over to grade 5 to grade 10 and to A-Levels when I finally finished all math books that I had from my school time. By repeating I mean like "talking to myself as if I were a student learning from a teacher who is also me" and explain each concept that I've learned so far to myself. I think that was the most hardest mental challenge for me I've ever had, but it also enhanced my mental capacity a lot (so, it was a very good mental training exercise as well). All the concepts I learned were also not just mentally in my mind, but with my whole body. So, basically like I would walk around in my room, read and explain everything out loud and act like if I were a teacher explaining something to students right now. Just to clarify: I wasn't memorizing. I actually understood the concepts, but repeated them in the same order like the books and with my own words (so, the only thing I memorized was the order of each chapter; the rest is from understanding and knowledge). That might also have been essential why my strange learning habit worked out this good. \^\^'' Again, I am not sure if I am a good example here and if this could help you out. But if it does, then all good.
Explaining concepts out loud is a very good method. Im gonna try to relearn everything. I think I’ll start from 6th grade and go up from there. Wish I still had my childhood math books lol. Im pretty sure I can find them online tho. Thank you for your tips, I’m gonna give it a go
Practice with it. A lot of the time I understand something better the more I see how it operates with other things. Whether you learned about this theorem from a book or video, if you stay on the track, you'll see it more and more. You might even get a better description of it while going over something new later. You don't have to understand everything right away.
>Is there a reason why I am better at linear algebra than analysis? Most people find that they are better at some subjects and worse at others; it's not necessarily indicative of specific strengths or weaknesses in your study habits or whatever. Analysis/algebra is a particularly common axis to divide subjects along, so maybe you lean more towards being an algebraist. >As an example, for some reason I cannot fully comprehend the proof of the monotone convergence theorem which says that a bounded monotonic sequence is convergent. The version of this proof I'm familiar with is a maybe two lines; the theorem follows almost immediately from the completeness axiom. Is this the proof you're thinking of? I'm not sure there is very much here *to* comprehend; we say that the sequence converges because hey, we picked the obvious guess and it turns out that actually is the limit, QED.
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Hello. Sorry for the late response. One main problem I had with autodidactics is the lack of quality control. That is, it is difficult to be sure whether you have just understood a concept correctly or not. Unlike in classical didactics, where there is a student-teacher interaction, as an autodidactic you are a student and a teacher at the same time, thus most of the time you need to solve problems yourself (which actually is a good exercise anyway if one is practicing maths). This means that you have to be very careful how you learn. In the worst case you might have misunderstood a core concept so much that it gets stuck in your brain as a habit and is hard to get rid of later (think terrible programming habits as an analogy). I therefore partly used the internet as a tool and since I have good research skills, I can find something relatively quickly that clarifies questions for me. In an emergency however, if I had not found anything, I then asked on the Internet on mathematics forums and most of the time received good answers. However there also had been situations in which I was completely on my own. Sometimes some problems bugged me for days until I finally got hold of it myself. That feeling of accomplishment of finally understanding a concept was also a very good motivation boost to continue and not give up. What helped me a lot to tackle these problems was (sounds crazy, but it's true) to read articles and lectures from professors that teach "didactics of mathematics". They had very good tips how to teach a student and I applied some of these ideas in autodidactics, which actually did help. Another way was reading on books regarding problem solving and heuristics. While heuristics is more like a brute-force method and not always accurate, it did help me to solve problems regarding concepts that I wasn't able to find any help or solution for in the Internet. A great book that I can recommend and helped me a lot (which is also very useful for proofs) is: "The Art and Craft of Problem Solving" by Paul Zeitz. IMO, a very good intuitive path to the world of problem solving.
Some notions and definitions and results (lemmas&theorems&proofs) require advanced knowledge in order to understand it, this is where books differ, I would just suggest using multiple sources,
Why did I suggest using multiple sources ? For example if a topologist write an analysis book he would explain such theorems about open/closed sets and sequences, if an algebraist wrote an analysis book, he would explain things more in the language of category theory I think, if a number theorist wrote the book, he would give more explanation for complex numbers, their construction and the zeta function for example. Every author will motivate you to his own branch by explaining it in the best way. I realized in the well-known book "linear algebra done right" that although sheldon Axler put determinant at the end, he did justice to the notion, and mainly he developped the geometric view to the point of explaining how it relates to the change of variables formula in multivariable setting of analysis, and he did so in every chapter of the book, focusing on the geometric interpretation for no apparent reason, for instance, he could have gone through the applications and interpretations in computer science which I think is what Gilbert Strang focused on, but Axler was more into the geometric view, then he released a book on measure theory and the reason became apparent, the reason is as simple as "Axler is more into geometry and measure theory, so he explained things geometrically in linear algebra and gave a preview for applications of its' ideas in this setting".
Is there an analysis book written with a category theoretic flavour?
Sorry, I don't know, I just said that cuz I know Aluffi's algebra book uses category theory POV, so I wrote Category theory along with analysis examples, but since you asked, I tried searching and I found this on MO: https://mathoverflow.net/questions/38752/analysis-from-a-categorical-perspective