Studying Share self-studying mathematics tips

[mnb96]

I found the original thread of micromass very useful, as it contains a lot of good advice for people willing to self studying math.
Personally, I would be very pleased to see a similar thread by micromass giving advice on how to choose a good textbook when self-studying math.

I believe that the typical default lists of supposedly "good" math books do not work for everyone.
From my own experience, whenever I want to learn a new subject, I always seem to find it difficult to identify a book that, for instance, suits my needs, contains just the level of formalism that I can easily digest, contains useful exercises, and things like that.

It is usually a personal choice, and it is not an easy one to do (at least for me).
General advice in this regard would be very very welcome.

 

[micromass]

I believe that the typical default lists of supposedly "good" math books do not work for everyone.

I know very well that they don't. Like you said, textbooks are a very personal thing. I will need to think on general guidelines on how to choose a textbook. Thank you for your suggestion!

 

[ohwilleke]

In terms of selecting textbooks, my approach is old school, but works reasonably well.

1. Go in person to your local college book store and narrow your choice to textbooks that a professor at some local college or university deemed good enough to assign to his students. This narrows your choices to typically 1-4 textbooks.
2. Exclude textbooks that don't have a significant answer set at the back.
3. Take serious time (maybe 30 to 45 minutes) examining the choices in detail and imagining yourself trying to understand concepts and do problems in a fairly early part of the book. Then choose one and don't look back.

 

[micromass]

2. Exclude textbooks that don't have a significant answer set at the back.

That eliminates some pretty terrific textbooks and won't work for many advanced subjects.

 

[Loststudent22]

What are your opinions of textbooks that do not have the solutions to the problems? I wanted to read through Kiselev's Geometry book 1 and 2 this summer but the book does not contain solutions to any of the problems.

 

[waddlingnarwhal]

What are your opinions of textbooks that do not have the solutions to the problems? I wanted to read through Kiselev's Geometry book 1 and 2 this summer but the book does not contain solutions to any of the problems.

I've had success teaching myself things from books without solutions. The process of figuring out if my response is correct is very instructive at times. If you're just studying the book by yourself, then you can ask online for help if you get stuck on a problem. There are an extensive number of sample pages available from the publisher (http://www.sumizdat.org/geom1.html). Maybe you can work through the first chapter and see if the book is right for you?

In terms of selecting textbooks, my approach is old school, but works reasonably well.

1. Go in person to your local college book store and narrow your choice to textbooks that a professor at some local college or university deemed good enough to assign to his students. This narrows your choices to typically 1-4 textbooks.
2. Exclude textbooks that don't have a significant answer set at the back.
3. Take serious time (maybe 30 to 45 minutes) examining the choices in detail and imagining yourself trying to understand concepts and do problems in a fairly early part of the book. Then choose one and don't look back.

I think this last piece of advice is well-chosen and very important. Don't spend too much time worrying if you have the "right" book. Just start working on math, and ask for help as you go along.

 

[Abtinnn]

I self-study mathematics and physics A LOT
I think one thing that can be dangerous with self-studying is that you read through the book, read about a concept, become confident with it but not deeply understand it.
I mean you think you get it, and you do some problems to reassure yourself that you get it. But then you come across some complex problem and you spend so much time trying to solve it, but you're unsuccessful because you haven't understood the concept in a right way.

And it is hard to "re-understand" a concept that you've learned wrong.

But I guess it happens in a regular classroom too...

 

[micromass]

What are your opinions of textbooks that do not have the solutions to the problems? I wanted to read through Kiselev's Geometry book 1 and 2 this summer but the book does not contain solutions to any of the problems.

Kiselev is an excellent choice. I suggest you definitely go with it! You can always ask here on the forum if you're not certain.

 

[IGU]

1. Go in person to your local college book store and narrow your choice to textbooks that a professor at some local college or university deemed good enough to assign to his students. This narrows your choices to typically 1-4 textbooks.
2. Exclude textbooks that don't have a significant answer set at the back.
3. Take serious time (maybe 30 to 45 minutes) examining the choices in detail and imagining yourself trying to understand concepts and do problems in a fairly early part of the book. Then choose one and don't look back.

I think your advice is a good illustration of how different things work for different people. It may work for you, but as a general approach I disagree strongly with all of it.

1. Books suitable for a classroom are not necessarily suitable for self-teaching, so a college book store is probably useless. Depending on the college their choices are not necessarily based on quality, but are just as likely to be a result of state rules and politics.
2. Answer sets are counterproductive for self-teaching. Rather look for books that don't have them, as peeking is too much of a temptation for most people. For self-teaching you want to work good problems on your own, taking what time is necessary. If it becomes important to ask for help, the bar to doing to should be high. You should, rather than look at answers, ask for hints at sites like this one or stackexchange. People are happy to help.
3. You don't want to take serious time looking at the books until you've narrowed your choices down considerably. The way you do that is to look at web sites like this one and stackexchange for people's advice. Read reviews on Amazon. Learn what books work for you, try to understand why, and look for other books with similar sounding reviews, and that are liked by the same people. Often you can count on a particular author you like to produce multiple books that work for you.

I disagree most strongly with your last bit of advice about not looking back. Rather you should regularly review whether you have made a good choice. Are you learning the material to your satisfaction? Does the author convey concepts in a way that works for you? Are there a sufficient number of good problems to work? Is your learning efficient? Can you get help when you need it?

One last thing is that there are more and more free resources available online. Some of them are excellent and go way beyond being just books. While these are new and have not withstood the test of time, they merit examination. Textbooks and lectures are an ancient learning modality, and there's no particular reason to believe they are the best way for any particular person to learn. Be a bit adventurous. The real goal is not to find the right textbook, but to learn the material in a way that works for you.

 

[heatengine516]

This summer I plan on self studying Godsil's Algebraic Graph Theory text.

 

[Samit]

Yes, Rudin is a difficult book. It's not really suitable for self-study because of these things. It's better for a class textbook so the professor can give some extra explanations. But you can of course always ask here if you have a problem with anything.

Can you pls. suggest a Simple book for Complex Analysis to start with while self-studying ?

 

[micromass]

What real analysis do you know?

 

[IDValour]

Here's a question that I feel has a valuable answer, though it may too be a lengthy one. Which well-known, widely used, or even well-liked textbooks should be avoided for those pursuing self-study? I feel, for example, that someone looking into real analysis may hear a lot of Baby Rudin, whereas this is no necessarily the best-choice for a beginner electing to self-study the topic. Which other books do you feel fall under this classification?

 

[micromass]

Here's a question that I feel has a valuable answer, though it may too be a lengthy one. Which well-known, widely used, or even well-liked textbooks should be avoided for those pursuing self-study? I feel, for example, that someone looking into real analysis may hear a lot of Baby Rudin, whereas this is no necessarily the best-choice for a beginner electing to self-study the topic. Which other books do you feel fall under this classification?

You are correct, this is a very important question. But there are so many bad books out there that should be avoided. I guess we can focus on the famous books. But there's the problem that books are really personal. So I don't feel comfortable saying a book is bad when some people really tend to like them. For some reason, I feel more comfortable recommending certain books though.

 

[IDValour]

Hm I don't really mean to say that books such as Baby Rudin are bad exactly, rather that they just seem inappropriate for self-study. I feel someone would benefit much more from Baby Rudin, should it be there first exposure, if they also had an instructor to go through it with them.

 

[micromass]

Hm I don't really mean to say that books such as Baby Rudin are bad exactly, rather that they just seem inappropriate for self-study. I feel someone would benefit much more from Baby Rudin, should it be there first exposure, if they also had an instructor to go through it with them.

Right. But I go further than you. I say that Baby Rudin is a bad book. I don't get why it is so popular. But I realize that I'm a minority here.

 

[IDValour]

Ah, I more meant with respect to what books are inappropriate for self-study. You mentioned you were uncomfortable with categorising books as bad, but I was not suggesting that you do that, but rather that you simply relate to us which ones you would advise against using for self-study. Apologies if this wasn't clear from my message.

 

[micromass]

Ah, I more meant with respect to what books are inappropriate for self-study. You mentioned you were uncomfortable with categorising books as bad, but I was not suggesting that you do that, but rather that you simply relate to us which ones you would advise against using for self-study. Apologies if this wasn't clear from my message.

OK, I'll see if I can do this. But in the meanwhile you can always ask whether a book is good or bad. Or you can follow my recommendations in the main thread (that are of course still incomplete).

 

[IDValour]

May I ask what you think is the best real analysis book for someone who has covered Spivak's Calculus to learn the topic in depth then? Also do you have an opinion on the numerous Olympiad style books by authors such as Andreescu and Zeitz?

 

[micromass]

Oh, but I have many favorite real analysis books. It depends on what you want.

First of all, Spivak is a very comprehensive book. I would already call it a real analysis book. So you'll likely won't need some "intros to real analysis" anymore. If you do want them, then here are some of my favorites:

1) Apostol "Mathematical analysis"
OK, this is a very dry book. And it's not fun to read. But it does contain a lot of very nice results and theorems. It is my go-to book when I want to review something basic in analysis. It has very good (difficult!) problems too.

2) Bloch "Real numbers and real analysis"
This is a lovely book. It proves everything. And then I really mean everything. It starts from just accepting the natural numbers axiomatically (and set theoretic notions) and then building the integers, rationals and reals. It even proves rigorously that the decimal notation works (not a nice proof though). And then it develops the notion of an "area" and proves that the integral really does measure the area". If you're in need for a book that derives everything carefully from axioms, then this is the book for you. Not easy though.

3) Tao's analysis
This is filled with intuition. A book from a great mathematician and it shows.

But you can also immediately start doing the fun stuff.

4) Carothers' "Real analysis"
This is my favorite math book of all time. You really can't find a better real analysis book than this. It is so immensely well-written. It does require you to know some real analysis already, but I guess that Spivak is enough for this.

Neither of the books I listen is an introduction to real analysis, I think that all of them assume (or should assume) some familiarity to real analysis already, but I think Spivak provides that adequately.

 

[IDValour]

Sorry if I'm being a bother but I was wondering if you could perhaps make me some further recommendations, tailored to my situation. I will hopefully be attending Cambridge University to study for a Maths Degree in the not too distant future - I have a list of the current 1st year courses here: http://www.maths.cam.ac.uk/undergrad/course/text.pdf and would very much appreciate it if you could suggest some texts for each of the courses (preferably ones that might extend somewhat beyond the scope given there even). If it helps any further you can access the examination papers here: http://www.maths.cam.ac.uk/undergrad/pastpapers/2014/ia/List_IA.pdf . A final resource might be the lecture notes available here: http://www.maths.cam.ac.uk/studentreps/res/notes.html. Thank you for the time taken to help me so far :)

 

[Abtinnn]

What is a good book on non-euclidean geometry?

 

[Loststudent22]

What are your thoughts on watching lecture series for self learning?

I came across a complete lecture series of linear algebra from princeton by Adrian Banner(who I actually supplement my calculus study with his book calculus lifesaver)

https://www.youtube.com/playlist?list=PLGqzsq0erqU7w7ZrTZ-pWWk4-AOkiGEGp

Do you think I would get enough just watching the video series and working the examples he does by pausing the video, or would a textbook be required also? Obviously a textbook would be better but the luxury of just watching a video a day would be quite nice also and less time consuming and not cost money. Plus the textbook they use has poor reviews.

 

[Loststudent22]

What is a good book on non-euclidean geometry?

Marvin Jay Greenberg textbook I have seen recommended and its on my list also as a book to read.

 

[micromass]

What are your thoughts on watching lecture series for self learning?

I came across a complete lecture series of linear algebra from princeton by Adrian Banner(who I actually supplement my calculus study with his book calculus lifesaver)

https://www.youtube.com/playlist?list=PLGqzsq0erqU7w7ZrTZ-pWWk4-AOkiGEGp

Do you think I would get enough just watching the video series and working the examples he does by pausing the video, or would a textbook be required also? Obviously a textbook would be better but the luxury of just watching a video a day would be quite nice also and less time consuming and not cost money. Plus the textbook they use has poor reviews.

My thoughts on video lectures can be found now in the main thread: https://www.physicsforums.com/threads/how-to-self-study-mathematics.804404/ I think it is the answer you want. Besides, there are also textbooks for free. For linear algebra, the best textbook is a free one: http://www.math.brown.edu/~treil/papers/LADW/LADW.html

 

[micromass]

Marvin Jay Greenberg textbook I have seen recommended and its on my list also as a book to read.

Yes, Greenberg is an excellent book. But there are very little proofs in the book. If anything, the proofs are done in the exercises. I would recommend to supplement Greenberg with Moise: https://www.amazon.com/Elementary-Geometry-Advanced-Standpoint-Edition&tag=pfamazon01-20 This is the exact opposite of Greenberg: many rigorous proofs, but not much historical and philosophical discussions.

For an easier book, I recommend Kiselev.

 

[micromass]

Sorry if I'm being a bother but I was wondering if you could perhaps make me some further recommendations, tailored to my situation. I will hopefully be attending Cambridge University to study for a Maths Degree in the not too distant future - I have a list of the current 1st year courses here: http://www.maths.cam.ac.uk/undergrad/course/text.pdf and would very much appreciate it if you could suggest some texts for each of the courses (preferably ones that might extend somewhat beyond the scope given there even). If it helps any further you can access the examination papers here: http://www.maths.cam.ac.uk/undergrad/pastpapers/2014/ia/List_IA.pdf. A final resource might be the lecture notes available here: http://www.maths.cam.ac.uk/studentreps/res/notes.html. Thank you for the time taken to help me so far :)

For vectors and matrices, I recommend linear algebra done wrong: http://www.math.brown.edu/~treil/papers/LADW/LADW.html It's an excellent resource and completely free. It contains about everything you need to know of linear algebra.

For groups, I recommend Anderson and Feil: https://www.amazon.com/dp/1584885157/?tag=pfamazon01-20 It is very suitable for self-study in my opinion. Another nice (and easier and cheaper) alternative is Pinter: https://www.amazon.com/dp/1584885157/?tag=pfamazon01-20

For differential equations, I recommend Ross: https://www.amazon.com/dp/0471032948/?tag=pfamazon01-20 Probably the best introduction to differential equations out there. It has both analytic solutions, approximation methods and theoretical results.

For probability, I absolutely adore the follow site: http://www.math.uah.edu/stat/ It has many quality information on probability, WITH applets. I think applets are absolutely essential to understanding probability: it's one thing to know the theoretical result, another to see it happening in practice! If anybody is interested, I have compiled all the information on the site in a LaTeX book. It's over 2000 pages long. A more traditional book would be Feller: https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20 But that can be pricy

For vector calculus, I recommend Hubbard: https://www.amazon.com/dp/0136574467/?tag=pfamazon01-20 I'm not really satisfied about this book, but it's the best one I've seen yet. For a more theoretical (and difficult) approach, you can check the second volume of the excellent analysis books of Zorich: https://www.amazon.com/dp/3540462317/?tag=pfamazon01-20

 

[IDValour]

Thank you so much for this, I really appreciate the time you've taken to help me. I'd certainly be interested in that LaTeX book if you're willing to share it, perhaps you could attach/link it when you update your other thread with more textbook recommendations?

 

[micromass]

Thank you so much for this, I really appreciate the time you've taken to help me. I'd certainly be interested in that LaTeX book if you're willing to share it, perhaps you could attach/link it when you update your other thread with more textbook recommendations?

Yes, but it's not finished yet. I have to reread everything (2000 pages) to make it flow more nicely. If anybody needs it quickly, then I can upload a preliminary version of course. Otherwise, I will just upload it to my thread when I'm done with it.

 

[IDValour]

Ah in that case please do take your time! I don't have an urgent need for it and am more than willing to wait for the finished product! :)