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Learning proofs can mean different things in different contexts. First, a few questions:

1. What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)
   
   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.
   
   
2. What's your current mathematical background?
   
   This may be especially true for things like logic and very elementary set theory.
   
   
3. What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?
   
   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.)
   
   
4. What resources are available to you for vetting your work?
   
   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.
   
   
5. Are you trying to learn the basics of mathematical proofs, or genuinely rigorous mathematical proofs?
   
   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?
   
   
6. What experience have you already had with proofs in particular?
   
   Have you had at least, for example, a geometry class that's proof-based?
   
   
7. How would you characterize your general writing ability?
   
   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.
   
   

• The book How to Prove It: A Structured Approach by Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as is How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  
• Since you've already taken calculus, it would be worth reviewing the topic using a more abstract, proof-centric text like Calculus by Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.
  
  
• In order to learn how to write mathematically sound proofs, it helps to read as many as you can find (at a level appropriate for your background and such). You can find plenty of examples in certain textbooks and other resources, and being able to work from templates of "good" proofs will help you immeasurably.
  
  
• It's like the old joke about how to get to Carnegie Hall: practice, practice, practice.
  
  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!

There would have been a time that I would have suggested getting a curriculum
text book and going through that, but if you're doing this for independent work
I wouldn't really suggest that as the odds are you're not going to be using a
very good source.

Going on the typical

Arithmetic > Algebra > Calculus

****

Arithmetic

Arithmetic refresher. Lots of stuff in here - not easy.


I think you'd be set after this really. It's a pretty terse text in general.

*****

Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

You can get both of these algebra texts online easily and freely from the search

chrystal algebra part I filetype:pdf

chrystal algebra part II filetype:pdf

I think that you could get the first (arithmetic) text as well, personally I
prefer having actual books for working. They're also valuable for future
reference. This filetype:pdf search should be remembered and used liberally
for finding things such as worksheets etc (eg trigonometry worksheet<br /> filetype:pdf for a search...).

Algebra by Gelfland

No where near as comprehensive as chrystals algebra, but interesting and well
written questions (search for 'correspondence series' by Gelfand).

Calculus

Calculus made easy - Thompson

This text is really good imo, there's little rigor in it but for getting a
handle on things and bashing through a few practical problems it's pretty
decent. It's all single variable. If you've done the algebra and stuff before
this then this book would be easy.

Pauls Online Notes (Calculus)

These are just a solid set of Calculus notes, there're lots of examples to work
through which is good. These go through calc I, II, III... So a bit further than
you've asked (I'm not sure why you state up to calc II but ok).

Spivak - Calculus

If you've gone through Chrystals algebra then you'll be used to a formal
approach. This text is only single variable calculus (so that might be calc I
and II in most places I think, ? ) but it's extremely well written and often
touted as one of the best Calculus books written. It's very pure, where as
something like Stewart has a more applied emphasis.

**

Geometry

I've got given any geometry sources, I'm not too sure of the best source for
this or (to be honest) if you really need it for the above. If someone has
good geometry then they're certainly better off, many proofs are given
gemetrically as well and having an intuition for these things is only going to
be good. But I think you can get through without a formal course on it.... I'm
not confident suggesting things on it though, so I'll leave it to others. Just
thought I'd mention it.

****

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

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.

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.

It only goes up through pre-calc. I wish it went to calculus.

I looked around for ways to study calculus that wouldn't blow my brains out. After a while I learned that calculus is kind of two things: applications of trig and vectors and limits and curves on one side, and real analysis on the other. Turns out I like the analysis part quite a bit, not the other part.

I found a couple of fabulous resources for calculus to prepare for higher math. Someday I truly hope to return to these so I can explore higher math on my own, but for now my focus is finishing school.

• Intro to Analysis PDF I was told by someone else here who knows a lot of higher math "wow that's a fantastic PDF" so maybe you'll enjoy it.
  
  
• Calculus Made Easy re-released text from about 100 years ago. Read the first chapter, seriously, it looks like an awesome book. It's on my nightstand, hope to get to it soon.

> To be honest, I do still think that step 2 is a bit suspect. The inverse of [;AA;]is [;(AA)^{-1};] . Saying that it's [;A^{-1}A^{-1};] seems to be skipping over something.

I realized how right you are when you say this after I reread the chapter on Inverse Matrices in my book. I am using Introduction to Linear Algebra by Gilbert Strang btw. I'm following his course on MIT OCW.

The book saids: If [;A;] and [;B;] are invertible then so is [;AB;]. The inverse of a product [;AB;] is [;(AB)^{-1}=B^{-1}A^{-1};].

So, before I went through with step two, I would have to have proved that [;A;] is indeed invertible.

>Their proof is basically complete. You could add the step from A2B to (AA)B which is equivalent to A(AB) due to the associativity from matrix multiplication and then refer to the definition of invertibility to say that A(AB) = I means that AB is the inverse of A. So you can make it a bit more wordy (and perhaps more clear), but the basic ingredients are all there.

I will write up the new proof right here, in its entirety. Please let me know what you think and what I need to fix and/or add.

Theorem: if [;B;] is the inverse of [;A^2;], then [;AB;] is the inverse of A.

Proof: Assume [;B;] is the inverse of [;A^2;]

1. Since [;B;] is the inverse of [;A^2;], we can say that [;A^2B=I;]
   
   
2. We can write [;A^2B=I;] as [;(AA)B=I;]
   
   
3. We can rewrite [;(AA)B=I;] as [;A(AB)=I;] because of the associative property of matrix multiplication.
   
   
4. Therefore, by the definition of matrix invertibility, since [;A(AB)=I;], [;AB;]is indeed the inverse of [;A;].
   
   Q.E.D.
   
   Do I have to include anything about the proof being correct for a right-inverse and a left-inverse?
   
   > That's a great initiative! Probably means you're already ahead of the curve. Even if you get a step (arguably) wrong, you're still practicing with writing up proofs, which is good. Your write-up looks good to me, except for the questionability of step 2. In step 3 (and possibly others) you might also want to mention what you are doing exactly. You say "therefore", but it might be slightly clearer if you explicitly mention that you're using your assumption. You can also number everything (including the assumption), and then put "combining statement 0 and 2" to the right (where you can also go into a bit more detail: e.g. "using associativity of multiplication on statement 4").
   
   I haven't began my studies at university yet, but I sure am glad that I exposed myself to proofs before taking an actual discrete math class. I think that very few people get exposed to proof writing in the U.S. public school system. I've completed all of the Khan Academy math courses, and the MIT OCW Math for CS course is still very difficult. I basically want to develop a very strong foundation in proof writing, and all the core courses I will take as a CS major now, and then I will hopefully have an easier time with my schoolwork once I begin in the fall. Hopefully this prior knowledge will keep my GPA high too. I really appreciate all the constructive criticism about my proof. I will try to make them as detailed as possible from now on.
   

I took both precalc and calc 1 back to back (and we used Stewart's calc book for calc 1-3). To be honest, concepts like limits and continuity aren't even covered in precalculus, so it isn't like you've missed something huge by skipping precalc. My precalc class was a lot of higher level college algebra review and then lots and lots and lots of trig.

I honestly don't see how you'd need much else aside from PatricJMT and lots of example problems. It may be worthwhile for you to pick up "The Calculus Lifesaver" by Adrian Banner. It's a really great book that breaks down the calc 1 concepts pretty well. Master limits because soon you'll move onto differentiation and then everything builds from that.

Precalc was my trig review that I was thankful for when I got to calc 2, however, so if you find yourself needing calculus 2, please review as much trig as you can. If you need some resources for trig review, PM me. I tutored college algebra, precalc, and calc for 3 years.

Good luck!

I've heard people recommend Kiselev's Geometry, on a physics forum. Warning, though; Kiselev's Geometry series(in English) is translated from Russian.

Here's the link to where I got all these resources(I also copy-pasted what's in the link down below; although, I did omit a few entries, as it would be too long for this reddit comment; click the link to see more resources):

https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/

__

Note: Alternatively, you can order Kiselev's geometry series from http://www.sumizdat.org/

Geometry I and II by Kiselev

http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202

http://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210

> If you do not remember much of your geometry classes (or never had such class), then you can hardly do better than Kiselev’s geometry books. This two-volume work covers a lot of synthetic (= little algebra is used) geometry. The first volume is all about plane geometry, the second volume is all about spatial geometry. The book even has a brief introduction to vectors and non-Euclidean geometry.

The first book covers:

• Straight lines
  
  
• Circles
  
  
• Similarity
  
  
• Regular polygons and circumference
  
  
• Areas
  
  The second book covers:
  
  
• Lines and Planes
  
• Polyhedra
  
• Round Solids
  
• Vectors and Foundations
  
  > This book should be good for people who have never had a geometry class, or people who wish to revisit it. This book does not cover analytic geometry (such as equations of lines and circles).
  
  ____
  
  

  Geometry by Lang, Murrow
  

  
  http://www.amazon.com/Geometry-School-Course-Serge-Lang/dp/0387966544
  
  > Lang is another very famous mathematician, and this shows in his book. The book covers a lot of what Kiselev covers, but with another point of view: namely the point of view of coordinates and algebra. While you can read this book when you’re new to geometry, I do not recommend it. If you’re already familiar with some Euclidean geometry (and algebra and trigonometry), then this book should be very nice.
  
  The book covers:
  
  

• Distance and angles
  
  
• Coordinates
  
  
• Area and the Pythagoras Theorem
  
  
• The distance formula
  
  
• Polygons
  
  
• Congruent triangles
  
  
• Dilations and similarities
  
  
• Volumes
  
  
• Vectors and dot product
  
  
• Transformations
  
  
• Isometries
  
  > This book should be good for people new to analytic geometry or those who need a refresher.
  
  > Finally, there are some topics that were not covered in this book but which are worth knowing nevertheless. Additionally, you might want to cover the topics again but this time somewhat more structured.
  
  > For this reason, I end this list of books by the following excellent book:
  
  

  Basic Mathematics by Lang
  

  
  http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
  
  > This book covers everything that you need to know of high school mathematics. As such, I highly advise people to read this book before starting on their journey to more advanced mathematics such as calculus. I do not however recommend it as a first exposure to algebra, geometry or trigonometry. But if you already know the basics, then this book should be ideal.
  
  

• The book covers:
  
  
• Integers, rational numbers, real numbers, complex numbers
  
  
• Linear equations
  
  
• Logic and mathematical expressions
  
  
• Distance and angles
  
  
• Isometries
  
  
• Areas
  
  
• Coordinates and geometry
  
  
• Operations on points
  
  
• Segments, rays and lines
  
  
• Trigonometry
  
  
• Analytic geometry
  
  
• Functions and mappings
  
  
• Induction and summations
  
  
• Determinants
  
  > I recommend this book to everybody who wants to solidify their basic knowledge, or who remembers relatively much of their high school education but wants to revisit the details nevertheless.
  
  _____
  
  More links:
  
  https://math.stackexchange.com/questions/34442/book-recommendation-on-plane-euclidean-geometry
  
  Note: oftentimes, you can find geometry book recommendations( as well as other math book recommendations) in stackexchange; just use the search bar.
  
  __
  
  https://www.physicsforums.com/threads/geometry-book.727765/
  
  https://www.physicsforums.com/threads/decent-books-for-high-school-algebra-and-geometry.701905/
  
  https://www.physicsforums.com/threads/micromass-insights-on-how-to-self-study-mathematics.868968/

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:

1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.
   
   
2. Here's a more serious intro to 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:
   
   
3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.
   
   
4. Here's an intro to proper, rigorous 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.

Giving up won't help, especially with math. So your first order of business is not letting yourself off the hook. Keep working on it, even when it seems hard and confusing. Sometimes it takes many attempts before a concept finally 'clicks.'

Khan Academy is good, but you're going to need to do additional problems to really get the work to sink in. I recommend clicking through on the hints after you've done a problem to see how they explain it. A lot of times I've found those hints to be better explanations than what is in the video.

If you can't retain the concepts, you're going to have a hard time doing CS. You might also want to look into memory aids, like Anki flashcards, to practice and reinforce ideas as you go.

I started a small notebook where I write in concept outlines so I can come back and re-read them and reinforce them as I go.

I would recommend a YouTuber named James Tanton. He has an Exploding Dots series that is really basic, but is visually very helpful. There are loads of other people uploading info on YouTube, so you can pretty much always find someone explaining what you're learning. Watch different people to see how they approach things differently, until the ideas make sense to you.

Another online resource that will provide some help as you go along is the website Better Explained, though you'll need to poke around to find what you need.

I also read a book called A Mind for Numbers by Barbara Oakley. It's not a math textbook, but it's full of advice for how to train your brain to be better at math.

Those are the things I've been using that have helped. Good luck!

wildberryskittles recommended the classics but teaching methods have improved since then in my opinion.

You should revisit algebra, geometry, and trigonometry before tackling a book like Calculus Made Easy. For algebra, Practical Algebra: A Self-Teaching Guide seems like a great place to start. After that, head on to geometry with something like Geometry and Trigonometry for Calculus. The book Precalculus Mathematics in a Nutshell might also be helpful.

The only previous knowledge I really used when I took intro to proofs were some factoring methods that were helpful with proofs by induction, although they weren't necessary. That said, reviewing exponent/log laws, and certain methods of factoring couldn't hurt.

An intro to proofs course should be fairly self contained, meaning any necessary axioms and definitions should be covered in the course. Those examples that you gave are exactly the type of things that should be proven and not knowing them beforehand should be fine. The important thing is being able to understand and reproduce the proofs on your own, and with a bit of experience you will be able to intuitively reason whether a statement is true or false. This intuitive reasoning will also become much more important than memorizing later in the course when you come across statements you've never seen before that aren't immediately obvious.

I would recommend getting very comfortable with logic and basic set theory. I also highly recommend this book if you want some extra reading material (pdf). It's still one of my favorite math books. Hope that helps.

Math is essential the art pf careful reasoning and abstraction.
Do yes, definitely.
But it may be difficult at first, like training anything that’s not been worked.

Note: there are many varieties of math. I definitely recommend trying different ones.

A couple good books:

An Illustrated Theory of Numbers

Foolproof (first chapter is math history, but you can skip it to get to math)

A Book of Abstract Algebra

Also, formal logic is really fun, imk. And excellent st teaching solid thinking. I don’t know a good intro book, but I’m sure others do.

There's a lot of orders you could study mathematics in, and it's hard to say you should definitely pick one over the others.

One thing I can say pretty assuredly, though, is you should get a good background in algebra before you do much else. It's really the backbone of everything else. You can pick a bunch of different subjects after that, but study algebra first.

There are good online resources. Khan Academy is pretty good, as is Alcumus and Purple Math. Khan Academy has tests, and Alcumus is basically a big test.

Personally, though, I've learned way more from good books like this one than I tend to learn from websites.

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!

I'm in a similar process with software development, I've always had the passion, but mathematics always stopped me, now I'm very motivated!

​

These are the resources that have helped me, I hope and work for you

​

https://www.youtube.com/user/professorleonard57

The best teacher!

​

https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/dp/039916524X/ref=sr_1_2?crid=16HTDVAQL50FX&keywords=barbara+oakley&qid=1572824841&s=books&sprefix=barbara+o%2Cstripbooks-intl-ship%2C210&sr=1-2

The best book

​

http://www.math-aids.com/

Exercise page

​

And of course

Khan Academy!

Honestly, I highly recommend this book, and pretty much anything else by Arthur Benjamin. He's the real deal when it comes to mental math. Take it seriously, and do tons of practice problems. Feel free to go "fast" through the book the first time through, but go super slow the second time through and get everything super solid.

After completing the book you'll be able to do squares, multiplication, division, addition, subtraction pretty damn fast up to around 3-4 digits. With more practice you can eventually get as good as Prof. Benjamin (he doesn't leave anything out! Tells you the entire technique). By more, I mean years more, but hey, at least it's possible

Besides what has been said here, why don't you ask your parents to purchase you a fun math textbook? You'll have to do some research but why not just have some initiative and pick up your own algebra textbook and learn at your own pace? Maybe you might in interested in the Art of Problem Solving Series. You have an entire school of math teachers to ask for help if you get stuck somewhere. You have the internet (here being one of the places you can ask about anything math related; StackExchange is another good place). If I recall correctly, you can even "enroll" in online courses using edX that you can do on your own time. I often recommend to people Basic Mathematics because it covers everything that you should know math-wise before college. Some of the material might be advanced to you now but you can work through the book easily if what you claim about knowing all the material for your class is true.

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)

I graduated w/ degree in Math n' Physics but have been doing programming for startup for last 5+ years so many of my math skills got rusty.

While trying to get back into it went through several books and have found this to be the best if you're interested in more advanced mathematics: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094. It's not only been an excellent review but has fleshed out some areas I was weak (in higher level courses like complex analysis, topology, group theory the methodology of proofs was assumed and often not taught).

The explanations are solid, varied, and they go through each proof they present (often w/ exhaustive step-by-step details).

From there pick a domain you're interested in and pickup the relevant undergraduate (and maybe some graduate) level books/textbooks and see if you can pick it up.

Yeah either of those are easier. I don't like Fraleigh cause I think it lacks motivation (also the chapters on splitting/separable fields really suck) but I love Herstein. If you're set on cheap, this guy ain't too bad. If I were self studying though I would try to find a cheap older edition of Artin, as he's very example motivated, and it can sometimes be hard to wrap your head around all the abstraction without a class.

EDIT: Also you might want to find a cheap number theory text, since elementary number theory is probably the most accessible way to see groups and rings in action. And for "how do I prove xxx" questions I always recommend starting with this.

If you have a chance, I recommend checking out some textbooks on real analysis, which will guide you through the derivations and proofs of many theorems in calculus that you've thus far been expected to take for granted.

Some would recommend starting with Rudin's Principles of Mathematical Analysis, and it's certainly a text that I plan to read at some point. For your purposes, I might recommend Spivak's Calculus since it expects you to rigorously derive some of the most important results in calculus through proof-writing exercises. This was my first introduction to calculus during high-school. While it was overwhelming at first, it prepared me for some of my more advanced undergraduate courses (including real analysis and topology), and it seems to be best described as an advanced calculus textbook.

For classes like number theory and abstract algebra, I would suggest just picking up a book and attempting to read it. It will be hard, but the main prerequisite for courses like this is some mathematical maturity. That only comes with practice.

Realistically there is probably no preparation that you could have which would prepare you in such a way that a book on advanced mathematics would be super easy.

I like this number theory book
http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1348165257&sr=8-1&keywords=number+theory

I like this abstract algebra book
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_2?s=books&ie=UTF8&qid=1348165294&sr=1-2&keywords=abstract+algebra

> don't think that there is a logical progression to approaching mathematics

Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.

> Go to the mathematics section of a library, yank any book off the shelf, and go to town.

I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.

My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941

That is more than a summer's worth of work.

Sorry, agelobear, to be such a contrarian.

None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.

I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!

Run.

No, I'm only joking. I just took Calc, and the Calculus part is relatively easy, it's the Algebra/Trig components that trip people up. I'm not saying Calculus is easy, you'll still have to put the work in. But review algebra II and Trigonometry because they will make things a lot easier.

The following isn't a comprehensive list, just the stuff that I remember I had to know.

• For algebra, review polynomial operations, functions and graphing, exponents and radical rules.
  
  
• For trig, review the unit circle, how to find triangle angles and sides (particularly 45 degree and 30 degree angles), fundamental trig identities, and sin, cos, tan, etc.
  
  This was the book my school recommended for brushing up on Algebra and Trig
  
  For extra help, Prof Leonard and Patrickjmt were immeasurably helpful. They explained concepts much better than my professor.
  
  Good luck!
  
  

The best way to learn is take the class and find your deficiencies. Khan Academy is also great to get a base line of where you are. If you need help with calc. And precal, calculus lifesaver book is good.
lifesaver calculus amazon

I had to deal with the no internet thing for some time.
Find some place with free wi-fi(you are using phone?).
Download ebook/pdf reader, FBreader + PDF plugin is good (Assuming that you are using Android phone).
Install Firefox and this add-on Save Page WE, it also work for phones (tested with Android).

Then you can save pages from some of these web sites or Wikipedia:

• mathsisfun
  
  
• purplemath
  
  
• sosmath
  
  
• tutorial.math.lamar
  
  
• themathpage
  
  
  
  
  
  Poor man's library:
  
  http://gen.lib.rus.ec/
  http://b-ok.org/
  https://archive.org/
  
  Some books:
  
  [Harold Jacobs - Elementary Algebra] (https://www.amazon.com/Elementary-Algebra-Harold-R-Jacobs/dp/0716710471) (that's the most friendly and still not bullshit introduction I was able to find)
  
  [Basic Mathematics by Serge Lang] (https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877) - Make sure that you understand the first 100-150 pages from the first book if you want to start this one.
  
  Precalculus by David Cohen - Lang's Basic Mathematics is harder (and much more interesting) than this...
  
  Use the three web sites written above to download them. PDF for better performance (and some times quality) on cheap phone, epub/djvu for the smaller size.
  
  I think that free wi-fi will be easy to find. Reading books wouldn't be like watching videos but you will actually learn more. Compass, straightedge and protractor would be really useful.
  

I'm currently working through Munkres' book on Topology, and I am using the video lectures found here. I know these are in an annoying form factor, but, trust me, these are the only videos that go into any depth you will find on the internet. They use Munkres, too, which is a plus.

On the same site are video lectures for Algebraic Topology. For this, I definitely recommend buying Artin's "Algebra" (1st edition can be found cheaply, and I don't think there's really any significant difference from 2nd), and watch these video lectures by Harvard. Then, you can finally move on to the Algebraic Topology video lectures which uses the free textbook "Algebraic Topology" by Allen Hatcher.

Hope this helps.

Do not enroll in a precalculus class until you have a solid grasp on the foundations of precalculus. Precalculus is generally considered to be the fundamentals required for calculus and beyond (obviously), and a strong understanding of precalculus will serve you well, but in order to do well in precalculus you still need a solid understanding of what comes before, and there is quite a bit.

I do not mean to sound discouraging, but I was tutoring a guy in an adult learning program from about December 2017-July 2018...I helped him with his homework and answered any questions that he had, but when he asked me to really get into the meat of algebra (he needed it for chemistry to become a nurse) I found a precalculus book at the library and asked him to go over the prerequisite chapter and it went completely over his head. Perhaps this is my fault as a tutor, but I do not believe so.

What I am saying is that you need a good foundation in the absolute basics before doing precalculus and I do not believe that you should enroll in a precalculus course ASAP because you may end up being let down and then give up completely. I would recommend pairing Basic Mathematics by Serge Lang with The Humongous Book of Algebra Problems (though any book with emphasis on practice will suffice) and using websites like khanacademy for additional practice problems and instructions. Once you have a good handle on this, start looking at what math courses are offered at your nearest CC and then use your best judgment to decide which course(s) to take.

I do not know how old you are, but if you are anything like me, you probably feel like you are running out of time and need to rush. Take your time and practice as much as possible. Do practice problems until it hurts to hold the pencil.

If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.

I've read some good reviews of Basic Mathematics by Serge Lang. It should prepare the reader for calculus.

Otherwise, many online and free books are already available. Here you find a list of free books approved by the American Institute of Mathematics.

If you want to understand the WHY, then you need to read proofs and at least be familiar with basic concepts of logic. I've found this site really helpful. It's a source for definitions and proofs.

I once took a first-year course in logic, starting with the propositional calculus. All these years later, I still regard it as the most important thing I ever did. Proof-writing became [almost] easy after that. It wasn't always easy to put the pieces together, but at least I had a blueprint. I knew that if I could clearly define a contrapositive, or understand how set identities like DeMorgan's Laws were constructed, I was on much firmer ground. I highly recommend Discrete Mathematics and Its Applications. It's such an enormous and comprehensive text, in so many subjects, that I found myself referring back to it, for something, in almost every class of my undergraduate.

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.

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 is a great start. I think after that, you should focus on learning to think mathematically through practice instead of reading (at least, that's how I and most people learn best). Take classes or read and work through the textbooks of subjects that interest you. Discrete math would be a good place to start since it teaches proof techniques and basic probability and combinatorics; my class used this book which I thought was nice.

If you don't actually do the work, your thinking process isn't going to change.

Check out /r/compsci, /r/algorithms, and the subreddits in their sidebars.

Does this appear to be the same book to you?

Same main title, and guy - but a different subtitle, this looks more like the older one you linked, but if I can save a buck or two... d=

since you are a computer science student, you can start with proofs in Discrete Mathematics fo this you can look at Kenneth Rosen's book, it can help you with a lot of basic concepts, constructing proofs. Its a good book for those who want to go in algorithms or theoretical cs or a even want to work on pure maths problems. I had this same confusion I wanted to do maths but also cs with it. After this you can try "The art of computer programming"(this has 4 volumes) by Donald Knuth but CLRS is a must along with Rosen's if you want to take cs and maths side by side. If you want to explore further you can look at Design of Approximation Algorithms and Randomised Algorithms. These book can help you with concepts of probability, number theory, geometry, linear algebra etc. But then if you want pure math problems then search for them, go though different journals, SIAM and Combinatorica are really good ones, search them pick a problem you like, then find text relevant to problem and try to give better solutions.

I also recommend this timeless classic, Calculus Made Easy by Sylvanus P. Thompson. It's really great for developing intuition.

Source: I learned calc while doing precalc, just like OP is planning.

Gelfand's Algebra is interesting, encourages mathematical thinking, and has the additional advantage of being much more approachable than the books you've listed.

This is probably a much better place to start for someone who's interested in "starting from the basics."

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.

The Serge Lang book looks to be pretty expensive on Amazon, is it worth it?

Thank you for the recommendations, the Gelfand books look like they're worth checking in to!

This book and his videos: https://www.amazon.com/Calculus-Lifesaver-Tools-Princeton-Guides/dp/0691130884

I was good at calculus, but this book made anything I struggled to fully understand much easier. He does a good job of looking back at how previous work supports and and talks about how this relates to future topics.

For the very basics (and more), I can highly recommend you Professor Leonard on YouTube.

>What books would you recommend?

How about doing your own research?

Google.com -> book site:reddit.com/r/learnmath



Anyways, take a look at Basic Mathematics by Serge Lang. This is what I'm learning with right now, it's really great.

Mathematics, a learning map

Edit:

Ehm, or take a look at your own thread from a year ago.

https://www.reddit.com/r/learnmath/comments/46xdpp/learning_math_from_scratch_all_by_myself/

To piggy back off of danielsmw's answer...

> Fourier analysis is used in pretty much every single branch of physics ever, seriously.

I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.

If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.

A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.

If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.

Random links which may spark your interest:

• Fourier analysis also finds applications in number theory, for example in the Hardy-Littlewood circle method.
  
• Solving cubic equation using discrete Fourier transform
  
• The Wavelet transform
  
• Fourier optics
  
• Monstrous moonshine
  
• Fractional Laplacian/calculus

Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)

For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).

>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?

Read textbooks for mathematics students.

For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.

There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.

I highly recommend reading "Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand et. al. It helped me get a better understanding of how to write a proof as well as organize my own thoughts.

Here's the Amazon link: Mathematical Proofs: https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX

As long as you have a solid foundation in algebra (and basic trig), you should be fine. However, you have to put in the study time. If you want supplementary material, I'd recommend The Calculus Lifesaver, which was a tremendous help for me, although it only covers single-variable calculus (i.e., Calc I and II). The cool thing about this book is that its author (a Princeton University professor) also has video lectures posted online.

William Dunham has a great book,Journey through Genius: The Great Theorems of Mathematics, about this.

This book seems to have pretty good reviews

and this may also prove useful as well.

1. Go on Amazon, get a previous edition pre-calculus book, and work through the problems.
   
   https://www.amazon.com/gp/aw/ol/0495392766/ref=mw_dp_olp?ie=UTF8&condition=all
   
   
2. Also on Amazon 'Pre-calculus in a Nutshell'
   
   https://www.amazon.com/gp/aw/d/1592441300/
   
   
3. Khan Academy
   
   https://www.khanacademy.org/math/precalculus
   
   But, I assure you, you'll almost certainly have better luck in a structured class.
   
   
   
   

If you are serious about this, then the best way to self learn Math is to learn to read Math books. This is a valuable skill. It stops you from having to rely on websites/tutorials and frees you to really read the stuff you're interested in.

Generally you probably want a more "back to basics" approach that will cover basic stuff and act as an introduction (again) to the topic (without handling you as if you're a child). I recommend Discrete Mathematics with Applications. Epp does a good job of starting at the beginning (with logic) and building a decent foundation through connectives, conditionals, existentials, universals, etc. eventually leading into proofs.

Her writing style is very readable IMO but still dense enough to help you learn how to read Math books.

If you're self motivated enough then start there. Read a chapter. Do the problems. Be confused. Do more problems. Still confused? Read the chapter again. Do more problems. Repeat. Eventually finish the book.

The next one will be faster and easier because of the work you put in. Eventually you'll be 3-5 books down, and you'll feel you know quite a bit. Then read more.. realize the field is huge and you know nothing. Read more to solve this.

Repeat

????

Success(?)

For a good resource for reviewing precalculus and trig, I recommend Just-in-Time Algebra & Trigonometry for Calculus by Guntram Mueller and Ronald Brent. This book covers all of the precalc and trig topics you'll need to know for calculus, and it even presents them in the order you'll probably need to know them as you go through a calculus course, so you can work through this book as you progress through calculus.

I recommend proofs. How To Prove It is a good introduction, but there are also free resources available online.

Calculus Made Easy by Silvanus Thompson and edited by Martin Gardner is one of the best introductory texts for people without a calculus background. Highly recommend.

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.

Discrete math gives you a good foundation for learning higher math.

This one is a good book:

Discrete Mathematics with Applications by Susanna Epp.

It deals with logic, sets, relations, counting, proofs etc...you know the stuff you need in higher math. Seeing as you're a CS student, I am guessing Discrete Math would be much more important to you than Calculus.

You need Susanna Epp's Discrete Math. Her stuff can be ever so slightly rough at certain spots, but absolutely phenomenal everywhere else. But there's Book of Proof by Richard Hammack[FREE] to help you overcome these difficulties.

For a slightly different take on essentially the same material you could also try Mathematics: A Discrete Introduction by Scheinerman. Also a very gentle book.

Okay. So..


You speak of this book I assume. Which is intended to be used by students in H.S. Yet you are familiar with abstract algebra? I understand abstract algebra has many levels to it. But how far did you go? Was it so close that you were touching on topographies or statements?

I'm very confused here. You're concerned about your math. But yet you're reading a calculus prep book?

What is an IT college exactly? Are you a freshman or sophomore at a Uni? And it happens that you are referring to your department? Or are you referring to a technical college / school?

These questions are to satisfy my assumptions. Optional at best.



As a math major with a CS minor in my uni, which is something I'm in the process of. I am required pre-algebra, algebra, pre-calc, calc, calc 1, calc 2, calc 3, abstract algebra, linear algebra, discreet math, some general programming classes involving these prerequisite math courses, and some other math classes I cannot remember.

Abstract algebra, in my opinion is something of a higher level language. So this should explain my confusion here.

If you are interested in the history of mathematics I highly suggest reading Journey through Genius: The Great Theorems of Mathematics. It's a wonderful book that gives you insight about the persons behind the maths we take for granted today and what tools they had available at the time.

Your specific questions about ellipses and hyperbolas is not covered but parabolas and cones are mentioned.

Khan Academy is a good resource - I would also recommend the book A Mind For Numbers. The fact that math makes you anxious, while perfectly normal, is going to be a mental block when it comes to learning it. Math is like any other skill - you keep practicing, you get it wrong a bunch of times, and gradually your brain retrains a small portion of itself to do it. This process can be interfered with - you'll never learn a sport if you don't eat or sleep enough, and you won't learn a mental skill if your brain is subjected to enough stress.

You absolutely are capable of grasping the math. I do hope you believe this, and if there is any doubt I hope the book will give you some mental tools for managing that anxiety, as well as some high-level approaches on how to approach mathematical problems.

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.

From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.

Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.

From the link:

For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:

Solving inequalities and equations

Properties of functions

Composite functions

Polynomial functions

Rational functions

Trigonometry

Trigonometric functions and their inverses

Trigonometric identities

Conic sections

Exponential functions

Logarithmic functions

For Calc 2(applied) - I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:

Sequences and series

Binomial theorem

In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:

Vectors

Parametric equations

Polar coordinates

Matrices and determinants

As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.

https://www.amazon.com/Precalculus-Mathematics-Nutshell-Geometry-Trigonometry/dp/1592441300 is a good start. Reading through the principia mathematica could also be useful. https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica

/u/cm362084 already recommended The Millennium Problems by Keith Devlin, which is literally exactly what you're looking for. If you're interested in other great books about math, 2 I'd recommend are Journey through Genius by William Dunham and Fermat's Enigma by Simon Singh.

Journey through Genius is organized such that every other chapter is some important proof (detailed out step-by-step), and the remaining chapters provide the historical/biographical context for those proofs. There are some interesting stories included in the book such as how mathematicians in the middle ages would keep their techniques secret, since there was a chance that another mathematician would come to town and challenge them to a math duel.

Fermat's Enigma tells the story of how Andrew Wiles was able to prove Fermat's Last Theorem, which states that there are no integer solutions to the equation a^n + b^n = c^n for n>2. (This one was a century problem last century, but since it was solved, there was no need to list it as a Millennium Problem.) This is a bit more storytelling than actual math, though Singh doesn't shy away from going a little bit into detail about the underlying math.

The last book to consider is The Poincare Conjecture by Donal O'Shea. The Poincare Conjecture was one of the Millennium problems and was recently solved. I should point out that I can't recommend this book personally because too much of it went over my head. That says more about me than the book, though, so I don't want to leave it off the list just because I was too dumb to get it. :) I never took any classes in topology, so I may want to read up on that and give this book another shot.

Secrets of Mental Math May be helpful for filling in some gaps. Also A Mind for Numbers gives helpful meta learning info: how to study, etc.

“How to Prove it”. D. Velleman: Amazon US Link

Probably the best resource on the topic!

If you can read through gallian's book, I consider dummit and foote's book (http://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) as the best math textbook i've ever read. tons of examples, thorough treatment of material, and tons of exercises.

Start with 3 Blue 1 Brown's Essence of Calculus Series - https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

and follow the following books -

Calculus by Spivak - https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

Calculus Made Easy - http://calculusmadeeasy.org/

Follow all the concepts and solve the examples and exercises.

Feel free to ask the questions here or in mathsoverflow.

Last but not the least, PRACTICE, PRACTICE, PRACTICE........!

Spivak's Calculus is a great resource that I used for a real analysis class. The first exercise is something on par with proving that 1+1=2 and it goes on to build all of Calculus from there.

In addition to Khanacademy, this book is pretty good:

http://www.amazon.com/Precalculus-Mathematics-Nutshell-Geometry-Trigonometry/dp/1592441300/

You might want to look at the books by I. M. Gelfand as well (The Method of Coordinates, Algebra, and Trigonometry). If you do well with the Simmons, go through those books, which are more difficult but focus more on rigor.

If you give yourself a year to get up to speed on all that, nothing should stop you from going on to calculus.

For highschool level math I reccomend i.m.gelfands books, one of which is Algebra.

They're excellent for self-study, and provide you with many insights not found elsewhere afaik.

If you loved math before, don't let some bad grades convince you you're bad at it. Math isn't that hard to study on your own, without stressing out about what someone else thinks about your progress. If you're interested in some books to go with Khan Academy, I'd check these out:

Free online:

• http://openstax.org/details/algebra-and-trigonometry
  
• http://openstax.org/details/precalculus
  
• http://www.stitz-zeager.com/
  
• http://www.mecmath.net/trig/
  
  Dead tree books that cost money:
  
  
• Axler, Algebra and Trigonometry: http://amazon.com/dp/047047081X
  
• Stroud and Booth, Engineering Mathematics: http://amazon.com/dp/0831133279
  
• Gelfand and Saul, Trigonometry: http://amazon.com/dp/0817639144
  
  Note that in the US system, often "Algebra and Trigonometry" and "Precalculus" are taught out of the same book with the first course covering the earlier chapters and the second covering the later chapters. In some cases (like OpenStax) you'll see that there are separate books for the two subjects but most of the chapters are the same, with A&T including a bit of extra basic material and precalc including limits at the very end.
  

Friendly info:

"College Algebra" = Elementary Algebra.

College Level Algebra = Abstract Algebra.

Example: Undergrad Algebra book.

Example: Graduate Algebra book.

As others have said, learning proofs first is the way to go. Book of Proof is a free online book you could try working through.

If you're set on abstract algebra for some reason, A Book of Abstract Algebra seems perfect for your level. You could try to dive right into this if you wanted, but if so you should do the appendix first (Book of Proof would be a more in depth treatment of the content in the appendix of this book).

Pick up Calculus Made Easy .. check out its reviews on amazon. it will help you actually understand calculus instead of just memorizing techniques.

I took a Discrete Mathematics class in College, and this was the textbook the professor recommended:

http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=la_B001IGOE0C_1_1/189-4818032-4394023?s=books&ie=UTF8&qid=1382239523&sr=1-1

I can't honestly say I ever touched it, because the class was actually very easy, and you could study for it using only the professor's lecture notes.

MIT's OCW also has some material available (it includes video lectures, assignments and a textbook):

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010

If you're really ambitious, try this book by I. M. Gelfand:

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144

It will give you a deeper understanding than most trig books.

And here's the free version:

https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX

I read this book a few years ago, and it is pretty much the way I do any basic arithmetic in my head now. http://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=sr_1_1?ie=UTF8&qid=1333153637&sr=8-1

I find Gilbert Strang's Introduction to Linear Algebra quite accessible, and seems to be aimed towards the practical (numerical) side of things. His video lectures are also quite good, IMHO.

Mathematical Proofs: A Transition to Advanced Mathematics was my undergrad discrete text and I've gone back to it for review over the years, I think it's a very fine text especially for an introduction to proofs.

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

This book was very helpful to me.

Whoa, great questions, but I think you want a textbook, not a reddit post response. I used Dummit & Foote but it is probably a bit "heavier" than what you want/need at this point.

I've heard some good things about Michael Artin's book (http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635).

Here is a good book on trigonometry.

Here is one for algebra.

Here's another

How to Prove It is a nice introduction to writing proofs.

To learn basic proof writing I highly recommend How to Prove It by Velleman.

From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:

An Introduction to Graph Theory

Introduction to Topology

Coding the Matrix: Linear Algebra through Applications to Computer Science

A Book of Abstract Algebra

An Introduction to Information Theory

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.

This book gives a quick review of the important precalculus topics. Reading that carefully, and supplementing it with Khan Academy is what I'd do in your situation.

Working your way through a beginning discrete math class is kind of an overview of the history of math. But here are some stand-alone books on it. Writing quality varies.

The World of Mathematics

A History of Mathematical Notation. Warning: his style is painful.

Journey Through Genius

The Princeton Companion to Mathematics. A reference book, but useful.

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.

There are these videos and there is also this book. The book is better if you struggled the first time, and it includes a short section on number theory.

A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.

Is this the book you are recommending?

Why shouldn't everyone just begin with Baby Rudin?

https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Here you go OP.

I read this book https://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X and programming helped!

A mind for numbers. Give this a read.

https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra/dp/039916524X/ref=asc_df_039916524X/?tag=hyprod-20&linkCode=df0&hvadid=312710253827&hvpos=1o1&hvnetw=g&hvrand=9888540244076994554&hvpone=&hvptwo=&hvqmt=&hvdev=m&hvdvcmdl=&hvlocint=&hvlocphy=9003609&hvtargid=pla-404584213199&psc=1&tag=&ref=&adgrpid=61681020945&hvpone=&hvptwo=&hvadid=312710253827&hvpos=1o1&hvnetw=g&hvrand=9888540244076994554&hvqmt=&hvdev=m&hvdvcmdl=&hvlocint=&hvlocphy=9003609&hvtargid=pla-404584213199

How to Prove it by Velleman seems to be right up your alley.

This book is golden

https://www.amazon.com/dp/0817636773/

https://www.amazon.com/dp/B000HMQ9VU/

https://www.amazon.com/Precalculus-Mathematics-Nutshell-Geometry-Trigonometry/dp/1592441300/ref=sr_1_1?s=books&ie=UTF8&qid=1539990788&sr=1-1

http://www2.clarku.edu/~djoyce/trig/

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.

For getting more intuition on proofs I would suggest the following book
http://www.amazon.com/Nuts-Bolts-Proofs-Third-Introduction/dp/0120885093/ref=sr_1_1?ie=UTF8&qid=1311007015&sr=8-1

I think Rudin might be really tricky at your level, you can keep with it if you want, but I think Calculus by Michael Spivak would be much more approachable for you.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1311007057&sr=1-1