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How to Prove It: A Structured Approach, 2nd Edition?
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I'm 33 and I did not learn coding in school either.
<em>How to Prove It: A Structured Approach, 2nd Edition</em> by Daniel Velleman, is very good. He does reference computer coding, but no background in coding or proofing is required. Literally anyone can pick it up and get started.
I would recommend the book "How To Prove It".
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It helped me in my transition into proof based mathematics. It will teach common techniques used in proofs and provides a bunch of practice problems as well.
220 - do all your projects early and don't cheat. this class will teach you how to really "program". start the programs early and you will be fine.
250 - i recommend this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
this book is pretty good when combined with barrington's "teaching" style. this book doesn't have all the topics covered, but if you find yourself struggling to understand concepts i recommend ordering this book (or find a PDF online of it) and just work through every practice problem you can. answers for a bunch are in the back. i think this book does a better job explaining conceptually what everything means in the class.
good luck!
The common metaphor is like a sequence of dominoes.
If
(1) You line up all the dominoes so that each one would knock over the next one
and
(2) You knock over the first one
Then they will all fall.
The first one will fall because you knocked it over (2).
The second one will fall because you knocked the first one over (2), and because the first one will knock over the second one (1).
The third one will fall because you knocked over the first one (2), and because the first one will knock over the second one (1), and because the second one will knock over the third one (1).
Mathematically, if you want to prove some statement about every number, say greater than or equal to 0, then you have to prove it about 0 and about 1 and about 2, and so on.
So what you do is you prove that if a statement about each number is true then the statement about the next number is true (1)
And that the statement about zero is true (2).
Then you've proved the statement about all numbers.
You've proved it about 0 from (2)
And you've proved it about 1 because you've proved it about 0 from (2) and because you've proved that if you've proved it about 0 then you've proved it about 1 from (1).
And you've proved it about 2 because you've proved it about 0 from (2) and because you've proved that if you proved it about 0 then you've proved it about 1 from (1), and you've proved that if you proved it about 1 then you've proved it about 2 from (1),
and so on.
For a text, I recommend Velleman
I was literally in the same boat as you. Started at a community college, with the intent on going into engineering, and went into engineering having picked up math along the way. Ended up liking math a lot more than engineering, yet finished undergrad with both. Now, currently doing a PhD in pure math.
One book I’d recommend is How to Prove It. Accessible, yet definitely not easy. It really shows you what mathematics truly is. Definitely is a stepping stone into the world of mathematics, yet doesn’t require much background math.
Of course, no worries!
If you don’t mind suggestions, I highly recommend How to Prove It as a book to use when you want to learn proofs. I, along with many others, find this book to be very accessible (compared to other intro to proof writing books) and it also doesn’t rely on too much foundational math (e.g. you can probably read most of this without having taken calculus at all…from what I recall).
And yes, I agree that you shouldn’t rush things. Although don’t be too worried if you don’t understand 100% of the things you learn, especially the first time around. I think if you feel like you understand most things, you can move on to the next subject/topic and if you need, you can always come back to it later. Good luck and don’t forget to enjoy!
Get used to proof based mathematics. How to Prove It: A Structured Approach, by Daniel J. Velleman, would be a great start.
EDIT: Ok math that's useful for a STEM major, maybe forget about the proof based math unless you're considering mathematical physics. It's still a good book though.
You don't need linear algebra for a first class in calculus, but you will need it eventually if you want to move on to multivariable or differential equations.
Some ideas from linear algebra/calculus can be helpful in the other, but it's not necessary. You'll eventually see that a derivative (a key idea from calculus) is an example of a linear function (the center piece of linear algebra).
Proof based vs applicable comes down to your own goals. If you want to get deeper into math, you'll need to learn it with proofs. If all you want to do is something like physics, you might never need to see the proofs. A course with proofs would definitely be harder (especially since it's your first time), but you'd learn more.
That would count as algebra. Spivak essentially builds calculus from scratch, and you need significant amounts of regular high school algebra to do calculus. The first few chapters essentially go through proving all the algebra you'll need for the actual calculus. If you have a hard time with this, consider a book like this.
Most people do differential and integral calculus at the same time. I don't know much about any books besides Spivak and Apostol, the standard proof-based introductions.
I also tried to learn calculus through spivak and found it very difficult; I stopped at then 4th chapter and switched to an easier textbook. If it's your first time learning calculus choosing an easier and verbose text like Stewart may suite you better. It's important to remember Spivak's Calculus is more like a textbook on Analysis (the theory of calculus), which is what often comes junior or senior year for math majors/minors.
If you have already learned calculus I'd suggest the bookHow to Prove It which helps think of math in a more concrete way that can help with proofs, even though no calculus is presented. Also, remember that Spivak likely didn't intend for people to find his questions easy, so don't feel like you are unprepared if it takes a while to do a single question.
If you happen to have the UCLA edition of Friedberg's Linear Algebra (the one you'll likely use for 115A) already, there's a section at the end with an intro to proofs. This book is pretty popular at universities with a dedicated intro to proofs class, so it might be worth checking out; I read a bit of it before taking the upper divs. Hope that helps!
Advanced math is subjective. Discrete math is a lot of topics mixed together into one class. A little bit of logic, graph theory, set theory, number theory, modular arithmetic, combinatorics, introduction to proofs, algorithm analysis and some other stuff I might be missing. The only prerequisite for it is pre-calculus. The difficulty of the class is subjective some people find it hard and some people find it easy. If you can remember definitions and theorems and string them together to construct a proof you should be fine. How to prove it is recommended a lot as an intro to writing proofs.
Don't worry, a lot of people are struggling as noted above. The class average was 70% so it's very likely that half the class did worse than that. Also, don't forget 20% of the grade is from the online quizzes.
I was in a similar position when I took Math 347. Between Math 347 and CS 173 I found this book <em>How to Prove It: A Structured Approach</em> by Daniel J. Velleman. I'm not one to use hyperbole but the book is nothing short of phenomenal. The book might not cover specific topics but it does cover all of the basics of proof writing in detail and in way designed for novices. From basic logic to induction the book covers just about every technique required for an intro proof class. It also happens to be relatively inexpensive. You can check the reviews online if you don't believe me (or even ask your math major friends).
Oh man 2011 was probably the hardest MATA31 revision. Don't worry, about that midterm though, the course content is really different now, that was when CSC/MATA67 used to be merged with MATA31, so they did a lot more set theory/number theory in MATA31 than they do now. I doubt most people who took MATA31 (and did well) could even pass that midterm just because we don't learn that stuff in MATA31 anymore. If you're trying to get started on studying for MATA31 now, I actually recommend you don't learn MATA31 material. Instead, improve on your critical thinking skills which your high school has definitely not given you. "Find" a book called how to prove it and go through maybe the first two or so chapters which just introduce proofs, and start to build up your proof skills. Becoming comfortable with proofs will come in handy immensely for CSCA67, MATA37, and in a big chunk of MATA31.
I'm self-studying BR right now (working the exercises for Ch 6) with great success. As others have said, definitely make sure you are comfortable understanding and writing proofs because you will be doing a lot of that. If you're not there yet I recommend working through How to Prove It, which you are most definitely prepared for based on what you have already studied. If and when you do end up trying to tackle BR stop by /r/babyrudin or feel free to PM me if you have any questions, I'd be happy to help. I'm also happy to help with HTPI if you decide to go that route instead.
Yes, I would make sure you are really comfortable with proofs before tackling Rudin. If you're not that comfortable you might check out How to Prove It. If you think you are comfortable enough and want to try Rudin stop by /r/babyrudin. It's a group of us who are self-studying it. We're generally all at different points in the book but feel free to post any questions you might have once you get started.
If you are at all interested in pure math (which for the post part is math that only math majors will ever take) I recommend picking up a copy of How to Prove It. It is a great introduction to proofs, which is what "real" math is all about. You'll also learn about some basic set theory and other useful things that are the foundation of pure math. The best part is that it requires only a strong background in algebra so you could easily learn it alongside your normal calculus track. It's a great book and will change the way you think about math.
Well for one, if you haven't, you should talk to the disability services people at your school. At the very least, you can probably get extra time for exams.
Look up the stuff you're doing on Khan Academy. I was a math hater for 30 years before realizing that I might actually like it if I tried to learn it. One day of Khan Academy and I was hooked. I even quit my job to go back to school. If you don't like his style, just browse YouTube and find someone whose style you do like. Khan focuses a lot on intuition which is what you need to solve exam problems that are a little different than your homework.
If you have specific questions about how things work, you have reddit. Check the sidebar for links to a couple different math subreddits. There are answers to all of your why questions and very few of them are difficult to understand, although you may have to learn a few other things first. I still shit myself when I read a confusing definition or theorem in my math classes, but they're never out of reach. Usually there are like 2 sentences of knowledge missing in my brain and it almost feels silly once I figure out what they are.
What you're doing now probably won't impact your life outside of letting you finish your degree, but the fun barrier is breakable. You just need to fill in those knowledge gaps so that you have the necessary tools to solve all of those little puzzles that you're given. That means asking specific questions about specific topics.
If you actually want to like math, check out this book: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
That's what math is actually like and it's around your level.
If you want a thorough tutorial on proof techniques check out How to Prove It. In particular Chapter 3 goes through all the situations you might run into and how to deal with them. Also check out the chapter on induction.
It's been five years since I've taken that class so my advice would probably be useless. I looked up the textbook and it seems to have horrible reviews on amazon. I just wanted to add that if you decide that you hate the book or if you need something else to supplement the textbook "How to Prove It" is one of the best and is really cheap. Also, Dong is pretty cool professor and very detailed so his lectures should come in handy.
This book is usually suggested. My tip for getting into proofs is to use your intuition as much as possible to give ideas as to why something might be true. Coming up with a proof is then a process of translating your intuition into the language of the axioms and definitions you start with.
A nice book on proofs is How to Prove It. Maybe not entirely relevant for the number theory part of MATH 135, but it teaches all the basic techniques in a very intuitive way.
I highly recommend How to Prove It for a great introduction to "real", proof-based math, especially if you've never really had any exposure to it. I studied that book and then proceeded to study axiomatic set theory and then real analysis from baby Rudin. HTPI provided a great foundation for these more advanced topics such that I haven't had any problem studyhing them all on my own.
The first step is to become comfortable with proofs. It is extremely different than the type of math you likely did in calculus, linear, and diffeq. There is very little "carry out this set of steps until you have computed the answer". This is not what proofs are like, and it is not what mathematics is really about. I've heard this is a very good book for learning about proofs and proving: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
After that, you can begin studying specific topics. I would recommend starting with abstract algebra or analysis. To do this, just get the textbooks. There are lots of resources to answer the question "which textbook should I use" (see sidebar).
The analogy with dominoes is often used. Your base case is knocking over the first domino, and induction is the chain reaction that follows: the first domino knocks over the second, the second knocks over the third, and so on. You could say that proving the inductive step is analogous to showing that the dominoes are close enough together for one to knock over the next.
I usually think of it as some kind of abstract ladder - you show that it's possible to go from one rung to the next, so if you start off on the first rung you can go as far up the ladder as you like!
Velleman's How to Prove It is the classic introductory text to the art of proof construction.
You should check for an introductory proofs course. There are usually courses in geometry or discrete math (perhaps titled 'Mathematical Structures') which can prepare you for high level math. While DE or linear algebra are the obvious next step (and very fun courses), I think serious math students benefit from getting into proof courses as quickly as possible. You have all the necessary prerequisites.
This is the textbook for Math Structures at ASU (and a lot of other schools): http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
Edit: Grammar.
A little more advanced is Velleman's How to Prove It. It introduces the techniques for actually proving mathematical theorems (what mathematicians actually do), rather than just solving pre-established mathematical problems.
It's a fantastic book, and teaches one how to think about mathematics in a very different way. In a lot of cases your university-level maths courses will assume you know how to prove theorems without ever really teaching you how to do it. This book fills that gap very well.
Math takes a lot of time to polish out, and a lot of the fundamental stuff is spread out through the ages. So, all in all, original texts in math generally suck, newer resources have the advantage of hindsight to sharpen the teaching process and stimulate learning. This is all opposite of how it is in philosophy, music and humanities because the cultural context is an important aspect of the writing style and what is being said.
Having said that, Khan Academy is probably your best bet. But this will just help teach you calculus and diff eq, which are not very interesting at all. If you want to get a taste of real math you should try picking up this book, which is an introduction to how math really works.
Not to pile on, but as has been previously stated what you wrote is not a proof. I'm not going to focus on whether or not what you said is true or false because the larger problem is that it's not written as a proof structure-wise. By this I mean, proofs are written using logic. If you're really interested in proof writing and basic analysis I suggest this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1331568877&sr=1-1
- http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&qid=1329845873&sr=8-1
- Take 420 with Caginalp if at all possible.
- I got a C+ in 413 and I'm still here, an applied math major. Power through it.
Unless you're already comfortable with reading and, more importantly, writing proofs, it's probably best to start with something foundational. If, for example, you could tell me off the top of your head what an equivalence relation is then you might be OK, but in my experience they don't teach that kind of thing in basic calculus. Any book worth reading on topics like number theory or topology is going to presuppose this knowledge. On the plus side, Velleman's book will give you a working intuition for mathematical logic.
Not sure your point. Mendelson's text is mathematically rigorous & serves the purpose of OP's question . My point is OP's quesiton allows for a vague interpretation of book suggestions. A textbook that is an introduction to Calculus isn't the only book that OP can use as an exposure to proofs. This is also a good introductory proofs book.
Advice #2: Instead of reading a pop math book, buy and work through a beginning proofs book (this guy's pretty cheap, and I had a lot of fun with it.) It will help you get a feel of what real math is like beyond the easy stuff, and for the most part it should be accessible to a beginner.
Copying my answer from another post:
I was literally in the bottom 14th percentile in math ability when i was 12.
One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College
It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is extremely well organized. Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.
Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.
Also, there is a metric shitload of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)
I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.
Edit: other resources
Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through application of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.
This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions...)
Stuff that's more important than you think (if you're interested in higher math after your GED)
Trigonometric functions: very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.
Matrix algebra: Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.
Edit 2: Electric Boogaloo
Other good, cheap math textbooks
/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).
Search up Dover Mathematics on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math ~~Overflow~~ Exchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1,000 pages of Linear Algebra, Graph Theory, and Discrete Math text for $50. If you prefer paper to .pdf, this is probably a good route to go.
Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.
This is a great book. Totally worth your time. How to Prove It
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
- Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on MathStackexchange.
Good luck.
A lot of times with a course I was studying there was just something I didn't get and it took a while for it to click, then once it did the rest of the course was cake. I had this problem with exterior derivatives. The text we were using didn't introduce or motivate them well so I ended up in the profs office for an hour once and he helped motivate it and after that the course was no problem.
With proofs there's a big jump to get over and often times the profs don't motivate what you're doing really well. I found a good book on proof technique was helpful because it helped me understand what I was reading. It turns out for most undergrad math you can break a proof down into one of a few patterns and once you know the patterns when you're reading a text you know what to look for and recognize the flow of the proof.
This is a good text on the subject
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
“How to Prove it”. D. Velleman: Amazon US Link
Probably the best resource on the topic!
As a side recommendation these are all common books for picking up basic discrete math and proofs: Book of Proof How To Prove It Discrete Mathematics and Its Applications
Hammack is free online and also cheap in print, I worked through it on the side when I took the calculus sequence and really enjoyed it. I've never read Velleman but I've also only heard good things. Rosen is more discrete math in general and less emphasis on proofs. It's longer than the other two and probably harder but it covers a ton of useful stuff.
You could also probably look up pdfs of each online to see which ones you like if they interest you.
The first required one is Sets and Logic. Usually they use: How to Prove It: A Structured Approach, 2nd Edition https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_rE.eybZMYQ8NV If you want to take a peek before you decide to do math.
Check out this book: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
You can definitely handle it and it'll give you a feel for exactly what being a math major is like. It's a lot of fun too. This material is the basis for all higher math. The things you do in the future are just more specialized versions of this.
If you enjoy math because you like solving puzzles and figuring out how things work, you'll love real math. If you like the idea of constructing arguments, you'll love real math. If you enjoy math because you like applying things that you've memorized and having immediate answers to problems, you probably won't like real math.
The way real math is written and taught is really fantastic. It's nothing to be afraid of once you get the hang of it. Proofs have a fairly standard lingo that's written in an inclusive and relatively unintimidating way. "We'll do this", "Suppose that is true." The math itself can certainly be intimidating but the prose is very welcoming. That might not be as true for professional publications but it's definitely true for almost all learning materials. For learning new things, almost every piece of information comes with a label. Definition, Theorem, Proposition, Corollary, Lemma, Proof, Remark. It looks very robotic at first but it's done that way to help you and it makes it much easier to piece together knowledge. I'm at the point now where, if my professor doesn't give something a label, I assign it one myself.
The stuff you're learning now certainly comes in handy but it's more of a toolbox for doing math rather than math itself.
edit: This is the style guide I used when I was first learning to write proofs, in case anyone is curious where all of that nonsense came from: https://www.math.hmc.edu/~su/math131/good-math-writing.pdf
Just pick up an intro to proof book. Velleman's Book is well-regarded, or if you want something free you could read The Book of Proof.
When I took the course, these were some of the resources I found really helpful on top of the textbooks:
How to Prove it by Daniel J. Velleman. Used this book for additional digestible explanations on direct proofs and induction. Other topics are also in there that are covered in the course.
Discrete Mathematics An Open Introduction, 3rd edition by Oscar Levin has been thrown around here and there on this sub as a more digestible book on the topics in the class.
Discrete Math YouTube playlist by Kimberly Brehm has some A-1 content for almost every topic in the class, if not all. Used her videos a lot. I noticed she came out with a second playlist "Discrete Math II," which includes some things that are covered in 225.
CS 198-087 Introduction to Mathematical Thinking class at UC Berkley is a bit of an analog to CS225. While the topics covered aren't 1:1, I found the induction video and the combinations with repetition (i.e. stars and bars problems) super helpful.
> Here are the course descriptions, if it’s helpful:
These are helpful. Do you by any chance also know what textbooks have been assigned? Sharing those with us might help us at least as much as the lists of syllabus topics.
> What resources can I consult ahead of time?
Let me start by sharing something I'd already written before in a related subreddit about real analysis in particular, my attempt to answer the question "Why is Real Analysis so hard?":
>It can differ a lot depending on where you're taking classes, who the teacher is, what textbooks you're using, etc. But even taking those caveats into account, there are a few common (if not universal) traits that can make introductory analysis unique in the undergraduate curriculum: > >1. Analysis is (often) the first deep exploration of proofs at the undergraduate level. > >2. Analysis proofs are often of the form "in order to prove the assertion in this exercise, it would suffice to prove this preliminary step." (This itself is because so many assertions ultimately boil down to proving some inequality holds rather than establishing that equality holds.) However, especially before you develop intuition about it, it can lead an introductory student down a number of blind alleys before one finds a technique that's sufficient and true. > >3. Many of the most basic notions in analysis, like that of the limit, can require a bit of careful thought before one understands them in an intuitive way. The definition of limit, for example, involves lots of logical quantifiers, and keeping track of all the moving pieces can be tricky in a way that, say, counting subsets of a finite set is more straightforward. > >4. Introductory real analysis quite often explores how badly behaved a function can be, and such pathological functions are often unfamiliar and counterintuitive. For example: Thomae's function is continuous at precisely the irrational numbers; there are families of functions which are everywhere continuous but <em>nowhere</em> differentiable; there are a number of different notions for the convergence of sequences of functions. It can be enough to wrap one's mind around a notion of convergence of a sequence of functions, let alone to make distinctions between difference kinds of convergence of sequences of functions. > >4. A number of universities use analysis as a "weeder class", designed to be difficult in a way that ensures those who stick around really care about staying in the math department. > >Of course, knowing why analysis can be more challenging doesn't necessarily help one pass the class. But hang in there! To begin, explore all the resources available to help you through the class: go to every office hour you can, consider free tutoring that might be offered through the math department, and look into whether there are any study groups you might join—or start, for that matter. One thing I can guarantee you: you are not alone in your class in terms of struggling with the material.
So: what can you do to prepare? Here are a few recommendations:
- Understand that both real analysis and abstract algebra will be very proof-heavy courses.
The more familiar you are with reading, understanding, and writing rigorous proofs, the better. For most students following an American curriculum, their first introduction to proofs would be a class like discrete mathematics. Have you already taken that? Separately, how proof-based was your linear algebra class?
If your proof background is still limited so far, and if you're locked into taking analysis and algebra, then you'll want to become as fluent as possible in the structure of proofs themselves before worrying about how to prove results specific to those branches of mathematics. There are some standard book recommendations for improving your general skills in proofs - Velleman's <em>How to Prove It: A Structured Approach</em> - Pólya's <em>How to Solve It: A New Aspect of Mathematical Method</em> - Cummings' <em>Proofs: A Long-Form Mathematics Textbook</em>.
I'm not personally familiar enough with any of these to vouch for them myself, but I believe all have good reputations.
- In addition to being proof-heavy, these classes will likely presuppose familiarity with common mathematical notation, especially from logic and set theory.
For example, imagine you were given a statement like the following: - ∀S⊆R, ∃M∈R such that for all s∈S, s≤M.
Do you understand what this says? Can you understand, at least intuitively, whether it's true or false? And can you prove your answer is correct?
- Here on reddit, /u/AcademicOverAnalysis recently posted "The Real Analysis Survival Guide" to \/r/math, and you might find that useful.
From my own experience, his recommendation of persistence is especially apt. Many introductory analysis classes can include lots of material that seems like it won't be useful, so you may be inclined to skip over it. Resist any such inclination. As in math classes you've already taken, the ideas will build on each other. You don't want to skip over anything, then find yourself forced to play catch-up.
As a related matter, if you don't understand the importance or relevance of a particular idea, please ask someone! To give a concrete example, the axiomatic definition of the real numbers <strong>R</strong> may first seem like a digression. Soon, though, you'll appreciate that everything in calculus will basically follow as a corollary to the properties of R.
- Consistent with your course's academic integrity policies, take advantage of as many resources as possible.
This includes, but it not limited to, the following:
a. Instructor office hours, which can be absolutely invaluable
b. grader office hours
c. collaborating with fellow students in study groups
d. both your assigned textbooks and related textbooks, the latter of which you might have access to for free via a college or university library
e. online resources like Khan Academy and others (though it might be harder to find detailed guidance there for analysis and abstract algebra than for more introductory material)
f. online communities like reddit or Math Stack Exchange, so long as you do so in ways your class or school won't consider to be cheating
g. YouTube videos and channels, though for these subjects that might be limited to videos of course lectures
h. free or paid tutors
- You mentioned that this will be an online program, which presents its own unique challenges.
If you won't be on-campus for your class, that may also limit your access to other typical resources, from meeting in-person for study groups to access to the university's physical library to finding in-person tutoring. You'll have to lean more heavily on other available resources (like an instructor's Zoom-based office hours) and find alternatives to in-person results.
- Give some consideration to how you'll be sharing your work online.
Sharing documents online in general, and math documents in particular, can be complicated for logistical reasons alone. How can you share your work as easily as possible? If you'll be using either pen-and-paper-with-digital-pictures, or perhaps writing directly on a tablet, please be conscientious about your handwriting. It doesn't have to belong on /r/handwritingporn (SFW, despite the name), but it has to be legible.
You might have more options if you already have some familiarity with and fluency in the math typesetting program LaTeX, its parent TeX, or related siblings and cousins (XeTeX/XeLaTeX, ConTeXt, LuaTeX/LuaLaTeX, etc.). Using something in the *TeX ecosystem is made simpler by using free tools like Overleaf to draft, compile, and share your documents.
Caveat: I would think carefully about trying to learn LaTeX simultaneous with learning analysis and abstract algebra, since that might add too much to your workload to do all at once. That's especially true for your abstract algebra class, since it appears to have a highly geometric emphasis. (The reference manual for the standard LaTeX-native graphics package, Ti*k*Z and PGF, is over 1,300 pages long!)
Using a more general purpose markup language like Markdown might also work, especially since it's basically integrated into sites like reddit. But Markdown isn't nearly as powerful as LaTeX for math-specific content.
If you can provide more information about your background, especially regarding proof-centric classes, then I might be able to get even more detailed. But for now, I hope this gives you a starting point. Namely: (a) do as much as you can to prepare yourself for proofs in particular, (b) respect that these are likely to be challenging classes, so you will need to remain consistently diligent throughout, and (c) reach out for as many resources as you can, consistent with your class and university policy regarding academic integrity.
I expect these classes will be a lot of work, but you're going to learn so much from them. Good luck!
Struggling for a dream might be worth it, because a lot of the issues that are had with upper level mathematics is that it is hard to "practice" since proofs are a little harder to repeat over and over. I had a professor who compared learning proofs to the struggle that some kids have learning basic Algebra introducing variables.
If you love math and want to stick with it, you might try some self study over the summer learning proofs from a set theory basis: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
or another one I hear great reviews about: https://www.amazon.comsss/How-Solve-Mathematical-Princeton-Science/dp/069116407X/ref=sr_1_1?crid=R64KJDSPG8RM&keywords=how+to+solve+it&qid=1647905717&s=books&sprefix=how+to+solve+it%2Cstripbooks%2C81&sr=1-1
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At the end of the day, if you are really in this for a job, start thinking about how you would apply what you are learning to a job? Could you move in a more applied mathematics area (programming, data science, analytics, informatics, engineering, etc. ) if you goal is moving into a profession?
Book of Proof by Richard Hammackis available for free as a PDF; it is a great resource.
How to Prove It by Daniel Velleman is not free, but is another great resource.
How to Prove It was the book I learned from and I enjoyed it. It seems that that’s the general consensus for others as well.
I suggest starting with:
https://www.amazon.ca/How-Prove-Structured-Daniel-Velleman/dp/0521675995
This will remind you of the foundation of math and how predicate logic dictates syntax, and obviously how to prove things.
If you need to work through functions again, do this first, but then I would immediately jump into analysis and I suggest this great book:
https://www.springer.com/gp/book/9781461599906
After that, it's really just about picking the right textbooks. Obviously my advice is more tailored for pure maths.
In the mean time, pick up a textbook on proofs, and start learning about logic and reasoning. That will help you build the foundation to take upper level math classes, and also get a better taste of how higher studies of mathematics will be different from calculus.
Here is an example. https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_2?dchild=1&keywords=how+to+prove+it&qid=1603559367&sr=8-2
Check out How to Prove It: A Structured Approach by Daniel J. Velleman.
you can find all the information here: https://fas.calendar.utoronto.ca/section/Mathematics
At uoft you're required to take 2 majors or 1 specialist or 1 major and 2 minors. The math major is at a much lower level than the math specialist, and you would need to combine the math major with something else. If you take the math specialist, then you'd just be doing math. The math major is similar to math majors at most other schools. The only schools with anything similar to the math specialist is waterloo/mcgill/ubc.
I'm assuming you know Stewart pretty well. It may give you a good foundation, but the specialist stream courses (157, 257, etc) require a different type of thinking than the calculus that is taught in Stewart, so it might not be of any help at all. You're still in high school correct? You're lucky to have lots of time to learn. If you found Stewart interesting, you'll find "real" math even moreso, I recommend picking up "How to Prove It" by Daniel Velleman ( https://www.amazon.ca/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=asc_df_0521675995/?tag=googleshopc0c-20&linkCode=df0&hvadid=293014842916&hvpos=&hvnetw=g&hvrand=16646358249913893663&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9000827&hvtargid=pla-449384309962&psc=1 ) and going through the textbook for 157, Calculus by Spivak ( https://www.amazon.ca/Calculus-4th-Michael-Spivak/dp/0914098918 ). You can find pdf versions of these books online if you look hard enough.
Sadly, I can't think of a title in discrete math or introduction-to-proofs that I can recommend. A common recommendation for the latter category, which I haven't read myself but has a good reputation, is the following:
- <em>How to Prove It: A Structured Approach</em>, 2nd edition by Daniel J. Velleman.
Another book which has a good reputation is
- <em>How to Solve It: A New Aspect of Mathematical Method</em> by George Pólya.
The book even has its own Wikipedia article!
These, however, are both about proofs as their own technique. I wish I could provide a recommendation for books on discrete math, introduction to set theory, and the related topics I mentioned above. You might consider something like
(This title also has its own Wikipedia article, too.)
but I'd defer to others for recommendations on textbooks for these prerequisite concepts and principles useful to an analysis student.
>Also, when should I start my real analysis? Can i study it with the calculus or after completing calculus?
I'd consider taking real analysis after completing the introductory sequence in calculus, possibly including multivariable calculus, linear algebra, and an introduction to differential equations. I'd also wait until after you've had a good introduction to mathematical proofs, something most universities and colleges present in a class on discrete mathematics.
If you jump into analysis completing at least one-variable differential and integral calculus, as well as a class with a strong proof-based component, you're likely to find yourself in over your head.
First, most analysis classes assume the students are already familiar with ideas like convergent sequences, limits, continuity, differentiability, and integration. This is all presented again in a much more rigorous way, but it's typical for an analysis class to lean heavily on prior interaction with such topics.
Second, analysis is a heavily proof-based class, and learning how to read, understand, and write proofs is its own skill set. Trying to acquire fluency in proofs by taking an analysis class, despite no prior formal encounters with proofs, will make analysis considerably more challenging for you.
I hope this helps some. Good luck!
Logic and problem solving?
Time to work on the pure math muscle. Proofs tend to be pretty helpful with this. It brings rigour to your logic and reasoning.
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Some possible resources for introduction to proofs (each by itself is more or less enough. Just options):
A great book to "get it":
A book meant for 7th to 10th graders who are aiming to start USAMO, etc.:
https://www.amazon.com/Art-Problem-Solving-Vol-Basics/dp/0977304566
https://www.amazon.com/Art-Problem-Solving-Basics-Solutions/dp/0977304574
Make sure to also purchase the solutions manual.
Great book. Almost every book except Calculus on Art of Problem Solving is the king of mathematics for textbook learning. (there's better books for Calculus and beyond in the market)
Another option is (free):
http://www.people.vcu.edu/~rhammack/BookOfProof2/index.html
Or learn number theory instead. What better site than to use a site from Brown Univ with its free textbook:
http://www.math.brown.edu/~jhs/frint.html
Or you can learn from the standardized Discrete Math textbook (Rosen's). Very good but very frustrating. Really frustrating. I'm sure even more so without a teacher. And I'm saying this as someone who used to participate in math competitions when I was in middle/high school. Not an easy textbook but truly the defacto book for intro to Discrete Math. It's the Apostol's Calculus of Intro Discrete Math [though I would endorse Spivak over Apostol anyday. Apostol is dry for starters who haven't been exposed to really dry books].
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Other than those, once you get accustomed to such train of thought and you really want to train your mind, I do recommend looking at pure math courses. Many of those are like the dumbbells of logic and reasoning.
Generally, from my experience, there's a few different ways of thinking that can be helpful for learning (there's far more but just wanted to abstract out a few very different though process ones from my experience):
- the abstract way (Algebra)
- the classic rigor and tedius step by step way (Analysis)
- the what-is-going-on way (Topology)
- the this is fun typical puzzle question (Discrete/Number Theory)
Hmm...sorry but a lot of your post shows a lack of mathematical rigor and philosophical understanding of the terms you say. Not trying to offend you, but you really want to practice on proofs.
> Let me see if I understand you OP. You are asserting that by adopting a position where a positive claim (and BTW a claim that something does not exist or does not work is still a positive claim even though the claim involves a negative) must be justified and supported, such as the position of non-belief in the existence of Gods (for or against), or a person is innocent until proven guilty, "harms discourse and is dishonest"? Really?
Except, this is exactly what the burden of proof is? Any claim, positive or negative, must be proven. Yes, even unicorns existing. This has been discussed at length throughout math and philosophy so I don't see how you think (unless you're ignorant) otherwise. Atheist conflict the burden of proof as a legal tenant and one from an epistemological essence. Legal wise, this is more as "innocent until proven guilty" but in no way does that mean x person didn't do it.
Deeper discussion here: https://www.reddit.com/r/philosophy/comments/72o984/the_natural_world_is_all_there_is_as_far_as_we/
>Any claim that purports to be of knowledge has a burden of proof.
>
>Any claim that limits itself merely to belief does not have a burden of proof.
>
>It makes no difference if the claim is theistic (gnostic or agnostic) or naturalistic (strong or weak), nor does it make any difference if it's a claim that a particular thing exists or is true, or that a particular thing does not exist or is not true, or anything else really for that matter. If it's a claim that purports to be of knowledge, it has a burden of proof, and if it's merely a belief, it does not.
Your version of the burden of proof (taken from rational wiki) has no basis in math nor philosophy. Do not get information from rational wiki. Get a copy of many proofs based mathematical books and start from there by actually proving problems.
Again from stack: https://philosophy.stackexchange.com/questions/678/does-a-negative-claimant-have-a-burden-of-proof
>I would say that generally, the burden of proof falls on whomever is making a claim, regardless of the positive or negative nature of that claim. It's fairly easy to imagine how any positive claim could be rephrased so as to be a negative one, and it's difficult to imagine that this should reasonably remove the asserter's burden of proof.
>
>Now, the problem lies in the fact that it's often thought to be extremely difficult, if not actually impossible, to prove a negative. It's easy to imagine (in theory) how one would go about proving a positive statement, but things become much more difficult when your task is to prove the absence of something.
>
>But many philosophers and logicians actually disagree with the catchphrase "you can't prove a negative". Steven Hales argues that this is merely a principle of "folk logic", and that a fundamental law of logic, the law of non-contradiction, makes it relatively straightforward to prove a negative.
Any claim, false or positive requires to be proven. Whether I say for all natural numbers in set N there exists no element such that N^N <= N^2. Or I state the inverse "for all natural numbers in set N there exists an element such that N^N <= N^2. The burden of proof is on me.
> Or OP, would you just accept that the grobbuggereater exists because I give witness to this existence?
I truly wish my professors were as simple-minded....so many hours could have been saved by proving negative statements in Mathematics and theoretical computer science. However, yes. Philosophically speaking, to claim grobbugereater does not exist requires proof. Grobbugereater is an idea x, where the probability is x / |r| where r is the set of all ideas. as r tends to infinity the probability of grobbugereater existing tends to 0. Thusly, since grobbugereater has no epistemological evidence then we can conclude his probability of existing is infinitely small. This is how you prove grobbugereater does not exist.
One of your claims (presumably) is that induction is better than deduction. That somehow science is far better than math, philosophy, theism, or any other deductive method. Such a claim is metaphysical and cannot be proven via induction thusly a contradiction.
I find it odd, that so many people who use rational claims lack mathematical rigor. Honestly dilutes the topic into a mindless debate and petty insults. Here is a good read to strengthen your skills:
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
I'm an IT student. I did quite well in single variable calculus but did quite poorly in multivariable calculus. I've just had a course in abstract algebra, and now I'm having a course in complex analysis and another in real analysis. The course in algebra was my first encounter with mathematical rigor. I had most trouble with writing proofs (the reading part was okay). I hadn't had any reasonable training in proofs. I'm planning to buy this book when I become less busy.
Because you already have a university background in math, I don't recommend that you start from the bare basics. The basics are already covered in a rigorous manner in introductory books of algebra and analysis. I'm still an undergrad student, so take my advice with a grain of salt.
The course in real analysis that I'm having uses Rudin's "Principles of Mathematical Analysis" and it's a difficult read. I spend quite some time on some pages before I finally digest the material. But many things that are normally taken for granted and/or grasped intuitively are actually proved here. That is the reward.
Finally, I'd also like that you check out these lecture notes. They are really useful, especially for review.
Usually high school level geometry isn't so great. For example, students usually learn "two column proofs", which isn't really what proofs are about. A proof is an argument which is meant to demonstrate completely and fully that claim is true. It's not something that's meant to be systematic and rigid, but instead something which explains, from one human to another, why something is true.
It's hard to speak about this more abstractly, so I'm just going to give you an example. You may have heard that the square root of 2 is an irrational number, meaning that you can't write the square root of 2 as a fraction of integers. Here's the proof that fact.
Assume for contradiction the square root of 2 is rational, and let sqrt(2) = a/b, where a and b are integers, and b is not 0. We can also assume this fraction is in "simplest form", meaning that a and b share no common factors. This is ok to do, because if a and b had any common factors, we could simple cancel them out.
Now because sqrt(2) = a/b, it follows 2 = a^2 / b^2, and thus a^2 = 2b^2. This means that a^2 is even. It turns out this means a is even, because the square of an odd number is always odd, and the square of an even number is always even (I encourage you to check this for yourself! Remember, a number is even if you can write it as 2n for some integer n, and a number is odd if you can write it as 2n+1 for some integer n).
So because a is even, we let a = 2n. Then because a^2 = 2b^2, substituting a= 2n gives that 4n^2 = 2b^2, and thus 2n^2 = b^2. This means b^2 is even, and thus b is even. But then a and b are both even, which contradicts the assumption we made at the start that a and b have no common factors. This means our original assumption, that the square root of 2 is rational, must be wrong, so the square root of 2 is irrational!
All higher math is "proof based", meaning that instead of focusing on how to compute things and answer specific questions, you learn how to read, understand, and write proofs about specific fields of math. The way to learn proofs is to start doing them. There are lots of books which teach interesting mathematics for those who are new to proofs. Here are some of my recommendations:
- The book of proof and how to prove it are two fairly standard recommendations for those interested in learning just proofs.
- Analysis I is a really nice introduction to proofs in the field of "analysis". This book in particular focuses on proving the theorems of calculus.
- A book of abstract algebra is a good introduction to proofs in the field of "algebra" (algebra involves the study of symmetry, among other things). If you choose to start with this book, read the appendix first.
While I linked amazon, you should be able to find pdfs for all of these books online if you look in the right places. I recommend you give them all a download, start reading, and see which appeals to you most. It's going to be slow and hard at first, don't let that discourage you, if you push through it will be extremely worth it.
1) Learn everything in this book
2) http://www.cs.toronto.edu/~david/csc165/resources/csc165_notes.pdf
Learn the information of chapter 4 in this book, in particular the stuff from page 90 to the end of the chapter
If you are interested in proofs, you might want to look at Velleman's How To Prove It. Another one people seem to like is The Book of Proof. Do what you want, the world is yours.
I've heard the book How To Prove it is pretty good. Also I'd recommend the Art of Problem Solving books as well for algebra and the likes. (It seems to go over stuff you'd learn in 7th grade, but written at a level adequate for adults).
I would also recommend sites like www.expii.com and www.brilliant.org
Khan academy also has a problem generator iirc.
Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.
If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.
If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).
If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.
If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.
If you mean what are the fundamental concepts (as in more than one) then it is almost certainly logic and sets as others mentioned. But that is literally the bare minimum. From there you have to expand into functions and relations and start thinking of mathematical objects the same way you would think of classes and objects in OOP. There's a lot of similarity in those ways of thinking, down to the idea that in logic you create statement forms that act as "classes" or templates of statements that are "instantiated" by the "caller" supplying variables to convert them from statement forms into concrete claims that can be evaluated "at run time" so to speak.
So for example, the statement "there is a number that when added to x results in zero" (the additive inverse axiom) is converted into logical form ∃y(x+y=0) which is then "instantiated" for evaluation by providing values for x and y. The claim is not testable until instantiated. And it is true for some domains but not others, e.g. it is false for the domain of natural numbers and whole numbers because negative numbers do not exist in those domain sets, but it is true for integers and reals because those sets contain negatives.
Excellent easy to read book (easy to read, but take it slow to understand it) is How to Prove It which covers exactly these topics, plus functions, relations, proof techniques, etc. Another one I want to read but haven't had time yet is Basic Concepts in Mathematics which covers a lot of the same material but I think is a step further than the former book.
Here is a massive map that shows how a ton of math fields interact and tries to link them in terms of building on each other: http://i.stack.imgur.com/jGMXs.gif
Also if you haven't already studied it check out discrete math. It teaches the bare basics of logic and set theory and Boolean algebra. It also teaches a crapton of methods of counting objects which ties into logic and set theory, and notice that map lists counting as just about the most fundamental concept.
Pick a book like how to prove it and see if you like it, if you do then you'll probably enjoy pure mathematics.
If you are mostly looking for a book on proofs, How to Prove It might be what you want.
You might want to check out Stein and Shakarchi's book Complex Analysis http://press.princeton.edu/titles/7563.html. This book is a bit hard but iirc doesn't require you to have had real analysis before hand. I would highly recommend that you work through a proof based book before hand though. Often times this will be a course book but something like https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995?ie=UTF8&*Version*=1&*entries*=0 that should also get the job done.
Or you can go the traditional route like other people mentioned of getting about a semester's worth of real analysis under your belt. The reason why this is usually the suggested path is because it's not expected that you are 100% competent at writing proofs in the beginning of real but you are in complex.
I didn't take any lower division classes here but the upper division classes are pretty great. I haven't really had any bad professors and they seem to be a lot better at teaching than the professors in my upper division physics courses were.
The quarter system isn't bad. I think it's actually a good pace and the courses that have more than 10 weeks of content are 2 or 3 quarters long, which is great because it means you're not stuck in a class that you hate for very long.
The difficulty depends entirely on the professor, but I haven't had a class that was super difficult and uncurved. Curves always seem fair for the difficulty of the class. Finals are usually fair but midterms really suck because they're only 50 minutes long. You will probably do horrible on a few of them before you figure out a way to make it work. We have a much cushier path to upper division than most schools. Instead of being dumped into linear algebra or real analysis and having to learning how to do proofs, we have an intro to proofs and logic prerequisite and another class where you essentially just practice proof techniques that you will use in analysis later. I loved it because it let me focus on the material in my more challenging classes without having to figure out the mechanics and techniques of general proof writing.
One thing to keep in mind is that upper division math is nothing whatsoever like the math that you're probably used to. You essentially start over and learn things correctly, and you usually have to pretend that you don't know anything that you've learned over the past 14 years of math classes outside of basic arithmetic and algebra. You will be writing paragraphs in plain English with occasional math symbols. It's all about taking definitions and theorems that you know and using them to argue that other theorems are true. It's a lot more fun than it sounds. If you want to get a feel for what it's going to be like, check out this book:
http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
It's easy to find elsewhere. You don't need to know anything to get started and it's actually really fun to work through. This was the textbook for my intro to proofs class.
Not quite a textbook, but will cover everything covered in mathematical logic classes. First few chapters are more relevant to non-math majors.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
>Never done any proofs, but I will ask my math teacher if he can recommend any books or resources that will help me learn.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I'd recommend How to Prove It as a way to familiarize yourself with the core language of more abstract mathematics. I'd also start working through Spivak's Calculus, which is a ground-up, rigorous treatment of calculus starting from the axioms of a Dedekind-complete ordered field.
The rigor of Spivak's Calculus will provide a good contrast to AP Calc.
This is a phenomenal book.
I'm a graduate student, I've probably written a million proofs in my life and I am still finding it useful.
I watched these when I was taking it and it was pretty helpful to understand concepts. Not really the same material, and also not covered in the same way, but you can leave them playing in the background or something.
As for reading material, this book is really, really good:
http://www.amazon.ca/How-Prove-It-Structured-Approach/dp/0521675995
You can probably find a copy online somewhere.
I think I got an 89 in that course despite getting 50% on the first midterm because I misread the questions :/
What topic you should focus on is something you should ask yourself. Now you have an opportunity to explore what interests you, and that is what you should do. For some reason it has become standard to teach calculus before any other interesting math topic. But as you should be aware there are many interesting mathematical topics that don't have calculus (or linear algebra) as a prerequisite. I'll give you some ideas for other topics you can study. These topics are generally fun, challenging, and should be accessible to you. This is by no means a comprehensive list, it is just some things that came to my mind while writing this.
- Elementary number theory, where famously many problems are easy to state and hard to solve, like Fermat's last theorem.
- Geometry. You could take a look at a book that is aimed at an undergraduate course in classical geometry.
- Basic group theory. Groups are what mathematicians use to model symmetry. Thus it should not be surprising that groups crop up everywhere in mathematics! If you can find a good exposition (!) I'm in no doubt that you can learn some group theory.
- Graph theory. If eight-year-olds can do it, then so can you.
- What is a proof? What are 'implications', 'equivalences', etc.? Reading a book like 'How to Prove It' or following a course like Introduction to Mathematical Thinking will be extremely valuable in learning and understanding mathematics of any kind.
Again, these are just ideas for what you can do. I didn't add calculus or linear algebra to the list because you are already aware of these topics. Whatever you end up doing with your summer, I wish you the best of luck!
I took a class that was specifically dedicated to writing proofs, so I haven't personally read this myself, but I've heard a lot of people say good things about it.
Me and you are just about in the same place in terms of mathematical progression. At the moment, I am going through the following two books
How to prove it by Velleman
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
proof logic and conjecture by Wolf
http://www.amazon.com/Proof-Logic-Conjecture-Mathematicians-Toolbox/dp/0716730502
Both are interesting and provide a great experience for self study. A seriously good introduction to higher mathematics. Gives a quick run down of propositional and first order logic then kicks the proofs into gear, all while remaining intuitive! I'd seriously recommend them both, but I'd choose Velleman if I had to make a choice. Good luck! I hope this helps :)
How to prove it by Velleman is very clear.
You could check out <em>How to Prove It: A Structured Approach</em> by Velleman.
check out this mahfukka http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=tmm_pap_title_0
This book helped me out a bit: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- However, even though I have a background in programming, I felt it moved rather quickly, especially after about halfway through the book.
Not specific to calculus, but Daniel Velleman's How to Prove It is an excellent and thorough introduction to the practice/art of proof-writing, and it requires only a high school mathematics education as a prerequisite.
I recommend proofs. How To Prove It is a good introduction, but there are also free resources available online.
Math major courses after the first year consist of proof based courses. If you already have interest in proof writing and argumentation then I suggest you go for it. If not then you try it out. Firstly, you should brush up on basic logic. Check out the quick introduction at berkeley. Once that's cleared, go complete a book such as How To Prove It. Now, you should have the most basic tools needed to tackle and solve problems in future courses in analysis, algebra, topology, etc. The time required to learn and write good proofs is steep. It takes constant feedback and solving numerous problem to get a knack of. I would say that you should get used to proof writing in a semester. Overall, learning to write proofs isn't very difficult but learning the material for the courses and developing a solid theoretical understanding is. In regard to your concerns about time commitment with courses, it depends on how far you are willing to go to obtain a career in mathematics and to do deep research.
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