From 3.5 billion Reddit comments

ProductGPT

Try the custom AI to help you find products that Reddit loves.

13 points

·
17th Apr 2018

Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.

If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.

While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the *why-didn't-I-think-of-that* trick that solves it.

1 point

·
15th Feb 2021

Ok then I suggest you add this to the next collection of books you read. https://www.amazon.com/dp/1493927116/ref=cm_sw_r_cp_apa_fabc_GVMMR707YDDAX1ZQZ9R7

If anything you misunderstood what you calc professor thougt you but you where never lied to about division by zero.

1 point

·
31st Mar 2018

Your question is pretty vague because studying "mathematics" could mean a lot of things. And yes, your observation is correct: "There are a lot of Mathematical problems which are extremely difficult". In fact, that's true for a lot of people as well. So I suggest that you choose a certain field and delve into that.

For proof based subjects, the most basic to start with is Real Analysis. I recommend Stephen Abbott's Understanding Analysis as it is a pretty well-explained book.

1 point

·
9th Feb 2016

Understanding Analysis by Stephen Abbott is my favorite introductory analysis text. I think it does a better job of motivating the things it teaches about than other texts.

It doesn't fit your other criterion, however. It does a lot of asking students to prove important results as exercises. For instance, the squeeze theorem is one of his first exercises on limits of sequences.

He does, however, prove most of the big results himself, and you can use his proofs of them as a model for how you should approach proving the other things he asks for.

51 points

·
6th May 2019

For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.

​

If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.

2 points

·
10th Feb 2021

Read this book if you haven't already. This is in my opinion the gold standard, and should close all the holes you feel you have in your game.

2 points

·
8th Oct 2020

I think you'll just bore yourself out of resolve if you go too far back. Going forward could be a better thing to do. I learned to prove from this book, which I fully recommend to CS students. After that, you can go on to learn real analysis with Abbott or Spivak, and linear algebra with Axler or Treil.

If you want to brush up your algebraic skills, you might want to look for and work through a book about inequalities, they're extremely useful in analysis, and, as far as I can tell, only taught to a really trivial level in the usual algebra classes. You might want to ask for some resources on this and, if you do, please tell me.

1 point

·
7th Jun 2022

I'm not a math major, just a hobbyist, but I've heard good things about Stephen Abbott's "Understanding Analysis" or David Bressoud's "A Radical Approach to Real Analysis" for real analysis.

If you want textbook recommendations, this video recommends stuff for a more pure math perspective, whereas this guy stresses his view that applications and motivated examples are the best way to learn a subject instead of theory and formalisms.

N.B.: I haven't looked at any of these textbooks yet, so I can't personally vouch for any recommendations, but they seem pretty legit.

1 point

·
5th May 2022

Yes, you should read through

this book. It is basically what's going to be in Calc 2. If you can't finish this book in a month you should just give up the accelerated course.

​

Thanks!

Wish you luck!

1 point

·
9th Apr 2022

>To know the answer to that question of whether allowing proofs in mathematics curriculums is too much for kids, we would have to look at psychological studied and papers in mathematics education.

Well, to me kids can handle some simple proofs. Generalising shouldn't be that difficult.

https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 is a very concise book about real analysis.

You can use a few of the proofs there as reference.

1 point

·
17th Mar 2022

ayoo, based alert 😳😳😳

1 point

·
31st Dec 2021

The more information you can provide, the more useful our replies can be. So, for example:

What's your previous experience with analysis and/or topology in particular?

What's your experience with rigorous, proof-based math more generally?

What's the syllabus for the class you're about to take? Most specifically, what textbook will the class use?

What background is expected of students for your class?

Is your class considered an undergraduate- or graduate-level class or sequence of classes?

How soon before your class begins?

I'm asking because what to recommend will very much turn on what you're preparing for.

If this is an introductory undergraduate class in analysis using a book like, say, <em>Understanding Analysis</em> by Stephen Abbott, then

*Principles of Mathematical Analysis*("baby Rudin") will be of relative to Abbott's text.Point-set topology might be more abstract than what you'll be doing, though metric space topology more narrowly could prove useful.

If you're comfortable with proofs, great! That gives more flexibility in what we could recommend. Otherwise, though, learning how to read, understand, formulate, and write proofs may be the most important skill to cultivate.

If the class is introductory, then that would suggest a

*completely*different set of recommendations than if it's graduate-level (e.g., covering material like measure theory and Lebesgue integration).The more time you have before classes begin, the more opportunity you'd have to do some work beforehand. Unless you're not taking this class for

*quite*a while, though, it'd be unrealistic to expect you could make much headway in advance on something like baby Rudin.

You get the idea: *worthwhile* recommendations by us are context-dependent.

As an aside, I also have to wonder: was I the only one who saw the document you linked on point-set topology and immediately did an <em>It's Always Sunny in Philadelphia</em> doubletake regarding the title's font? (The fonts don't *quite* match, but still!)

Anyway, I hope this clarifies matters, though perhaps only in the sense of what additional information you could provide us in order for us to give you more bespoke advice. Good luck!

1 point

·
28th Apr 2021

When learning how to prove things you might regard as valid some arguments that aren't. Systems like Coq don't let you do that. I think that's one of the reasons why this book worked well for me (the other one being that I found the problems in the book to be quite engaging).

To be fair, at the time I was actually interested in dependently typed languages rather than in pure maths, and I wasn't new to functional programming nor to figuring out how to make obscure software work on my computer, but still... I quite like that book.

Also, the logic of this book doesn't allow proofs by contradiction nor contrapositive ones, but I think you can pick that up working through the first chapter or two of an analysis book, like Abbott's.

Another caveat if you use the first book I mentioned is that you'll have to pick up your proof writing style from somewhere else since Coq proofs are quite unsightly.

1 point

·
9th Oct 2020

I recommend this Calculus Textbook by David Guichard. It is a short and concise textbook on Calculus, plus it is ** FREE**. You don't need to go through a thousand page doorstop like Stewart or other books like that to understand Calculus at that level.

I ** personally** find Stewart's text to be too colorful, verbose, and all over the place but if you're just aiming to pass your course, it is enough.

If you want more understanding, read Understanding Analysis by Stephen Abbott.

1 point

·
9th Jun 2017

This free pdf book should help you: Proof, Logic, and Conjecture - The Mathematician's Toolbox

It's really well written (I like it better than Velleman's How to Prove It.) After this you should go through something easier than Rudin, like Spivak Calculus. Then you can try a real analysis book, but try using Abbott or Pugh instead; I hear those books are much better than Rudin.

1 point

·
24th Jan 2017

I found 'Understanding Analysis' by Stephen Abbott ( https://www.amazon.co.uk/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ) to be super helpful/enlightening post Real Analysis insofar that it helped me build an intuition and understanding for some of the key ideas. Earlier today someone highly recommended this book as well: 'A Story of Real Analysis' http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ (download link on the right). I had a quick glance through it and it seems pretty good.

1 point

·
3rd Apr 2016

There's a lot of different material on this subject. In fact, I feel like I see more introductory books on this than any other subject.

Which book you should use depends on your level of familiarity with the material. But I would recommend an analysis book. If you've never used one before, this will be difficult for you, but it's worth it.

My favorite introductory analysis book is Understanding Analysis by Stephen Abbott.

There's a free book online by a guy named Trench.

The classical choice analysis book is Baby Rudin which is a good choice, in my opinion, if you have a little experience with analysis already or if you have a decent amount of mathematical maturity.

Edit: I just realized you need this stuff tomorrow. You're not going to get a book on Amazon in one night but it's possible to find copies of Rudin and Abbott online if you look hard enough.

$100 - $200

< $50

$50 - $100