#1 in Science & math books
Use arrows to jump to the previous/next product
Reddit mentions of How to Prove It: A Structured Approach, 2nd Edition
Sentiment score: 83
Reddit mentions: 119
We found 119 Reddit mentions of How to Prove It: A Structured Approach, 2nd Edition. Here are the top ones.
or
- Cambridge University Press
Features:
Specs:
| Height | 8.95 Inches |
| Length | 6.1 Inches |
| Number of items | 1 |
| Weight | 1.1684499886 Pounds |
| Width | 0.9 Inches |
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
Precalculus
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.
Paul's Online Math Notes
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
OverflowExchange 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.
> Are the deep mathematical answers to things usually very complex or insanely elegant and simple when you get down to it?
I would say that the deep mathematical answers to questions tend to be very complex and insanely elegant at the same time. The best questions that mathematicians ask tend to be the ones that are very hard but still within reach (in terms of solving them). The solutions to these types of questions often have beautiful answers, but they will generally require lots of theory, technical detail, and/or very clever solutions all of which can be very complex. If they didn't require something tricky, technical, or the development of new theory, they wouldn't be difficult to solve and would be uninteresting.
For any experts that happen to stumble by, my favorite example of this is the classification of semi-stable vector bundles on the complex projective plane by LePotier and Drezet. At the top of page 7 of this paper you'll see a picture representing the fractal structure that arises in this classification. Of course, this required a lot of hard math and complex technical detail to come up with this, but the answer is beautiful and elegant.
> How hard would it be for a non mathematician to go to a pro? Is there just some brain bending that cannot be handled by some? How hard are the concepts to grasp?
I would say that it's difficult to become a professional mathematician. I don't think it has anything to do with an individual's ability to think about it. The concepts are difficult, certainly, but given time and resources (someone to talk to, good books, etc) you can certainly overcome that issue. The majority of the difficulty is that there is so much math! If you're an average person, you've probably taken at most Calculus. The average mathematics PhD (i.e., someone who is just getting their mathematical career going) has probably taken two years of undergraduate mathematics courses, another two years of graduate mathematics courses, and two to three years of research level study beyond calculus to begin to be able tackle the current theory and solve the problems we are interested in today. That's a lot of knowledge to acquire, and it takes a very long time. That doesn't mean you can't start solving problems earlier, however. If you're interested in this type of thing, you might want to consider picking up this book and see if you like it.
You've been posting lots of vague and confused questions about sequences, derivatives, and cardinality to /r/math. You also have a habit of inventing our own terminology without motivating it or even acting as if folks should naturally understand it.
It seems clear from these threads (including this one) that you're confused about some fundamental ideas surrounding sequences, cardinality, (un)countability, and differentiation. You're getting so-so responses because folks have to nail down what exactly you're asking/thinking before they can attempt to answer and that requires a ton of effort.
If you really want to learn this stuff, I'm of the mind that you need to spend less time posting ill-formed questions on /r/math and instead make sure you understand the fundamentals. For example, here some of the first things one typically learns when studying countability:
Obviously (3) implies the first two, but each is progressively more difficult to prove for someone approaching these ideas for the first time. The latter two require some version of the axiom of countable choice, for example, which isn't something most newcomers would think to deploy unless they had encountered it before.
They do, however, answer your question: if we have a countable set and "glue on" a countable number of countable collections of new numbers, the resulting set will still be countable.
I strongly recommend you buy and read Daniel Velleman's How to Prove It. It will help you organize your thoughts better and help you get comfortable with the "standard" mathematical terminology and notation. Topic-wise it covers basic set theory and the last chapter is all about infinite sets, cardinality, (un)countability, and so on.
Here are some screenshots from Amazon's "Search Inside the Book" to show you what you can expect by the end of the book:
You could read Timothy Gowers' welcome to the math students at Oxford, which is filled with great advice and helpful links at the bottom.
You could read this collection of links on efficient study habits.
You could read this thread about what it takes to succeed at MIT (which really should apply everywhere). Tons of great discussion in the lower comments.
You could read How to Solve It and/or How to Prove It.
If you can work your way through these two books over the summer, you'll be better prepared than 90% of the incoming math majors (conservatively). They'll make your foundation rock solid.
I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.
The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.
You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.
But it's probably best to get a good foundation before embarking on Real Analysis and the like:
Discrete Mathematics with Applications by Susanna Epp
How to Prove It: A Structured Approach Daniel Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Book of Proof by Richard Hammock
That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.
Traditionally, a mathematical proof has one and only one job: convince other people that your proof is correct. (In this day and age, there is such a thing as a computer proof, but if you don't understand traditional proofs, you can't handle computer proofs either.)
Notice what I just said: "convince other people that your proof is correct." A proof is, in some sense, always an interactive undertaking, even if the interaction takes place across gulfs of space and time.
Because interaction is so central to the notion of a proof, it is rare for students to successfully self-study how to write proofs. That seems like what you're asking. Don't get me wrong. Self-study helps. But it is not the only thing you need. You need, at some point, to go through the process of presenting your proofs to others, answering questions about your proof, adjusting your proof to take into account new feedback, and using this experience to anticipate likely issues in future proofs.
What you're proposing to do, in most cases, is the wrong strategy. You need more interactive experience, not less. You should be beating down the doors of your professor or TA in your class during their office hours, asking for feedback on your proofs. (This implies that you should be preparing your proofs in advance for them to read before going to their office hours.) If your school has a tutorial center, that's a wonderful resource as well. A math tutor who knows math proofs is a viable source of help, but if you don't know how to do proofs, it's hard for you to judge whether or not your tutor knows how to do proofs.
If you do self-study anything, you should not be self-studying calculus, linear algebra, real analysis, or abstract algebra. You should be self-studying how to do proofs. Some people here say that How to Prove It is a useful resource. My own position is that while self-studying can be helpful, it needs to be balanced with some amount of external interactive feedback in order to really stick.
The heart of conceptual mathematics (i.e., mathematics that isn't just computation and carrying out algorithms) is mathematical proof. I suggest you work through the book How to Prove It. This will give you the tools to self work through other textbooks (not that it will suddenly be easy).
Learn math at a more "fundamental" level, and that will test if you love it. For me, I didn't love math until I took a class on proofs and real analysis. One of the books we used was "How to Prove it", and to this day it's my favorite textbook ever. How do we know anything in mathematics? Which rules do we follow and how do we know they are true? This starts from basic logic and truth tables, and works its way up to some really complicated stuff. It's not as fancy as complex integrals and PDE's, but I would say it's a more fundamental form of mathematics and the basis for all other subjects in the field.
I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.
The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:
How to prove it
Number theory
After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.
At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.
There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.
I double majored in math and CS as an undergrad and I enjoyed math more than CS. I'm a graduate student right now planning on doing research in a mathy area of CS. Everything I write below comes from that perspective.
But if you're interested in really digging in and understanding some math at an advanced undergraduate level (analysis, abstract algebra, topology, etc.) then I don't think there is any substitute for books.
One of the most fun things I did when I was first learning about proofs was proving the basic facts about algebra from axioms. Where I first read about these ideas was the first chapter of Spivak's Calculus. This would be a very high level book for an 18 year old, but if you decide to look at it, don't be afraid to take your time a little.
Another option is just picking up an introduction to proof, like Velleman's How to Prove It. This wil lteach you the basics for proving anything, really, and is a great start if you want to do more math.
If you want a free alternative to that last one, you can look at The Book of Proof by Richard Hammack. It's well-written although I think it's shorter than How to Prove It.
Or How to Prove It by Velleman.
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:
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:
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] (https://math.stackexchange.com/questions).
Good luck.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
I picked up a book a couple years ago called How to Prove It.
It has helped me develop a greater appreciation for logic and proofs. I wish I took this stuff more seriously when I started programming. A little bit of knowledge of boolean algebra can help tremendously.
Is it really such a big step from du Sautoy's explanation to the formal proof? I don't think so, but maybe I'm biased. I bet there are books on elementary number theory that don't assume much of any background that you could understand. If you're interested in proofs in general, you might enjoy Velleman's How to Prove It.
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
Not sure what sort of thing you're trying to prove, but there are a few good books on techniques for proof that you'll end up using if you go into higher math. I like How to Prove It by Velleman. It's geared towards students finishing high school math who are planning to do math at the university level, so it might be the sort of thing you're looking for.
I know the answer to this.
First, though: arithmetic and all that, through calculus, is not math.
True math is the discovery of properties of ideas. One interesting example is the fact that there is a hypothetical machine that is proven to be able to do everything a (real) computer can do, but that there are many things that it can never do. Therefore, there are questions that can never be answered by a computer, no matter how powerful.
If you actually want to know about the beauty, you need to see it for yourself. As I recall, How to Prove it is pretty decent.
I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.
Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).
Alternately, any introductory book on mathematical analysis will have a section on sentential logic. 'How to Prove It' by Velleman is a great intro, and comes with a link to a web tool to practice!
I'd echo what /u/Odnahc has said.
Struggling in Intro the Proofs isn't he end of the world. I struggled in proofs and still ended up with a BS and MS in Math, however, I bought this book and self studied proofs over the Summer and made sure I had a stronger foundation.
The courses normally taken after proofs (Advanced Calculus and Modern Algebra) usually spend the first class reviewing proofs to make sure students have a handle of the material. After that though, you're expected to know the stuff. And honestly, you'll be doing lot of work trying to understand the new material and you're really going to struggle if you're fighting proof writing instead of the new ideas.
Proceed with caution. Definitely speak to your advisor.
Here is the ooh page on Statisticians:
http://www.bls.gov/oco/ocos045.htm
A job straight out of college might see you as a research assistant. I could see you getting a job at Mathematica perhaps. Try to get a SAS certificate before you graduate, a working knowledge of R, and if you feel like tackling it a programming language good for numerical analysis.
Have you taken a course on Regression? I'd consider that, and perhaps even trying to take a Mathematical Statistics Course, if it is offered. You can try to see if you university would allow you to take a class online, or try a Semester Abroad at a university that has that class.
My background: I am an Economist that uses Statistics heavily, and works with Statistical methods often (ie: econometrics). I love it.
Your plans on studying Calc 2 and Linear Algebra are great. That is perfect.
My pay after 10 years is likely to be 100k-150k.
Before you start your first semester at the graduate level know the following things really well: Set theory, integration, matrix algebra, and proofs.
Get this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- read it before you study linear algebra, and maybe even some Calculus. It doesn't require a heavy Math background and will save you a lot of frustration later on.
Try one (or a few) of these:
http://www.amazon.ca/Thinking-Mathematically-J-Mason/dp/0201102382
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/052171978X/
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/
www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/
https://www.coursera.org/course/maththink
tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.
Are you proficient in reading and writing proofs?
If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.
After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.
You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.
The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).
And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_other_apa_Sn9NBbDH6MYPX not sure it this is exactly what you're asking for(might be more than you're asking for?) but this helped me a lot.
You cannot go wrong with How To Prove It: A Structured Approach by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_3?keywords=how+to+prove+it&qid=1558195901&s=gateway&sr=8-3
I saw that book highly recommended, and after going through it myself a while ago I highly recommend it as well. When I do proofs I still maintain the mental model and use some of the mechanics that I learned from this book. You don't even have to read the whole thing in my opinion. Pick it up, work through a few pages per day, and stop when you feel like moving onto another subject-specific book, like Understanding Analysis.
Oh, and you might already know this, but do as many practice problems as you can! Learning proofs is all about practice.
Since nobody else has recommended it, I always recommend the book How to Prove it by Daniel J. Velleman for learning proofs. I always found proofs to be kind of black magic until I read that, which totally demystified them for me by revealing the structure of proofs and techniques for proving different kinds of statements. One of the best things about it is that it starts from square one with basic logic and builds from there in way that no prior knowledge is required beyond basic algebra skills.
You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Discrete Mathematics with Applications by Susanna S. Epp
How to Prove It: A Structured Approach Daniel J. Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Combinatorics: A Guided Tour by David R. Mazur
Read this book: How To Prove it
I recommend you start studying proofs first. How to Prove It by Velleman is a great book for new math students. I went through the first three chapters myself before my first analysis course, and it made all the difference.
As you are taking a class than combines analysis and calculus, you might benefit from Spivak's book Calculus, which despite it's title, is precisely a combination of calculus and real analysis.
If your intent is to take a class like analysis, you really should look into something like logic.
Daniel Velleman wrote an excellent little book called How to Prove It: A Structured Approach. It's actually designed for High School level students, but it works through the subject incredibly well.
Here's an Amazon link to the book:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1333383091&sr=1-1
Perhaps rather than concentrating on these particular proofs you should look at something like How To Prove It.
I think everyone is on point for the most part, but I'd like to be the devil's advocate and suggest a different route.
Learn logic, proof techniques and set theory as early as possible. It will aid you in further study of all 'types' of math and broaden your mind in a general sense. This book is a perfect place to start.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The best part is, when you start doing proofs you realize you've been thinking about math all wrong (at least I did). It's an exercise in creativity, not calculation.
In my mind, set theory & calculus are necessary pre-requisites to probability anyway, and linear algebra means much more once you have been introduced to inductive proofs, as well.
How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.
How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).
Read and work through this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&qid=1301319337&sr=8-1
Learning proofs can mean different things in different contexts. First, a few questions:
The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.
As a general rule, I'd say that working on proof-based contest questions that are just beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.
This may be especially true for things like logic and very elementary set theory.
Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)
Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's orders of magnitude more effective when you have someone who can guide you.
Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone else?
Have you had at least, for example, a geometry class that's proof-based?
Proofs are all about communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.
---
With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.
Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.
---
How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!
It's common to have some difficulty adjusting from lower-level courses with a computational emphasis to upper-level courses with an emphasis on proof. Fortunately, this phenomenon is well known, and there are a number of books aimed at bridging the gap between the two types of courses. A few such books are listed below.
I hope that helps!
Have I got the book for you, op
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_2EmtDbDDEMPG9
How To Prove It
What this expressions says
First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.
This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)
What does ∀x∀yP(x,y) mean?
This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.
What does ∀xP(x,x) mean?
This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.
So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?
Proof by contradiction
We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.
So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)
not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.
So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.
not P(a,a)
Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.
Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)
This happens in two stages:
First we instantiate y
∀xP(x,a)
Then we instantiate x
P(a,a)
The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:
∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.
Hopefully that makes sense.
Recommended Resources
Wilfred Hodges - Logic
Peter Smith - An Introduction to Formal Logic
Chiswell and Hodges - Mathematical Logic
Velleman - How to Prove It
Solow - How to Read and Do Proofs
Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics
You will first want to learn fundamental logic and set theory before diving into topics like analysis, algebra, and discrete topics. You will need an understanding of a rigorous proof -- not the hand-wavey kind of proof we've seen in our introductory calculus courses. This book is very readable and will prepare you for advanced mathematics. I've seen it work for many students.
After you're finished with it, you may want to study analysis which will build up the Calculus for you. If you don't care for calculus anymore, consider reading an abstract algebra text. Algebra is pretty fun. You can also pick a discrete topic like graph theory or combinatorics whose applications are very easy to see.
There are many ways to go, but in all of them you will absolutely need a a basic understanding of the use of logic in a mathematical proof.
How To Prove It. Read through the reviews. It's the best book for learning propositional and predicate logic for the first time.
Sorry, the solution is to do lots of proofs.
There's more to it, but honestly it's more of a thing that you have to read a book about rather than a message on reddit. How are you learning about this right now? Is it part of a course or self-study? I personally found How to Prove It to be a very useful textbook. Doesn't require any particular knowledge, and it builds out a nice foundation in logic and set theory.
“How to Prove it”. D. Velleman: Amazon US Link
Probably the best resource on the topic!
How to Prove It
It's cheap, highly rated, starts with the basics, and as the title says, shows you how to prove it!
You received A's in your math classes at a major public university, so I think you're in pretty good shape. That being said, have you done proof-based math? That may help tremendously in giving intuition because with proofs, you are giving rigor to all the logic/theorems/ formulas, etc that you've seen in your previous math classes.
Statistics will become very important in machine learning. So, a proof-based statistics book, that has been frequently recommended by /r/math and /r/statistics is Statistical Inference by Casella & Berger: https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
I've never read it myself, but skimming through some of the beginning chapters, it seems pretty solid. That being said, you should have an intro to proof-course if you haven't had that. A good book for starting proofs is How to Prove It: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
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.
Спасибо за ссылку. Я обязательно это проверю. Думаю, надо было быть более конкретным. Я читал книгу, которая учит своих читателей, как строить математические доказательства. В книге дается очень общий обзор этих тем, которые я перечислил выше. Я проверю ссылку, но если вы знаете книгу на русском языке, которая учит строить математические доказательства студентам, которые начинают изучать продвинутую математику, напишите Мне пожалуйсте.
Вот книга Для справки. (в случае, если вы знаете английского языка).
How to Prove it - A structured approach
Hey mathit.
I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!
Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.
I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.
Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.
Do you agree with my choices? What else do you recommend?
I found online courses to be ineffective, I prefer books.
What's your opinion, mathit?
Cheers and many thanks in advance!
How to Prove It is only 20 bucks.
So here are some options I recommend:
You can find all the textbooks I mentioned online, if you know what I mean. All of these assume you haven't seen math in a while, and they all start from the very basics. Take your time with the material, play around with it a bit, and enjoy your summer :D
EditL this article describes one way you can go about your studies
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
I'm going to recommend the book How to Prove It. Its all about learning the logic for proofs and strategies for writing proofs. Its one of those books that you work through slowly and complete all the exercises. Its recommended around here a-lot. I'd also suggest using the search feature if you ever want to look for other recommended books because those threads come up often around here.
Best wishes.
For proof writing techniques I highly recommend Velleman's "How to Prove It" link
I used Susanna Epp's Discrete Mathematics text and rather enjoyed it. Velleman's How To Prove It is also quite good.
http://www.abebooks.com/servlet/SearchResults?bi=0&bx=off&ds=20&kn=Epp+discrete+applications+3&recentlyadded=all&sortby=17&sts=t
How to Prove It: A Structured Approach by Daniel J. Velleman http://www.amazon.com/dp/0521675995/ref=cm_sw_r_udp_awd_ff3Vtb08QR4FZ
Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:
In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).
AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.
Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.
Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:
math on irc.freenode.net
Here are two possible routes, one minimal, one less-minimal:
Minimal
Less-minimal:
NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.
The exercises in Spivak’s Calculus (amzn) are the best part of the book.
/u/WelpMathFanatic You’ll probably have a better (more efficient, more enjoyable) time if you take a course, or otherwise find someone to help you face to face. But if you’re studying by yourself you might want to look at a book about writing proofs, such as Velleman’s [How to Prove It](https://amzn.com/0521675995) or Hammack’s [Book of Proof*](https://amzn.com/0989472108). (Disclaimer: I haven’t read either of these.)
I would recommend the following two books:
The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
This book was very helpful to me.
This book is really good. I read and worked through it the summer before I had an introductory analysis course, and I felt much more prepared than I would have without the book.
Here's the link just in case:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=tmm_pap_title_0?ie=UTF8&qid=1369627123&sr=8-1
Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.
I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.
Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.
Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.
Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.
Example,
Linear Algebra for freshmen: some books that talk about manipulating matrices at length.
Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler
Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman
Basically, math is all interconnected and it doesn't matter where exactly you enter it.
Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.
Books you might like:
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Building Proofs: A Practical Guide by Oliveira/Stewart
Book Of Proof by Hammack
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al
How to Prove It: A Structured Approach by Velleman
The Nuts and Bolts of Proofs by Antonella Cupillary
How To Think About Analysis by Alcock
Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash
Problems and Proofs in Numbers and Algebra by Millman et al
Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi
Mathematical Concepts by Jost - can't wait to start reading this
Proof Patterns by Joshi
...and about a billion other books like that I can't remember right now.
Good Luck.
I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.
If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.
If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.
Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.
After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.
After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.
Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).
If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!
TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres
To learn basic proof writing I highly recommend How to Prove It by Velleman.
Math isn't going to be like the math classes you've already taken. It's a lot of writing and logic and very little calculating. If you go for mathematical sciences, you'll probably take more classes that involve calculations, but you won't make it that far if you can't handle proofs.
Check out this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The full book is first-page googleable. If you find that material interesting, you'll probably enjoy being a math major.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
helps with the first part of the class. the stuff after that I would suggest just having good google-fu.
Book of proof is a more gentle introduction to proofs then How to Prove it.
​
No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.
​
An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.
You should absolutely not give up.
None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."
As for specific recommendations, make the most of this summer. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!
How to Prove It is a nice introduction to writing proofs.
Thanks for the answer!
Glad to hear about Spivak! I've heard good things about that textbook and am looking forward to going through it soon :). Are the course notes for advanced algebra available online? If so, could you link them?
Is SICP used only in the advanced CS course or the general stream one, too? (last year I actually worked my way through the first two chapters before getting distracted by something else - loved it though!) Also, am I correct in thinking that the two first year CS courses cover functional programming/abstraction/recursion in the first term and then data structures/algorithms in the second?
That's awesome to know about 3rd year math courses! I was under the impression that prerequisites were enforced very strongly at Waterloo, guess I was wrong :).
As for graduate studies in pure math, that's the plan, but I in no way have my heart set on anything. I've had a little exposure to graph theory and I loved it, I'm sure that with even more exposure I'd find it even more interesting. Right now I think the reason I'm leaning towards pure math is 'cause the book I'm going through deals with mathematical logic / set theory and I think it's really fascinating, but I realize that I've got 4/5 years before I will even start grad school so I'm not worrying about it too much!
Anyways, thanks a lot for your answer! I feel like I'm leaning a lot towards Waterloo now :)
How to Prove it by Velleman seems to be right up your alley.
I may be in the minority here, but I think that high school students should be exposed to statistics and probability. I don't think that it would be possible to exposed them to full mathematical statistics (like the CLT, regression, multivariate etc) but they should have a basic understanding of descriptive statistics. I would emphasize things like the normal distribution, random variables, chance, averages and standard deviations. This could improve numerical literacy, and help people evaluate news reports and polls critically. It could also cut down on some issues like the gambler's fallacy, or causation vs correlation.
It would be nice if we could teach everyone mathematical statistics, the CLT, and programming in R. But for the majority of the population a basic understanding of the key concepts would be an improvement, and would be useful.
Edit At the other end of the spectrum, I would like to see more access to an elective class that covers the basics of mathematical thinking. I would target this at upperclassmen who are sincerely interested in mathematics, and feel that the standard trig-precalculus-calculus is not enough. It would be based off of a freshman math course at my university, that strives to teach the basics of proofs and mathematical thinking using examples from different fields of math, but mostly set theory and discrete math. Maybe use Velleman's book or something similar as a text.
How comfortable are you with proofs? If you are not yet comfortable, then read this: How to Prove It: A Structured Approach
Get the book [How to Prove It: A Structured Approach by Daniel J. Velleman] (http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995) it will teach you how to write, and I think more importantly, read proofs.
Probably something on coursera; however, I really recommend this gem of a book, http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1458411780&sr=1-1&keywords=how+to+prove+it .
Like one of the other's suggested, learning more about proofs is probably what you're interested in since that's where these rules and equations come from. I've seen this book recommended a few times. It should give you a better understanding of how math is formed.
Read this book: http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/ref=asap_bc?ie=UTF8
And work through this book in its entirety: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1463201632&sr=1-1
Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.
You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.
Two other books you may be interested in instead, that teach the same kinds of things:
Introduction to Mathematical Thinking which he wrote to use in his Coursera course.
How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.
Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.
What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.
Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics without any freaky looking symbols, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/phil110-gmh/text/c01_3-99.pdf
Honestly if I were starting out I would love that last link, it looks fantastic actually.
Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.
You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres' Topology, Dummit and Foote's Abstract Algebra, and/or Rudin's Principles of Mathematical Analysis would be good places to go.
Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.
How to Prove It by Vellemen is a superb introduction to what proofs are, and how to make them.
Keep in mind certain proof based courses can be frustrating to some students (discrete math and real analysis) as these classes often make formal concepts students may understand intuitively. Abstract Algebra or Topology may give you a more accurate idea of your feelings towards math.
Your mileage with certifications may vary depending on your geographical area and type of IT work you want to get into. No idea about Phoenix specifically.
For programming work, generally certifications aren't looked at highly, and so you should think about how much actual programming you want to do vs. something else, before investing in training that employers may not give a shit about at all.
The more your goals align with programming, the more you'll want to acquire practical skills and be able to demonstrate them.
I'd suggest reading the FAQ first, and then doing some digging to figure out what's out there that interests you. Then, consider trying to get in touch with professionals in the specific domain you're interested in, and/or ask more specific questions on here or elsewhere that pertain to what you're interested in. Then figure out a plan of attack and get to it.
A lot of programming work boils down to:
As a basic primer, you might want to look at Code for a big picture view of what's going with computers.
For basic logic skills, the first two chapters of How to Prove It are great. Being able to think about conditional expressions symbolically (and not get confused by your own code) is a useful skill. Sometimes business requirements change and require you to modify conditional statements. With an understanding of Boolean Algebra, you will make fewer mistakes and get past this common hurdle sooner. Lots of beginners struggle with logic early on while also learning a language, framework, and whatever else. Luckily, Boolean Algebra is a tiny topic. Those first two chapters pretty much cover the core concepts of logic that I saw over and over again in various courses in college (programming courses, algorithms, digital circuits, etc.)
Once you figure out a domain/industry you're interested in, I highly recommend focusing on one general purpose programming language that is popular in that domain. Learn about data structures and learn how to use the language to solve problems using data structures. Try not to spread yourself too thin with learning languages. It's more important to focus on learning how to get the computer to do your bidding via one set of tools - later on, once you have that context, you can experiment with other things. It's not a bad idea to learn multiple languages, since in some cases they push drastically different philosophies and practices, but give it time and stay focused early on.
As you gain confidence there, identify a simple project you can take on that uses that general purpose language, and perhaps a development framework that is popular in your target industry. Read up on best practices, and stick to a small set of features that helps you complete your mini project.
When learning, try to avoid haplessly jumping from tutorial to tutorial if it means that it's an opportunity to better understand something you really should understand from the ground up. Don't try to understand everything under the sun from the ground up, but don't shy away from 1st party sources of information when you need them. E.g. for iOS development, Apple has a lot of development guides that aren't too terrible. Sometimes these guides will clue you into patterns, best practices, pitfalls.
Imperfect solutions are fine while learning via small projects. Focus on completing tiny projects that are just barely outside your skill level. It can be hard to gauge this yourself, but if you ever went to college then you probably have an idea of what this means.
The feedback cycle in software development is long, so you want to be unafraid to make mistakes, and prioritize finishing stuff so that you can reflect on what to improve.
ummarycoc has a good point. Snoop around his room and see if he already has How To Prove it: A Structured Approach. Someone bought this book for me and I return to it frequently.
The deck corresponding to the intellectual property book has ~325 cards.
The deck corresponding to IEA has ~400 cards.
The deck corresponding to linear algebra has ~1000 cards. That seems weird to me, since I feel that I make fewer cards for math books; most of the extra time comes from doing a lot of scratch work. Weird. In addition to timing, I've more recently started keeping track of how many Ankis I've added each section, so maybe I'll have more insight there later. We'll see.
And please message me when you start doing math! If you're looking towards advanced mathematics (beyond calculus/linear algebra-for-engineers), I recommend starting with either Mathematics for Computer Science (review) or, if you really have no interest in doing that, How to Prove It.
Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).
I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.
This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.
Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:
Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.
I've been studying How To Prove It by Daniel Velleman for a few months now and I don't know if it's the best book, but it's really good and it has opened my mind in so many ways. Plus, it's really cheap for a textbook.
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.
Start with a book like this:
http://www.amazon.com/books/dp/0521597188
or this:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.
You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.
This is a PDF of the third edition of the above book.
This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.
The math subreddit is primarily undergrads talking about various topics. Make a point of just hanging out and reading stuff. If you don't understand something just tell us and they'll do their best to help out.
Hang out on the math stack exchange and ask questions about things you do not understand while trying to help with things you do understand.
Hope that helps!
check out this mahfukka http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=tmm_pap_title_0
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&qid=1342068971&sr=8-1&keywords=spivak%27s+calculus
This book starts with basic properties of numbers (associativity, commutativity, etc), then moves onto some proof concepts followed by a very good foundation (functions, vectors, polar coordinate). Be forewarned that the content is VERY challenging in this book, and will definitely require a determined effort, but it will certainly be good if you can get through it.
A more gentle introduction to Calculus is http://www.amazon.com/Thomas-Calculus-12th-George-B/dp/0321587995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069166&sr=1-1&keywords=thomas%27+calculus and it is a much easier book, but you don't prove much in this one. Both of these can likely be found online for free. Also, if you want to get a decent understanding I recommend, http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069253&sr=1-1&keywords=how+to+prove+it or http://www.people.vcu.edu/~rhammack/BookOfProof/index.html the latter is definitely free.
You may also need a more introductory text for trig and functions. I can't find the book my school used for precalc, hopefully someone else can offer a good recommendation.
Also, getting a dummies book to read alongside was pretty helpful for me, and Paul's online notes(website) is very nice.
> I assume I ought to check it out after my discrete math class? Or does CLRS teach the proofs as if the reader has no background knowledge about proofs?
Sadly it does not teach proofs. You will need to substitute this on your own. You don't need deep proof knowledge, but just the ability to follow a proof, even if it means you have to sit there for 2-3 minutes on one sentence just to understand it (which becomes much easier as you do more of this).
> We didn't do proof by induction, though I have learned a small (very small) amount of it through reading a book called Essentials of Computer Programs by Haynes, Wand, and Friedman. But I don't really count that as "learning it," more so being exposed to the idea of it.
This is better than nothing, however I recommend you get very comfortable with it because it's a cornerstone of proofs. For example, can you prove that there are less than 2 ^ (h+1) nodes in any perfect binary tree of height h? Things like that.
> We did go over Delta Epsilon, but nothing in great detail (unless you count things like finding the delta or epsilon in a certain equation). If it helps give you a better understanding, the curriculum consisted of things like derivatives, integrals, optimization, related rates, rotating a graph around the x/y-axis or a line, linearization, Newton's Method, and a few others I'm forgetting right now. Though we never proved why any of it could work, we were just taught the material. Which I don't disagree with since, given the fact that it's a general Calc 1 course, so some if not most students aren't going to be using the proofs for such topics later in life.
That's okay, you will need to be able to do calculations too. There are people who spend all their time doing proofs and then for some odd reason can't even do basic integration. Being able to do both is important. Plus this knowledge will make dealing with other math concepts easier. It's good.
> I can completely understand that. I myself want to be as prepared as possible, even if it means going out and learning about proofs of Calc 1 topics if it helps me become a better computer scientist. I just hope that's a last resort, and my uni can at least provide foundation for such areas.
In my honest opinion, a lot of people put too much weight on calculus. Computer science is very much in line with discrete math. The areas where it gets more 'real numbery' is when you get into numerical methods, machine learning, graphics, etc. Anything related to theory of computation will probably be discrete math. If your goal is to get good at data structures and algorithms, most of your time will be spent on discrete topics. You don't need to be a discrete math genius to do this stuff, all you need is some discrete math, some calc (which you already have), induction, and the rest you can pick up as you go.
If you want to be the best you can be, I recommend trying that book I linked first to get your feet wet. After that, try CLRS. Then try TAOCP.
Do not however throw away the practical side of CS if you want to get into industry. Reading TAOCP would make you really good but it doesn't mean shit if you can't program. Even the author of TAOCP, Knuth, says being polarized completely one way (all theory, or all programming, and none of the other) is not good.
> From reading ahead in your post, is Skiena's Manual something worth investing to hone my skills in topics like proof skills? I'll probably pick it up eventually since I've heard nothing but good things about it, but still. Does Skiena's Manual teach proofing skills to those without them/are not good at them? Or is there a separate book for that?
You could, at worst you will get a deeper understanding of the data structure and how to implement them if the proof goes over your head... which is okay, no one on this planet starts off good at this stuff. After you do this for a year you will be able to probably sit down and casually read the proofs in these books (or that is how long it took me).
Overall his book is the best because it's the most fun to read (CLRS is sadly dry), and TAOCP may be overkill right now. There are probably other good books too.
> I guess going off of that, does one need a certain background to be able to do proofs correctly/successfully, such as having completed a certain level of math or having a certain mindset?
This is developed over time. You will struggle... trust me. There will be days where you feel like you're useless but it continues growing over a month. Try to do a proof a day and give yourself 20-30 minutes to think about things. Don't try insane stuff cause you'll only demoralize yourself. If you want a good start, this is a book a lot of myself and my classmates started on. If you've never done formal proofs before, you will experience exactly what I said about choking on these problems. Don't give up. I don't know anyone who had never done proofs before and didn't struggle like mad for the first and second chapter.
> I mean, I like the material I'm learning and doing programming, and I think I'd like to do at least be above average (as evident by the fact that I'm going out of my way to study ahead and read in my free time). But I have no clue if I'll like discrete math/proving things, or if TAOCP will be right for me.
Most people end up having to do proofs and are forced to because of their curriculum. They would struggle and quit otherwise, but because they have to know it they go ahead with it anyways. After their hard work they realize how important it is, but this is not something you can experience until you get there.
I would say if you have classes coming up that deal with proofs, let them teach you it and enjoy the vacation. If you really want to get a head start, learning proofs will put you on par with top university courses. For example at mine, you were doing proofs from the very beginning, and pretty much all the core courses are proofs. I realized you can tell the quality of a a university by how much proofs are in their curriculum. Any that is about programming or just doing number crunching is literally missing the whole point of Computer Science.
Because of all the proofs I have done, eventually you learn forever how a data structure works and why, and can use it to solve other problems. This is something that my non-CS programmers do not understand and I will always absolutely crush them on (novel thinking) because its what a proper CS degree teaches you how to do.
There is a lot I could talk about here, but maybe such discussions are better left for PM.
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.
To lean from a book like that of Artin's, you need to get a few basics down:
How to Prove It: A Structured Approach by Daniel Velleman
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
I'm not doing this for a class. It's just that I have been drawing a blank whenever there was a question "Show that" or "Prove that". So now I'm working through Velleman's How to prove it. There are answers for some problems but not all. Not for this one. This question is in chapter three and before that there has been covered some logic and set theory. Nothing fancy like rings and abstract stuff.
I like your suggestion that a<b<0 implies 0<-b<-a, and squaring a negative number of course gets a positive number. Wouldn't that be enough for a proof?
My two cents
If you want to improve your skills you can do two things in the short term -- read and practice.
I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
The great thing is that both include exercises.
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.
https://www.amazon.ca/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1501343615&amp;sr=8-1&amp;keywords=how+to+prove+it
Learn the information of chapter 4 in this book, in particular the stuff from page 90 to the end of the chapter
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
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
These two are great in that they kind of take a CS approach to proofs by using the idea of a pattern:
Proof Patterns https://www.amazon.com/dp/3319162497/ref=cm_sw_r_cp_apa_i_0Y44BbDC5HWYB
How to Prove It: A Structured Approach, 2nd Edition https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_C244Bb8QY01F2
This is great for understanding limitations and the history of the development of proof techniques:
Reverse Mathematics: Proofs from the Inside Out https://www.amazon.com/dp/0691177171/ref=cm_sw_r_cp_apa_i_M044BbM0BMAE3
As has been suggested, you should look into mathematical logic, modern symbolic logic, which includes propositional and quantificational logic, relational logic, and perhaps even modal logic. Copi is great for a classical treatment of modern and Aristotelian logic. Gensler is great for learning translations, and Smullyan is great for mathematical logic.
You might also look into set theory and hott or type theory, but only after you've approached some of the other stuff. I can't emphasize logic enough if you want to really understand proof theory. Perhaps even check out computability and complexity theory. A lot of these topics in theoretical computer science run into the limitations and possibilities of these and other ideas. You'll then realize as others have suggested, that yes there are some fundamentals taken for granted, but real math is difficult because of the intuition needed in approaching a proof, and the possibility that you could be working on a problem that is unanswerable.
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!
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.
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&amp;ie=UTF8&amp;qid=1331568877&amp;sr=1-1
This is the book I used at university. I thought it was pretty good. Velleman's book is also popular. I've heard good things about this book, but I've not read it.
For whichever professor you have for Math 42, I highly recommend you get this book: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It definitely saved me a ton. It’s straight to the point, and not as dry as most textbooks can be. Math 32 will be a bit more work, but in my experience just start homework early and don’t be afraid to go to professor office hours and ask questions. Even if they seem distant during class, most professors do appreciate students who make the effort to ask questions. If you need free tutoring in any of your classes, contact Peer Connections. Specifically for math, I believe MacQuarrie Hall room 221 offers drop-in tutoring for free as well! And for physics, Science building room 319 has free drop-in tutoring.
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&amp;*Version*=1&amp;*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.
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.
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.
Sounds like you are about 4 years behind me (Future physics PhD candidate). Glad to know you have discovered Dover books, they really are great and so cheap. It also sounds like you know what you're doing so good job, keep at it and you might make a good case for graduate school (if that's your destination). But I will warn you that upper division mathematics courses are different. I have seen so many people who think they are really great at mathematics up to vector calculus and then get completely shit on by more abstract courses like real analysis, abstract algebra and topology. The reason for this is that it requires more formalism and is very rigorous as far as proofs go. You'll eventually learn that math is all about making sure you have checked every possible condition in order to move on. I think something you will need is mathematical logic before you tackle abstract courses. If you do collect textbooks (like I do) then I would also recommend this textbook. It teaches you how to think like a mathematician and the logic behind proofs. I think a mathematics logic course is essential to students and it's a shame many mathematics students don't go through a formal logic course before they tackle advanced courses. Of course, some don't need it but unless you are brilliant, I would recommend it (Even if you are brilliant it would be a easy read). Just dig deep and focus and good luck with your future work. Mathematics and Physics are two beautiful subjects and it's always great to talk to future mathematicians or physicists(or any aspiring scientist in that case!) and help them get inspired or motivated!
P.S. Funny story, I had a friend who thought it would be funny to make people believe that Euler is pronounce "you-ler" with the argument that Euclid is pronounced "you-clid". It was pretty funny seeing people believe him.
If your Calculus is rusty before Rudin read Spivak Calculus it is great intro to analysis and you will get your calculus in order. Rudin is going to be overkill for you. Also before trying to do proofs read How to prove it It is a great crash course to naive set theory and proof strategies. And i promise i won't bore you with math any more.:D
If you do decide on it, definitely read this beforehand:
http://www.amazon.ca/How-Prove-It-Structured-Approach/dp/0521675995
Don't pay too much attention to the other replies - if you really want to take Math 145/146 it's possible, it will just be a lot of work.
My marks were good in high school (but not 95+) and my score on the Euclid was terrible (in order to enrol without an override you need 80+ on the Euclid). The thing to know is these courses have heavy emphasis on proofs, so the summer before coming I worked my way through the first half of a book on proofs and ended up doing relatively well in these courses.
You can certainly do it, but you have to be really dedicated.
If you are mostly looking for a book on proofs, How to Prove It might be what you want.
"I'm also sure that due to my limited educational resources, self-directed study will be a huge part. Any suggestions on which books are must reads to gain competency in CS?"
Here are a few good choices for the more theoretical areas of computing:
http://www.amazon.com/Algorithms-4th-Edition-Robert-Sedgewick/dp/032157351X/ref=sr_1_1?ie=UTF8&amp;qid=1408406629&amp;sr=8-1&amp;keywords=algorithms+4th+edition
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1408406673&amp;sr=8-1&amp;keywords=how+to+prove+it
You'll also want to look at a decent discrete mathematics book. Sadly the book I used as an undergrad was rubbish, so I don't have a good recommendation.
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.
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.
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.
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.