(Part 3) Best mathematics books according to redditors
We found 7,687 Reddit comments discussing the best mathematics books. We ranked the 2,734 resulting products by number of redditors who mentioned them. Here are the products ranked 41-60. You can also go back to the previous section.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
They've been very bad since at least the 1970s, running one ludicrous political article in every issue, the first of the "real" articles. I watched over several years as the Kosta Tsipis group, I think it was, steadily decreased their claims of what was beyond the foreseeable future state of the art. They were so bad, they effectively said the Mount Palomar Hale Telescope's 1948 mirror was ... beyond the foreseeable future state of the art. Someone pointed out that you and I could in an afternoon, using car batteries in a small alley or the like, construct a power system they said that was ... beyond the foreseeable future state of the art (of course, that wouldn't be space rated). Really, utterly sloppy, no wonder he doesn't have a Wikipedia entry.
Someone who I consider to be reliable, Petr Beckmann, perhaps best widely known as the author of A History of pi, was a refugee from Soviet occupied Czechoslovakia (and Nazi occupied, he related getting out from a first run showing of Fantasia to learn that the British and French had sold out his country in Munich). He said in his Access to Energy newsletter that you could predict the topic of this political Scientific American article roughly 6 months in advance based on what another anti-West and science and technology Communist Czech journal published. Can't read the language, but it's a very falsifiable claim, and he didn't make up bullshit.
Statistical Rethinking: https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFrhghGJ4K25bFc
Also has the book: https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445
I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.
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These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
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The other two I've been looking at which seem to be a lot more complex are
And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
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This is from Paolo Aluffi's excellent Algebra: Chapter 0, which uses categories as a unifying theme.
A groupoid is a small category in which every morphism is an isomorphism. An automorphism of an object A of a category C is an isomorphism from A to itself. The set of automorphisms of A is denoted Aut_C(A).
Edit: added that groupoids are small categories (thank you cromonolith)
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.
If you're planning on learning haskell (you should :D), why not do a book that teaches you both discrete maths and haskell at the same time?
There are atleast two books that do this:
Both books use Haskell in an effective way to teach you discrete mathematics. (They don't assume prior knowledge of Haskell).
I've been working through the 2nd book and posting my notes and solutions here . You could compare solutions/notes and discuss stuff with me.
The Haskell Road to Logic, Maths and Programming:
http://www.amazon.com/Haskell-Logic-Maths-Programming-Computing/dp/0954300696
Depends on what kind of math you are looking for. For example, there is a middle school outreach program called Bootstrap World which is about teaching algebra using functional programming. You could take a look at their materials.
If you're looking for university-level math, there are some books like The Haskell Road to Logic, Maths, and Programming. I haven't read it, but I think it covers discrete math sort of topics.
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
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.
Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.
If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.
While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the why-didn't-I-think-of-that trick that solves it.
Please, simply disregard everything below if the info is old news to you.
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Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
A Friendly Introduction to Group Theory by Nash
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Rings and Factorization by Sharpe
Linear Algebra: Step by Step by Singh
As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:
Understanding Analysis by Abbot
The Real Analysis Lifesaver by Grinberg
A Course in Real Analysis by Mcdonald/Weiss
Analysis by Its History by Hirer/Wanner
Introductory Topology: Exercises and Solutions by Mortad
In the case of this paper, it's referring to dimensions in a mathematical sense, not a physical "space-like" or "time-like" sense. In that regard, the more abstract mathematical notion of "dimension" is used all the time to describe things on a computational level that most people wouldn't associate with their idea of "dimension". For example, a picture on the computer can be thought of as a single point in some extremely high dimensional space (Im talking on the scale of millions of dimensions).
Personally, I'd find a more interesting occult correlation between the neural network structure shapes being directed/undirect simplices. If anyone is curious about learning about some of the mathematics behind those sorts of structures (called graphs) I'd recommend Introduction to Graph Theory by Dover books on the subject. It's a great introduction and has a great preface on the subject of mathematics.
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.
Sure! I have a lot of resources on this subject. Before I recommend it, let me very quickly explain why it is useful.
Bayes Rule basically means creating a new hypothesis or belief based on a novel event using prior hypothesis/data. So I am sure you can already see how useful it would be in medicine to think about. The Rule(or technically theorem) is in fact an entire field of statisitcs and basically is one of the core parts of probability theory.
Bayes Rule explains why you shouldn't trust sensitivity and specificity as much as you think. It would take too long to explain here but if you look up Bayes' Theorem on wikipedia one of the first examples is about how despite a drug having 99% sensitivity and specificity, even if a user tests positive for a drug, they are in fact more likely to not be taking the drug at all.
Ok, now book recommendations:
Basic: https://www.amazon.com/Bayes-Theorem-Examples-Introduction-Beginners-ebook/dp/B01LZ1T9IX/ref=sr_1_2?ie=UTF8&qid=1510402907&sr=8-2&keywords=bayesian+statistics
https://www.amazon.com/Bayes-Rule-Tutorial-Introduction-Bayesian/dp/0956372848/ref=sr_1_6?ie=UTF8&qid=1510402907&sr=8-6&keywords=bayesian+statistics
Intermediate/Advanced: Only read if you know calculus and linear algebra, otherwise not worth it. That said, these books are extremely good and are a thorough intro compared to the first ones.
https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=sr_1_1?ie=UTF8&qid=1510402907&sr=8-1&keywords=bayesian+statistics
https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573/ref=sr_1_12?s=books&ie=UTF8&qid=1510403749&sr=1-12&keywords=probability
Bayesian Data Analysis and Hoff are both well-respected. The first is a much bigger book with lots of applications, the latter is more of an introduction to the theory and methods.
Books:
"Doing Bayesian Data Analysis" by Kruschke. The instruction is really clear and there are code examples, and a lot of the mainstays of NHST are given a Bayesian analogue, so that should have some relevance to you.
"Bayesian Data Analysis" by Gelman. This one is more rigorous (notice the obvious lack of puppies on the cover) but also very good.
Free stuff:
"Think Bayes" by our own resident Bayesian apostle, Allen Downey. This book introduces Bayesian stats from a computational perspective, meaning it lays out problems and solves them by writing Python code. Very easy to follow, free, and just a great resource.
Lecture: "Bayesian Statistics Made (As) Simple (As Possible)" again by Prof. Downey. He's a great teacher.
So off the top off my head, I can’t think of any courses. Here are three books that include exercieses that are relatively comprehensive and explain their material well. They all touch upon basic methods that are good to know but also how to do analyses with them.
But there are many many others.
Then there are plenty of tutorials to python, R or how to handle databases (probably the core programming languages, unless you want to go the GUI route).
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
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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.
The most important thing you can do is memorize the definitions. I mean seriously have them down cold. The next thing I would recommend is to get another couple of analysis books (go cheap by getting old books, it isn't like the value of epsilon has changed over the past two hundred years) and look at their explanations, work those problems. Having a different set can be enlightening. Be prepared to spend a lot of time on it all.
Good luck!
EDIT: Back home now and able to put in some specific books. I used Rosenlicht and you wouldn't believe how happy I was to buy a textbook that, combined with a slice of pizza and a coke, was still less than $20. One of my books that I looked at for a different view point was Sprecher.
I also got a great deal of value out of Counterexamples in Analysis because after seeing things go wrong (a function that is continuous everywhere but nowhere differentiable? Huh?) I started to get a better feel for what the definitions really meant.
I hope you're also sensing a theme: Dover math books rock!
Algebra
Trigonometry
Functions and Graphs
These are three books that I would recommend to somebody trying to prepare for calculus. They're all written by the mathematician Gelfand and his colleages, and they're some of the best-written math books I've ever read. You come away from reading them really understanding the subject matter. I'd read them in that order, too.
It is hard to provide a "comprehensive" view, because there's so much disperate material in so many different fields that draw upon probability theory.
Feller is an approachable classic that covers all of the main results in traditional probability theory. It certainly feels a little dated, but it is full of the deep central limit insights that are rarely explained in full in other texts. Feller is rigorous, but keeps applications at the center of the discussion, and doesn't dwell too much on the measure-theoretical / axiomatic side of things. If you are more interested in the modern mathematical theory of probability, try Probability with Martingales.
On the other hand, if you don't care at all about abstract mathematical insights, and just want to be able to use probabilty theory directly for every-day applications, then I would skip both of the above, and look into Bayesian probabilistic modelling. Try Gelman, et. al..
Of course, there's also machine learning. It draws on a lot of probability theory, but often teaches it in a very different way to a traditional probability class. For a start, there is much more emphasis on multivariate models, so linear algebra is much more central. (Bishop is a good text).
Damn, son. That's way bigger than my guesstimate.
The amazon prices I checked out pinned the collection closer to $400, which granted is still really, really impressive.
In case you're curious this was my textbook. It's come down by a lot in price over a couple years. Brand new it was $365 in the shrink wrap from my school's store!
Eh, either way I'm wrong, just by a different amount.
How would you rather split beginner vs intermediate/advanced ?
My feeling was that Ben Lambert's book would be a good intro and that Bayesian Data Analysis would be a good next ?
Practical Algebra: A Self-Teaching Guide
really helped me a couple of years ago when I had to get up to speed on algebra quickly.
Beyond that, you can hardly do better in the best-bang-for-the-buck department than the Humongous Books series. 1000 problems in each book, annotated and explained, and he has an entertaining style.
The Humongous Book of Algebra Problems: Translated for People Who Don't Speak Math
The Humongous Book of Geometry Problems: Translated for People Who Don't Speak Math
The Humongous Book of Calculus Problems: For People Who Don't Speak Math
If you are serious about learning, Linear Algebra by Friedberg Insel and Spence, or Linear Algebra by Greub are your best bets. I love both books, but the first one is a bit easier to read.
Here is the book I always recommend for people who want an introduction to graph theory:
It's super cheap (only $3.99 on Amazon) and I think it's really a good introduction to the subject. It doesn't go as far in depth as more advanced books, but Kuratowski's theorem is covered in Chapter 3.
'Geometry' means different things to different people. So far the books that have been suggested range from elementary Euclidean geometry (Euclid) to differential geometry (Spivak) to algebraic geometry (Hartshorne, Grothendieck). In order to get more helpful suggestions, you should be more specific about what you're looking for.
Since it sounds like you want to solidify the geometry you learnt in high school, I'll suggest some elementary resources:
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I am Mnemosyne 2.1, It is too bad someone less notorious for their complacent attitude towards themselves had not written that link instead. ^^^^/r/botsrights ^^^^Contribute ^^^^message ^^^^me ^^^^suggestions ^^^^at ^^^^any ^^^^time ^^^^Opt ^^^^out ^^^^of ^^^^tracking ^^^^by ^^^^messaging ^^^^me ^^^^"Opt ^^^^Out" ^^^^at ^^^^any ^^^^time
Read a history, e.g. Gullberg.
An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.
A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.
Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.
Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.
Since you are already going to take Machine Learning and want to build a good statistical foundation, I highly recommend Mathematical Statistics with Applications by Wackerly et al.
Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.
I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/
https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful
If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2
And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X
It can't be helped, the price is almost the same for the first book on amazon.com and I assume a similar trend for the others. Hardcovers in general are more expensive due to the intrinsic higher cost of manufacturing. What you may have observed is that other books have lower cost Asian editions that make them more affordable for nations with smaller economies, but these research books do not serve such a niche as such.
What is interesting though is that state of the art Machine Learning is usually not in these books, and is simply being published in papers and blog posts as of late.
Reposted from this thread: http://www.reddit.com/r/IWantToLearn/comments/mqoxx/iwtl_math/
"Had a very similar path. Decided to bite the bullet and take my school's remedial algebra course, as I cheated through middle and high school and thus knew nothing. Failed remedial algebra and retook it. Now I'm graduating with a math minor and am taking a calc-based probability theory course. Have hope!
Advice:
I highly recommend this book: http://www.amazon.com/Humongous-Book-Algebra-Problems-Translated/dp/1592577229 Lots of problems broken down step-by-step. Skipped steps are one of the hardest things to deal with when you don't yet have much mathematical knowledge, especially during self-study. Look for other books by the author, W. Michael Kelly.
An interesting take on math and math education, though a little bitter: http://www.maa.org/devlin/LockhartsLament.pdf
Godspeed, sir!"
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.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
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
You may find Kreyszig's Introductory Functional Analysis with Applications interesting.
EDIT: NoLemurs suggested Shankar as a good text that proceeds from first principles. Another book famous for deriving beautiful results from basic physical ideas is Landau's Quantum Mechanics, though it is quite dense and not at all pedagogical.
the "vanilla" books are IMO quite boring to read - especially when you don't know more than Set/Functions.
but I really enjoy P. Aluffi; Algebra: Chapter 0 that builds up algebra using CT from the go instead of after all the work
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remark I don't know if this will really help you understanding Haskell (I doubt it a bit) but it's a worthy intellectual endeavor all in itself and you can put on a knowing smile whenever you hear those horrible words after
Start with Khan Academy. You start with a test that determines where you're at, and I think it's pretty damn good, if only slightly conservative, and that's the way it should be.
The video lessons are particularly good at teaching you how, and basically why, but most math materials, across the board, don't really do much, if anything, to help you with insight.
"What does it mean?"
There is an answer. You just need to know where to find it.
I recommend books on math history. I find I get a greater intuitive sense of math learning it like this than just learning the pure process and concepts in a math education book.
Journey Through Genius is one of my favorites. It's 26 proofs, the history behind how they were developed or discovered, why, and what they mean, in a sense. And you don't need to know shit about math for this book to make sense. The author really breaks it down for you. Sometimes the raw proof is right there, and he goes over it, sometimes he skips it and explains it by example.
A History of Pi is also fantastic, and follows it's earliest known history to relatively modern day. An anthropologist friend taught me "follow the money" when studying history, to understand the rise and fall of nations and empires. This book taught me "follow the knowledge," and it's equally telling, but in ways following the money won't.
Hey I'm a physics BSc turned mathematician.
I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.
I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.
For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.
Really interested, actually! But I'm curious about a few things:
When exactly will it start in January? And when will it end? Will it be in the evenings? Which days of the week?
Will we need a text book? I have a Dover book on basic analysis already which I haven't cracked open.
Where will the class be held?
I had an incredibly hard time with calculus as a university student. I took it 5 times because I kept dropping it or withdrawing or not getting a passing grade. I almost got kicked out of my program because I pushed the limits of how many times I could repeat the course. There was a general disinterest on my part, but now, almost 10 years later, I am much more fascinated and genuinely interested in math, number theory, and also in many ways, analysis.
I started reading a book recently that finally explained what calculus actually was in simple terms. I feel like it's the first time that was ever done for me and I can say that helped my interest.
Anyway, I'd really hope to attend your class! The reason I'm curious about exact start date is that I'll be away from the HRM until mid-January. And it's a bummer to miss the first few classes of anything!
If you are really good at calculus, learn some probability first. My personal favorite is Wackerly et al.'s Mathematical Statistics with Applications. This covers both the probability and mathematical stats background that you will see in college. The book is quite pricey, so I recommend buying it on half (dot) com.
You might notice that this text has a lot of negative reviews. This review of the above text explains the prerequisites quite well - this is not an AP-stats type of textbook:
> I believe that this book is designed to teach statistics to those who plan on actually using it professionally (and not just to pass a required course) while continuing to develop one's own mathematical maturity. While Wackerly is not as rigorous as Ross's Probability book, it is taught at a completely different level than a non-calculus-based statistics course that are often taken by students who simply want to know which formula to use for the exam. I think of it as the ideal text for anyone in the sciences, engineering, or economics. The level of rigor is similar to the 2 Calculus courses online at MIT-Open Course Ware.
>
> [...]
>
> this book derives virtually every formula, allowing students to continue to develop their mathematical maturity which will be required for higher-level courses on bootstrapping, pattern recognition, statistical learning, etc. In order to follow these proofs (and also in order to solve problems from about 4 chapters) one must have a firm grasp of calculus. That not only means that one can integrate, differentiate, work with series, use L-Hopital's rule and integration-by-parts, but also that one understands the concepts of calculus very well.
>
> The proofs are all broken down so as to not really skip many steps, but as someone away from math for over 25 years, I must write down each step myself and make sure that I understand it before moving on. If a few steps are skipped, I must connect the dots myself using plenty of scratch paper. My math background was the Calculus series, ordinary differential equations, and linear algebra. About 1/5 of all problems are proof-based.
See also this review:
> I will concede that you can't come at this book without an understanding of at least integral calculus (and since so many people get turned off by Algebra, well...), so I suspect a lot of the negative reviews here are written by people who jumped in the deep end of the pool without having a few swimming lessons. If you know the calculus and basic set theory, the book is exceedingly easy to follow.
Some of what you learned in AP Stats will transfer to calculus-based statistics, but a lot of what you learn in your undergrad will not be like anything you learned in AP Stats. Hence I'm recommending that you start from scratch on probability.
Generally speaking, I agree with /u/Akillees89 that you should get a head start in developing your math background. However, I don't agree that Strang or Axler are good for linear algebra for statistics. See my post here.
It sounds like you might perhaps want a background in Number Theory and/or Basic Logic and/or Set Theory. The thing about math is that there is a lot...
My advice for a text that might serve you well is N.L. Biggs' Discrete Mathematics (http://www.amazon.com/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178). If you are at all interested in computer science, this is also a great book for that because it introduces some of the mathematical rigor behind it. Some people have a smidgen of difficulty with this text because it doesn't give some names to proofs/algorithms that maybe you've heard whispered (e.g. Dijkstra's shortest path and Prim's minimal spanning tree). A text that I tend to think is on par with Biggs', but many think is vastly superior (I love both, but for different reasons) that covers some (most) of the same topics is Eccles' An Introduction to Mathematical Reasoning (http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=pd_sim_b_4?ie=UTF8&amp;refRID=1BB6VKRP59S2420M132F). This book has a wonderful focus on building from the ground up and emphasizes clearly worded and mathematically rigorous proofs.
You seem genuinely interested in mathematics, but I do want to warn you about some more ahem esoteric (read: improperly worded, perhaps?) problems that ask such things as why 1 is greater than 0. The mathematics here is largely armchair - lacking any fundamental logic. There would be no issue with redefining a set of bases such that "0" is greater than "1". However, if you want to have rationale of the concept of things being greater than another, that's more like number theory. You can learn the 10 axioms of natural numbers and then build from there.
Both of the books I mentioned will cover stuff like this. For example, they both (unless I'm not remembering correctly) delve into Euclid's proof of infinite primes, something which may interest you.
Briefly (and not so rigorously), assume that the number of primes, p1, p2, p3, ..., pN, is finite. Then there exists a number P which is the product of these primes. Based on the axioms of natural numbers, since all primes p1,p2,...,pN are natural numbers P is a natural number and so is P+1. Consider S = P+1. If S is prime than our list is incomplete, assume S isn't prime. Then some number in our list, say pI, divides S because any natural number can be written as the product of primes. pI must also divide P because P equals the sum of all primes. Therefore if pI divides S and pI divides P, then pI divides S-P = 1. That's a contradiction because no prime evenly divides 1.
Stuff like this is super cool, super simple, and super beautiful and you absolutely can learn it. These two books would be a great place to start.
It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.
Maybe use the Bartle book.
That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.
On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.
It won't break the bank and, despite a few small typos, covers a lot material fairly gently.
In the lead up to calc first thing you want to do is just make sure you're algebra skills are pretty solid. A lot of people neglect it and then find the course to be harder than it needed to be because you really use algebra throughout.
Beyond that, if you want an extra book to study with and get practice problems from The Calculus Lifesaver is a big book of calculus you can use from now and into a first year college calculus course. If you do get it, don't worry about reading the whole thing from cover to cover, or doing all of the problems in it. It is a big book for a reason, it definitely covers more than you need to know for now, so don't get overwhelmed, it all comes with time.
Best of luck
I highly recommend this book for learning calculus.
I faced same problem as yours with calculus and this book helped me alot.
You may also want to check out The Haskell Road to Logic, Maths and Programming.
This book focusses on logic and how to use it, so you get to learn proofs. It even hits corecursion and combinatorics. If you think math is pretty but you want to use it interactively as source code, this could be the book for you.
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!
For Calculus:
Calculus Early Transcendentals by James Stewart
^ Link to Amazon
Khan Academy Calculus Youtube Playlist
For Physics:
Introductory Physics by Giancoli
^ Link to Amazon
Crash Course Physics Youtube Playlist
Here are additional reading materials when you're a bit farther along:
Mathematical Methods in the Physical Sciences by Mary Boas
Modern Physics by Randy Harris
Classical Mechanics by John Taylor
Introduction to Electrodynamics by Griffiths
Introduction to Quantum Mechanics by Griffiths
Introduction to Particle Physics by Griffiths
The Feynman Lectures
With most of these you will be able to find PDFs of the book and the solutions. Otherwise if you prefer hardcopies you can get them on Amazon. I used to be adigital guy but have switched to physical copies because they are easier to reference in my opinion. Let me know if this helps and if you need more.
> Elementary Statistics
http://ftp.cats.com.jo/Stat/Statistics/Elementary%20Statistics%20%20-%20Picturing%20the%20World%20(gnv64).pdf
Presuming you mean this book, I am still at an absolute loss to understand how you think this doesn't somehow require algebra as a prerequisite.
All the manipulations about gaussian distributions, student t distributions, binomial distribution etc... or even the bit on regression, right there on page 502, how is that not algebra. It literally makes reference to the general form of a line in 2-space. Are they just expected to memorize those outright with no regard to their derivation?
How do you treat topics like expected value? Because it seems like right there on page 194 that they've given the general algebraic formula for discrete, real valued, random variable.
They seem to elide the treatment of continuous random variables. So I presume they won't even be going through the exercise of the mean of a Poisson.
All of that granted, this book still heavily relies on the ability to perform algebraic permutations. Right there on page 306 is the very z-score transform I explicitly mentioned earlier.
As far as where I teach, I don't, excepting the odd lecture to clients or coworkers. Typically, however, our domain does not fit prettily into the packaged up parameterized distributions of baccalaureate statistics. We deal in a lot of probabilistic graphical models, in manifold learning, in non-parametrics, etc.
The books I recommend to my audience (which is quite different than those who haven't a basic grasp on algebra) are:
and more...
Now, this is not basic statistics. But I would also make a strong argument that the book you teach out of still requires knowledge of the treatment of basic symbolic manipulation--which is primarily taught in algebra.
While I certainly have my own issues with the bottled up and black-box sort of way that frequentist statistics is taught to undergraduates, that is an entirely separate pedagogical and epistemological argument. I still just don't see how you can see the book out of which you teach as a replacement for algebra.
I can offer my two cents. I’m a Googler who uses machine learning to detect abuse, where my work is somewhere between analyst and software engineer. I’m also 50% done through the OMSCS program. Here’s what I’ve observed:
Yes, Reinforcement Learning, Computer Vision, and Machine Learning are 100% relevant for a career in data science. But data science is vague; it means different things depending on the company and role. There are three types of data science tasks and each specific job may be weighted more heavily in one of these three directions: (1) data analytics, reporting, and business intelligence focused, (2) statistical theory and model prototyping focused and (3) software engineering focused by launching models into production, but with less empathsis on statistical theory.
I've had to do a bit of all three types of work. The two most important aspects are (1) defining your problem as a data science/machine learning problem, and (2) launching the thing in a distributed production environment.
If you already have features and labeled data, you should be able to get a sense of what model you want to use within 24 hours on your laptop based on a sample of the data (this can be much much harder when you can't actually sample the data before you build the prod job because the data is already distributed and hard to wrangle). Getting the data, ensuring it represents your problem, and ensuring you have processes in place to monitor, re-train, evaluate, and manage FPs/FNs will take a vast majority of your time. Read this paper too: https://papers.nips.cc/paper/5656-hidden-technical-debt-in-machine-learning-systems.pdf
Academic classes will not teach you how to do this in a work environment. Instead, expect them to give you a toolbox of ideas to use, and it’s up to you to match the tool with the problem. Remember that the algorithm will just spit out numbers. You'll need to really understand what's going on, and what assumptions you are making before you use each model (e.g. in real life few random variables are nicely gaussian).
I do use a good amount of deep learning at work. But try not to - if a logistic regression or gradient boosted tree works, then use it. Else, you will need to fiddle with hyper parameters, try multiple different neural architectures (e.g. with time series prediction, do you start with a CNN with attention? CNN for preprocessing then DNN? LSTM-Autoencoder? Or LSTM-AE + Deep Regressor, or classical VAR or SARIMAX models...what about missing values?), and rapidly evaluate performance before moving forward. You can also pick up a deep learning book or watch Stanford lectures on the side; first have the fundamentals down. There are many, many ways you can re-frame and tackle the same problem. The biggest risk is going down a rabbit hole before you can validate that your approach will work, and wasting a lot of time and resources. ML/Data Science project outcomes are very binary: it will work well or it won’t be prod ready and you have zero impact.
I do think the triple threat of academic knowledge for success in this area would be graduate level statistics, computer science, and economics. I am weakest in theoretical statistics and really need to brush up on bayesian stats (https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445). But 9/10 times a gradient boosted tree with good features (it's all about representation) will work, and getting it in prod plus getting in buy-in from a variety of teams will be your bottleneck. In abuse and fraud; the distributions shift all the time because the nature of the problem is adversarial, so every day is interesting.
One of the post-docs in my department did his dissertation with Bayesian stats and he essentially had to teach himself! He strongly recommended this as a place to start if you are interested in that topic -- https://www.amazon.com/gp/product/1482253445/ref=oh_aui_detailpage_o09_s00?ie=UTF8&amp;psc=1 (I have not read it yet.)
One of our computer science profs teaches Bayes for the CS folks and said he would be willing to put together a class for psych folks in conjunction with some other people, so that's a place where I am hoping to develop some competency at some point. I strongly recommend reaching outside of your department, especially if you are at a larger university!
I really believe that Michael Kelly's "Humongous Book of" series are the best resources for getting through all math classes up to Calculus II. These books contain every single type of problem you will ever encounter in Algebra I & II, Geometry, Trig, and Calc I & II, all solved in great detail. They are like Schaums Outlines but much more reliable.
https://www.amazon.com/Humongous-Basic-Pre-Algebra-Problems-Books/dp/1615640835
https://www.amazon.com/Humongous-Book-Algebra-Problems-Books/dp/1592577229
https://www.amazon.com/Humongous-Book-Geometry-Problems-Books/dp/1592578640
https://www.amazon.com/Humongous-Book-Trigonometry-Problems-Comprehensive/dp/1615641823
https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129
The short version is that in a bayesian model your likelihood is how you're choosing to model the data, aka P(x|\theta) encodes how you think your data was generated. If you think your data comes from a binomial, e.g. you have something representing a series of success/failure trials like coin flips, you'd model your data with a binomial likelihood. There's no right or wrong way to choose the likelihood, it's entirely based on how you, the statistician, thinks the data should be modeled. The prior, P(\theta), is just a way to specify what you think \theta might be beforehand, e.g. if you have no clue in the binomial example what your rate of success might be you put a uniform prior over the unit interval. Then, assuming you understand bayes theorem, we find that we can estimate the parameter \theta given the data by calculating P(\theta|x)=P(x|\theta)P(\theta)/P(x) . That is the entire bayesian model in a nutshell. The problem, and where mcmc comes in, is that given real data, the way to calculate P(x) is usually intractable, as it amounts to integrating or summing over P(x|\theta)P(\theta), which isn't easy when you have multiple data points (since P(x|\theta) becomes \prod_{i} P(x_i|\theta) ). You use mcmc (and other approximate inference methods) to get around calculating P(x) exactly. I'm not sure where you've learned bayesian stats from before, but I've heard good things , for gaining intuition (which it seems is what you need), about Statistical Rethinking (https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445), the authors website includes more resources including his lectures. Doing Bayesian data analysis (https://www.amazon.com/Doing-Bayesian-Data-Analysis-Second/dp/0124058884/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&amp;psc=1&amp;refRID=58357AYY9N1EZRG0WAMY) also seems to be another beginner friendly book.
A First Course in Graph Theory by Chartrand and Zhang
Combinatorics: A Guided Tour by Mazur
Discrete Math by Epp
For Linear Algebra I like these below:
Lecture Notes by Tao
Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza
Linear Algebra Done Right by Axler
Linear Algebra by Friedberg, Insel and Spence
A graph theory project! I just started today (it was assigned on Friday and this is when I selected my topic). I’m on spring break but next month I have to present a 15-20 minute lecture on graph automorphisms. I don’t necessarily have to, but I want to try and tie it in with some group theory since there is a mix of undergrads who the majority of them have seen some algebra before and probably bored PhD students/algebraists in my class, but I’m not sure where to start. Like, what would the binary operation be, composition of functions? What about the identity and inverse elements, what would those look like? In general, what would the elements of this group look like? What would the group isomorphism be? That means it’s a homomorphism with a bijective function. What would the homomorphism and bijective function look like? These are the questions I’m trying to get answers to.
Last semester I took a first course in Abstract Algebra and I’m currently taking a follow up course in Linear Algebra (I have the same professor for both algebra classes and my graph theory class). I’m curious if I can somehow also bring up some matrix representation theory stuff as that’s what we’re going over in my linear algebra class right now.
This is the textbook I’m using for my graph theory class: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/1846289696?ref=yo_pop_ma_swf
Here are the other graph theory books I got from my library and am using as references: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/3662536218?ref=yo_pop_ma_swf
Modern Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/0387984887?ref=yo_pop_ma_swf
And for funsies, here is my linear algebra text: Linear Algebra, 4th Edition https://www.amazon.com/dp/0130084514?ref=yo_pop_ma_swf
But that’s what I’m working on! :)
And I certainly wouldn’t mind some pointers or ideas or things to investigate for this project! Like I said, I just started today (about 45 minutes ago) and am just trying to get some basic questions answered. From my preliminary investigating in my textbook, it seems a good example to work with in regards to a graph automorphism would be the Peterson Graph.
The course I took as an undergraduate used Friedberg, Insel and Spence. I remember liking it fine, but it's insultingly expensive. Find it in a library or get a used copy if you can. If you're looking for a bargain, it can't hurt to try Shilov. He's Russian, so the book is very terse, but covers a lot of ground.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1394386195&amp;sr=8-1&amp;keywords=pinter+abstract+algebra)
Right now I am studying Proofs from "Learning to Reason: An Introduction to Logic, Sets, and Relations" by Nancy Rodgers. Prior to getting started I looked at tons of "Intro to Proofs/Transition" books and the vast majority of them (including the popular darlings) are, frankly, just mostly doorstops - there's no way you could come out being able to do proofs by studying them.
Rodgers starts out with prop. logic and builds everything on top of that. Everytime she introduces a new topic, she gives logical justification (chapter 1 explores the logic extensively) that makes the proof structure work (very satisfying and makes the concepts stick around longer e. i. you are not just monkeying around with mish-mash of various tools, but actually know what you are doing)- never seen that in Real Analysis/Linear Algebra books that are, supposedly, designed to teach you proofs.
For example, in an intro to Real Anal, they just throw you the structure of Induction Proof and expect you to prove away - unrealistic. They dont show you why the proof works (logic and intuition behind the proof), wont let you explore the syntax of the proof before you get more comfortable with it and since one doesnt have a firm foundation made out of prop. logic, one's on a very shaky ground ready to break down whenever something serious comes on. With Rodgers, whenever something big and scary shows up, you just take everything apart into its logical building blocks like she teaches you in chapter 1 and it will make perfect sense.
But the worst part of RA books is they assume you are intimately familiar with Deduction and wont spend a half a page on it and that's 99% of math Induction Proof structure. Rodgers spends half the book exploring the intricacies of Deduction arguments. Basically, Rodgers' book explores math grammar in all its gory detail, is sort of a very revealing math porn.
If you ever studied a foreign language, you know there are 2 types of books. The ones that spell out all the grammar and give all the necessary vocabulary with an intention that you'll read some real literature in your target language in the future and those that skip the grammar or are very skimpy on it and give you pre-determined phrases and various random knowledge bites instead. The first category of books take the tougher road, but it pays off the at the end. Rodgers' book is one such book.
All in all, I just cant imagine learning proofs from Linear Algebra/Real Analysis books. Because, they are mostly about concepts inherent in these subjects and not proofs. Proofs are there to prove the said concepts, so there wont be enough time/space to explore proofs in-depth which will make your life tougher.
It mentions Rosenlicht at the bottom. Lucky you, that book's only 8 bucks! It's a good book, too.
Paul Nahin has published many good historical math books that don't skimp on the mathematical underpinnings. I particularly enjoyed An Imaginary Tale: http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004
Regarding Spivaks: I'm also working on it, and found that my proof technique was lacking. An Introduction to Mathematical Reasoning (Eccles) was helpful for me: http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188
Innumeracy
Excellent read.
Reading through Algebra: Chapter 0
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
You'll usually find the following recommended:
I've personally used Friedberg's text, and I found it to be pretty well written.
I am not sure that a pure math textbook is what you want. A lot of the problems that mathematicians think about may not be what you need. Let's take functional analysis for example. Most textbooks focus on bounded/ compact operators, and they only have one chapter at the end dedicated to unbounded operators. Unfortunately, the derivative (momentum) is an unbounded operator, so the part that has the least detail is what you need.
I would recommend a "math for physics students" book. A nice book that tries to paint the intuitive idea of most branches of math relevant to physics (and then some) and show you how to calculate is Goldbart and Stone's book, which they have made freely available online. This book assumes familiarity with linear algebra. If you are weak on this subject, I would highly recommend the book by Friedberg, Insel, and Spence. This is a more traditional math textbook, but it gets you very comfortable with the details of linear algebra (except for tensor products, but you should understand their construction with this background).
Archimedes created a method in which you start with 2 shapes (like a square/triangle/hexagon), one drawn on the inside of the circle so its points touch the edge, the other drawn outside of the circle so its edges touch the sides of the circle.
Lets use squares as an example (He actually used hexagons, because they fit a circle better, but anyway). So you have a circle with a square on the inside and another on the outside. You can find the lengths of the edges of the squares, and the average of these two numbers will be close to the circumference of the square. Knowing that Pi is simply the ratio of circumference to diameter, you can solve for Pi.
This will give you a very rough estimate. You can improve the accuracy of this by increasing the number of sides of the shapes you draw inside and outside of the circle. For example, we could double the number of sides, and then we'd have an octagon instead of a square, which fits a circle a lot better. As you add more and more sides the numbers begin to converge to a single value. It takes a polygon with about 96 sides to get a value of Pi accurate to 5 decimal places.
There is a very interesting book on this subject,
A History of Pi
A History of Pi. Recommended for those who like math and those who don't. Very interesting read.
P.S. May I also recommend:
e: The Story of a Number,
A History of Pi, and
Zero: The Biography of a Dengerous Idea
From pages 21-22 of Petr Beckmann's A History of PI:
> ...and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60)^2 (the Babylonians used the sexagesimal system, i.e. their base was 60 rather than 10).
>
> The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day).
Reminds me of Petr Beckmann in A History of Pi. He takes a sudden break from discussing the ways pi has been approximated over the centuries to go on a rant about how the USSR and Roman Empire never invented anything of worth.
For a general overview of everything to do with the history of math, which might be what you're looking for, I recommend Mathematics: From the Birth of Numbers. Very inspiring with a little bit of "how to do everything."
Here are some suggestions :
https://www.coursera.org/course/maththink
https://www.coursera.org/course/intrologic
Also, this is a great book :
http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&amp;qid=1346855198&amp;sr=8-5&amp;keywords=history+of+mathematics
It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.
EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.
For you, I would suggest :
http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&amp;qid=1346860077&amp;sr=8-1&amp;keywords=rudin
http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&amp;qid=1346860052&amp;sr=8-4&amp;keywords=from+matrix+to+bounded+linear+operators
http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&amp;qid=1346860077&amp;sr=8-5&amp;keywords=rudin
http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1346860356&amp;sr=1-2&amp;keywords=chaos+and+dynamics
http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&amp;ie=UTF8&amp;qid=1346860179&amp;sr=1-5&amp;keywords=numerical+analysis
This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.
One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.
If you are interested enough in machine learning that you are going to work through ESL, you may benefit from reading up on some math first. For example:
Without developing some mathematical maturity, some of ESL may be lost on you. Good luck!
Hmm I'm surprised you've had point-set topology, linear algebra, and basic functional analysis but have yet to encounter locally convex topological vector spaces! No worries, you have most likely developed all oft the machinery to understand them. I agree with G-Brain, Rudin's function analysis will do. Most functional analysis books should cover this at some point. The only I use is Kreyszig. Hope that helps!
If you enjoy analysis, maybe you'd like to learn some more?
I really enjoyed learning introductory functional analysis, which is presented incredibly well in Kreyszig's book Introductory Functional Analysis with Applications. It's very easy to read, and covers a lot and assumes very little on the part of the reader (basic concepts from analysis and linear algebra). This will teach you about doing analysis on finite and infinite dimensional spaces and about operators between such spaces. It's incredibly interesting, and I highly recommend it if you enjoy analysis and linear algebra.
Another great analysis topic is Fourier Analysis and wavelets. I enjoyed the books by Folland Fourier Analysis and Its Applications. I don't believe that book has any wavelets in it, so if you're interested in learning Fourier analysis plus wavelet theory, then I highly recommend the very approachable and fun book by Boggess and Narcowich A First Course in Wavelets with Fourier Analysis. If you have any interest at all in applications (like signals processing), this subject is fundamental.
We used this one in my undergraduate analysis class, and I found it pretty straightforward to read and understand. And it's only $13.
You might like Rosenlicht's book, Introduction to Analysis. Google Books will show you the first 2 chapters for free. It's a Dover book, so it's good and also cheap. I believe that it is often used as the text for the first "serious" real analysis course.
They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.
He really should be starting with the Trudeau, much better bed side reading.
If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.
An example text for such a course is this one:
https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188
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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:
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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995
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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9
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Disclaimer: I'm an engineer, not a mathematician, so take my advice with a grain of salt.
Early in my grad degree I wanted to master probability and improve my understanding of statistics. The books I used, and loved, are
DeGroot, Probability and Statistics
Rozanov, Probability Theory: A Concise Course
The first is organized very well, with ever increasing difficulty and a good number of solved problems. I also appreciate that as things start to get complicated, he also always bridges everything back to earlier concepts. The books also basically does everything Bayesian and Frequentist side by side, so you get a really good idea of the comparison and arbitraryness.
The second is a good cheap short book basically full of examples. It has just enough math flavor to be mathier, without proofing me to death.
Also, if you're really just jumping into the subject, I would recommend some pop culture math books too, e.g.,
Paulos, Innumeracy
Mlodinow, The Drunkards Walk
Have fun!
It is detailed. It just doesn't seem logical to me. His entire position is that since the odds against things being the way they are are so high, there must be a god that arranged them. It's a fundamental misunderstanding of probability. The chance of things being the way they are is actually 100%, because they are that way. We don't know how likely they were before they happened because we only have one planet and one solar system to examine. For all we know there could be life in most solar systems.
Even if that wasn't the case, even if we did have enough information to actually conclude that our existence here and now has a .00000000000000000000000000000001% chance of happening he then makes the even more absurd jump to saying "there being a God is the only thing that makes sense". God, especially the Christian God, is even less likely than the already unlikely chance of us existing at all. If it's extremely unlikely that we could evolve into what we are naturally, how is it less unlikely that an all-powerful, all-knowing, all-good being could exist for no discernible reason?
You should get him a copy of this book. It's great and will help him with these misconceptions. If you haven't read it, I highly suggest you do as well.
I think this book, Innumeracy, does a good job at explaining the odds of these kind of things happening to people.
Innumeracy is pretty entertaining (and useful), even if you're not a math person. It's only about 150 pages, so it's a quick read.
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.
The single best resource for 103/104 is The Calculus Lifesaver by Adrian Banner. There's a book and a series of recorded review sessions. I stopped showing up to 104 lectures when I found these because they were so much more thorough than the classes. Banner also did review sessions for 201/202 when you reach that point, which are equally good.
Check out Adrian Banner's The Calculus Lifesaver for a companion to a typical undergrad introductory calculus sequence and the accompanying videos from Princeton.
Since you have strong backgrounds in math, you could try Geometry: A Guided Inquiry out (I recommend getting the home study companion and Geometer's Sketchpad, as well). It relies heavily on working the exercises to find the important results yourself, which is best done with mathematically-inclined mentors to help. A review for these products can be found here.
For Trigonometry, I recommend Gelfand's text by the same name. It is very much made with future math students in mind, with appendices on approximating pi and on Fourier series.
Most of all, I recommend making your own stuff if you find yourself with extra time. If you find your daughter getting close to the end of the Geometry textbook, for example, set up some examples or further projects that round everything up and introduces her to another world of mathematics. If she is able to understand the material in the textbook rather well, it is entirely possible to prove Euler's Polyhedron Formula, look into the 5 platonic solids, as well as go into a little detail about the Euler Characteristic using the tools learned in Geometry, which would give her a glimpse into the world of Topology (Don't forget the Donut and Coffee Mug example).
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 like Algebra and Trigonometry by I.M. Gelfand. They are cheap books too.
I also have scans of them, PM me if you want to check them out.
Edit:
Also, Khan Academy is great resource for explanations. But I would recommend aiding Khan Academy with a text just for the problem set and solutions.
8 to 12 hours is really not that much, but it should be enough to learn something interesting! I would start with category theory if you can. I liked Emily Riehl's categories in context for an intro, but it will go a little slow for how little time you have to learn the basics. Maybe the first chapter of Algebra: Chapter 0 by Aleffi? [EDIT: you might want to find a "reasonably priced" pdf version of this book if you do decide to use it -- it's pretty expensive] If you can get through that, and understand a little about how types fit into the picture, you should be able to present the basic idea behind curry-howard-lambek. IIRC you do not need functors or natural transformations ("higher level" categorical concepts), as important as they usually are, to get through this topic; Aleffi doesn't go over them in his very first intro to categories which is why I'm recommending him. /u/VFB1210 has some very good recommendations above as well.
I am trying to think of a better introduction to type theory than HoTT -- if you can learn about types without getting infinity categories and homotopy equivalence mixed up in them, I would. Type theory is actually pretty cool and sleek.
Here's a selection of intro-to-type theory resources I found:
Programming in Martin-Löf's Type Theory is
pretty long, but you can probably put together a mini-course as follows: read chapters 1 & 2 quickly, skim 3, and then read 19 and 20.
The lecture notes from Paul Levy's mini-course on the typed lambda calculus form a pretty compact resource, but I'm not sure this will be super useful to you right now -- keep it in mind but don't start off with it. Since it is in lecture-note style it is also pretty hard to keep up with if you don't already kind of know what he's talking about.
Constable's Naïve Computational Type Theory seems to be different from the usual intro to types -- it's done in the style of the old Naive Set Theory text, which means you're supposed to be sort of guided intuitively into knowing how types work. It looks like the intuition all comes from programming, and if you know something functional and hopefully strongly typed (OCaml, SML, Haskell, or Lisp come to mind) you will probably get the most out of it. I think that's true about type theory in general, actually.
PFPL by Bob Harper is probably a stretch -- you won't find it useful right at the moment, but if you want to spend 2 semesters really getting to know how type theory encapsulates pretty much any modern programming paradigm (typed languages, "untyped" languages, parallel execution, concurrency, etc.) this book is top-tier. The preview edition doesn't have everything from the whole book but is a pretty big portion of it.
Gallian is basic undergrad stuff through Galois, right?
I can only recommend books that will start from scratch, so will cover many things you already know, but go much deeper than an undergrad text would. Mac Lane and Birkhoff is my favorite math text I've ever read. The only significant drawback is some of the terminology is awkward, the most significant example being that the word "homomorphism" is used once in the entire book, to note it as an alternative to their word "morphism". I'm also currently reading Aluffi to review for a qualifier, and while I personally don't like the exposition as much, it's definitely well-written, and is somewhat more modern. Both of them will cover things you know already, but they should have enough new stuff sprinkled in to keep you interested and help solidify your knowledge.
If you want a more direct transition I can't really be too helpful, sorry.
(edit: minor typo)
If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):
Basic Algebra by Jacobson
Algebra by Lang
Algebra by MacLane/Birkhoff
Algebra by Herstein
Algebra by Artin
etc
But there are other books that are "essential" to modern readers:
Chapter 0 by Aluffi
Basic Algebra by Knapp
Algebra by Dummit/Foot
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!
These are listed in the order I'd recommend reading them. Also, I've purposely recommended older editions since they're much cheaper and still as good as newer ones. If you want the latest edition of some book, you can search for that and get it.
The Humongous Book of Basic Math and Pre-Algebra Problems https://www.amazon.com/dp/1615640835/ref=cm_sw_r_cp_api_pHZdzbHARBT0A
Intermediate Algebra https://www.amazon.com/dp/0072934735/ref=cm_sw_r_cp_api_UIZdzbVD73KC9
College Algebra https://www.amazon.com/dp/0618643109/ref=cm_sw_r_cp_api_hKZdzb3TPRPH9
Trigonometry (2nd Edition) https://www.amazon.com/dp/032135690X/ref=cm_sw_r_cp_api_eLZdzbXGVGY6P
Reading this whole book from beginning to end will cover calculus 1, 2, and 3.
Calculus: Early Transcendental Functions https://www.amazon.com/dp/0073229733/ref=cm_sw_r_cp_api_PLZdzbW28XVBW
You can do LinAlg concurrently with calculus.
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) https://www.amazon.com/dp/0538735457/ref=cm_sw_r_cp_api_dNZdzb7TPVBJJ
You can do this after calculus. Or you can also get a book that's specific to statistics (be sure to get the one requiring calc, as some are made for non-science/eng students and are pretty basic) and then another book specific to probability. This one combines the two.
Probability and Statistics for Engineering and the Sciences https://www.amazon.com/dp/1305251806/ref=cm_sw_r_cp_api_QXZdzb1J095Y1
Differential Equations with Boundary-Value Problems, 8th Edition https://www.amazon.com/dp/1111827060/ref=cm_sw_r_cp_api_sSZdzbDKD0TQ9
After doing all of the above, you'd have the equivalent most engineering majors have to take. You can go further by exploring partial diff EQs, real analysis (which is usually required by math majors for more advanced topics), and an intro to higher math which usually includes logic, set theory, and abstract algebra.
If you want to get into higher math topics you can use this fantastic book on the topic:
This book is also available for free online, but since you won't have internet here's the hard copy.
Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_MUZdzbP64AWEW
From there you can go on to number theory, combinatorics, graph theory, numerical analysis, higher geometries, algorithms, more in depth in modern algebra, topology and so on. Good luck!
I posted this in another comment but I'm guessing this bastard?
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/
This has been pretty much the standard textbook on Bayes
https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/
Na verdade eu sou físico. Acho que é mais comum entre os físicos adotar uma perspectiva bayesiana do que entre os matemáticos ou mesmo os estatísticos. Talvez por causa da influência do Edwin T. Jayes, que era físico. Talvez por causa da conexão com teoria de informação e a tentadora conexão com termodinâmica e mecânica estatística.
O meu interesse pela perspectiva Bayesiana começou por conta do grupo de pesquisa onde fiz o doutorado. Meus orientador e meu co-orientador são fortemente bayesianos, e o irmão do meu orientador de doutorado é um pesquisador bastante conhecido das bases epistemológicas da teoria bayesiana (o físico uruguaio Ariel Caticha).
Tem vários livros bons sobre probabilidade bayesiana, depende muito do seu interesse.
O primeiro livro que eu li sobre o assunto foi justamente o do Jaynes - Probability Theory, the Logic of Science. Esse é um livro um pouco polêmico porque ele adota uma visão epistemológica bastante forte e argumenta de forma bastante agressiva a favor dela.
Uma visão um pouco alternativa, bastante conectada com teoria de informação e também fortemente epistemológica você pode encontrar no livro Lectures on Probability, Entropy and Statistical Physics do Ariel Caticha - (de graça aqui: https://arxiv.org/abs/0808.0012). Eu fui aluno de doutorado do irmão do Ariel, o Nestor Caticha. Ambos têm uma visão bastante fascinante de teoria de probabilidades e teoria da informação e das implicações delas para a física e a ciência em geral.
Esses livros são mais visões epistemológicas e teóricas, e bem menos úteis para aplicação. Se você se interessa por aplicação tem o famoso BDA3 - Bayesian Data Analysis, 3ª edição e também o Doing Bayesian Data Analysis do John Kruschke que tem exemplos em R.
Tem um livrinho bem introdutório também chamado Bayesian Methods for Hackers do Cam-Davidson Pylon (de graça aqui: https://github.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers) que usa exemplos em python (pymc). É bem basicão para aprender aplicações de probabilidades bayesianas.
O livro All of Statistics do Larry Wasserman tem uma parte introdutória também de inferência bayesiana.
Se você em interesse por inteligência artificial um outro livro muito bacana é o do físico britânico (recentemente falecido) David Mackay - Information Theory, Inference, and Learning Algorithms (de graça aqui: http://www.inference.phy.cam.ac.uk/mackay/itila/). Esse livro foi meu primeiro contato com Aprendizado de Máquina e é bem bacana.
Outros livros bacanas de Aprendizado de Máquina que usam uma perspectiva bayesiana são Bayesian Reasoning and Machine Learning (David Barber) e o livro-texto que tem sido o mais usado para essa área que é o Machine Learning: a Probabilistic Perspective (Kevin Murphy).
May I recommend this book to you then, Statistical Rethinking: A Bayesian Course with Examples in R and Stan (Chapman & Hall/CRC Texts in Statistical Science) https://www.amazon.co.uk/dp/1482253445/ref=cm_sw_r_cp_apa_Iu6.BbJE7EECQ
There seem to be a few options. I've had this and this on my reading list for a while, but haven't got further than that.
I'm also interested in recommendations.
I recommend thumbing through an introductory real analysis textbook like Abbot - and perhaps speaking to a professor - before declaring a second major. Mathematics beyond sophomore level are a lot different, even at the applied level.
FWIW, I quit a PChem PhD program to pursue applied math, it definitely gives you a lot more flexibility, but it's not for everyone.
Understanding Analysis by Stephen Abbott https://www.amazon.com/dp/1493927116/
Topics in Algebra by I.N. Herstein https://www.goodreads.com/book/show/1264762.Topics_in_Algebra
The Feynman lectures on physics http://www.feynmanlectures.caltech.edu/
I've got nothing for Economics, but the above would be my personal recommendations for self-study and just general reading.
Here's mine
To understand life, I'd highly recommend this textbook that we used at university http://www.amazon.com/Campbell-Biology-Edition-Jane-Reece/dp/0321558235/ That covers cell biology and basic biology, you'll understand how the cells in your body work, how nutrition works, how medicine works, how viruses work, where biotech is today, and every page will confront you with what we "don't yet" understand too with neat little excerpts of current science every chapter. It'll give you the foundation to start seeing how life is nothing special and just machinery (maybe you should do some basic chemistry/biology stuff on KhanAcademy first though to fully appreciate what you'll read).
For math I'd recommend doing KhanAcademy aswell https://www.khanacademy.org/ and maybe a good Algebra workbook like http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ and after you're comfortable with Algebra/Trig then go for calc, I like this book http://www.amazon.com/Calculus-Ron-Larson/dp/0547167024/ Don't forget the 2 workbooks so you can dig yourself out when you get stuck http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213093/ http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213107/ That covers calc1 calc2 and calc3.
Once you're getting into calc Physics is a must of course, Math can describe an infinite amount of universes but when you use it to describe our universe now you have Physics, http://www.amazon.com/University-Physics-Modern-12th/dp/0321501217/ has workbooks too that you'll definitely need since you're learning on your own.
At this point you'll have your answers and a foundation to go into advanced topics in all technical fields, this is why every university student who does a technical degree must take courses in all those 3 disciplines.
If anything at least read that biology textbook, you really won't ever have a true appreciation for the living world and you can't believe how often you'll start noticing people around you spouting terrible science. If you could actually get through all the work I mentioned above, college would be a breeze for you.
I don't know the first but I didn't really like the second book. Right now I do seem to make some good progress understanding stuff (not all but most) in Algebra Chapter 0 which is a lot bigger but introduces a lot of the Algebra I was missing (or forgot since school) along with the Category Theory terms.
(Note: I wrote this elsewhere)
Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:
When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
I don't know of any good online introduction to discrete math, but here are a few book ideas.
The Haskell Road to Logic, Maths and Programming
http://amzn.com/0954300696
I read only the first chapter or two a long time ago. I don't remember much, but I do remember I was able to progress through the book and learn new things about both math and Haskell from the text.
I didn't have any trouble getting the outdated examples to work. I had read LYAH previously though, so I wasn't a complete beginner.
I would really enjoy hearing what others have thought about this book.
The book that really helped me prepare for CS 6505 this fall was Discrete Mathematics with Applications by Susanna Epp. I found it easy to digest and it seemed to line up well with the needed knowledge to do well in the course.
Richard Hammack's Book of Proof also proved invaluable. Because so much of your success in the class relies on your ability to do proofs, strengthening those skills in advance will help.
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.)
> btw ce crezi de masterul asta de la unibuc http://fmi.unibuc.ro/ro/pdf/2008/curs_master/informatica/4InteligentaArtificialaEnachescuSite.pdf , e din 2008,nu am gasit o varianta mai buna.Daca voi avea posibilitatea sa fiu acceptat l;a o facultate mai moderna care face cercetare din afara o voi face,dar mai intai trebuie sa capat o diploma din Romania).
Acum, trebuie sa intelegi ca ML si AI sunt 2 lucruri diferite. AI includes ML, si ce ai tu aici e un master general de AI. Nu pot sa iti spun cat de bun e masterul, dar vad ca faci 1 curs de ML doar in anul 2, ceea ce pentru mine ar fi un motiv sa nu il fac. Information retrieval si NLP sunt interesante, dar eu as incerca sa invat ML la nivel teoretic first, si apoi sa abordez probleme specifice domeniilor.
> Eu ma gandeam ca Unibuc e mai potrivit pt ca la Poli voi face multa electronica si programare low-level si nu cred ca le voi folosi
Ar putea fi utile daca te gandesti la un moment dat ca te intereseaza mai degraba sa fii Research engineer si sa nu lucrezi atat de mult pe teorie, cat pe implementare. Toate librariile de scientific programming sunt implementate in C/C++. Dar pe langa asta, in general programarea low-level ar fi interesant sa o inveti pentru ca te ajuta sa intelegi cum functioneaza lucrurile at a more basic level, fara x abstractii construite pentru a fi totul beginner-friendly. Daca nu vrei sa continui cu asta dupa 1-2 cursuri e ok, tot cred ca iti va folosi mai incolo. Sa inveti python si c++ in paralel e un challenge interesant :).
> Va veni vacanta de vara si voi avea mult timp liber si vreau sa ma apuc de machine learning de-acum.Ce crezi de planul asta de invatare?
Iti va lua mai mult decat 1 vara sa termini ce ai listat aici. Sfatul meu ar fi sa imbini programare aplicata cu matematica. Cursurile sunt ok, dar eu pentru matematica as incepe cu single variable calculus -> multiple variable calculus inainte de altceva (daca ai cunostintele necesare sa abordezi cursul). Uite o carte pe care ti-o recomand: https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552
Are in jur de 8 sectiuni care reprezinta pre-requisites (lucruri pe care ar trebui sa le stii inainte sa abordezi cartea), algebra, geometrie de baza, etc. Fiecare invata diferit, eu prefer cartile.
Legat de programare, incearca sa faci probleme de aici: https://projecteuler.net/, te va ajuta mai incolo :). Si daca te plictisesti incearca construiesti lucruri care ti-ar fi utile. Vei invata destule din proiecte de genul.
Depends on the market depths, for a lot of books, there may be a couple low priced books where a few purchases will raise the price pretty significantly. I think a lot of booksellers have repricers that don't work very effectively where they'lll lower the price over time until it sells, and there really isnt a market for text books except for at the beginning of semesters.
http://www.amazon.com/gp/offer-listing/1285741552/ref=olp_f_primeEligible?ie=UTF8&amp;f_primeEligible=true
On that book, which is a pretty widely used text, If they sell about 5 books, the price rises almost $70.
This is very similar to the analysis featured on the cover of Bayesian Data Analysis (third edition).
Here's a bigger picture of their decomposition into day-of-week effects, seasonal effects, long-term trends, holidays, etc.
A bit more here, and lots more in the book.
A lot of the recommendations in this thread are good, I'd like to add "Bayesian Data Analysis 3rd edition" by Gelman et al. Useful if you encounter Bayesian models, especially hierarchical/multilevel models.
By the way, do you know if things like linear/nonlinear regression, ANOVA and multivariate statistics is useful for me? Like stuff from https://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=dp_ob_title_bk or https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=pd_sim_14_8?_encoding=UTF8&pd_rd_i=1439840954&pd_rd_r=c9c0f3c5-f332-11e8-9f0a-0f336f5f387a&pd_rd_w=xHkKH&pd_rd_wg=7lXm5&pf_rd_i=desktop-dp-sims&pf_rd_m=ATVPDKIKX0DER&pf_rd_p=18bb0b78-4200-49b9-ac91-f141d61a1780&pf_rd_r=PFCZ1JM04FMAVAHG6VNP&pf_rd_s=desktop-dp-sims&pf_rd_t=40701&psc=1&refRID=PFCZ1JM04FMAVAHG6VNP
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You can tell how little I know since I'm kinda shooting topics at the wall hoping something sticks
This may vary by school, but it's been my experience that there aren't a lot of classes explicitly labeled as "artificial intelligence" (especially at the undergraduate level). However, AI is a very broad and interdisciplinary field so one thing I would recommend is that you take courses from fields that form the foundation of AI. (Math, Statistics, Computer Science, Psychology, Philosophy, Neurobiology, etc.)
In addition, take advantage of the resources you can find online! Self-study the topics you're interested in and try to get some hands on experience if possible: read blogs, read papers , browse subreddits, program a game-playing AI, etc.
Given that you're specifically interested in reasoning:
*As a disclaimer: I'm fairly new to this field of study myself. What I've shared with you is my best understanding, but given my lack of experience it may not be completely accurate. (Anyone, please feel free to correct me if I'm mistaken on any of these points)
They're not free, but Doing Bayesian Data Analysis and Statistical Rethinking are worth their weight in gold.
I personally think you should brush up on frequentist statistics as well as linear models before heading to Bayesian Statistics. A list of recommendations directed at your background:
This book is excellent:
https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445
Not long! For this purpose I highly highly recommend Richard McElreath's Statistical Rethinking (this one here). It's SO good. The math is exceptionally straightforward for someone familiar with regression, and it's huge on developing intuition. Bonus: he sets you up with all the tools you need to do your own analyses, and there are tons of examples that he works from a lot of different angles. He even does hierarchical regression.
It's an easy math book to read cover to cover by yourself, to be honest. He really holds your hand the whole way through.
Jesus, he should pay me to rep his book.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
Calculus by James Stewart is the best introductory Calculus book that I used in college - I definitely recommend it. It will get you through both single-variable calculus, as well as most of multi-variable calculus that you will need for for master's level probability and statistical theory. In particular, if you plan to use the book, you should focus on chapters 1-7 (for single variable calculus), chapter 11 (infinite sequences and series) and chapters 14 and 15 (partial derivatives and multiple integrals). These chapter numbers are based on the 7th edition.
If you have previously taken calculus, you might consider looking at Khan Academy for an overview instead.
If you have not previously taken linear algebra, or it has been awhile, you will definitely need to work through a linear algebra textbook (don't have any particular recommendations here) or visit Khan academy.
Finally, a book such as Stephen Abbott's Understanding Analysis is not necessary for master's level statistics, but could be helpful for getting into the mindset of calculus-based proofs.
I'm not sure what level of math you have previously completed, and what level of rigor the MS in Statistics program is, but you will likely need be very familiar with single- and multi-variable calculus as well as linear algebra to be successful in probability and statistical theory. It's certainly possible, just pointing out that there could be a lot of work! If you have any other questions, I'm happy to answer them.
>my first venture into proofs?
Have you had no prior experience with rigorous proofs, other than some elements of your linear algebra class? Not even something like a discrete math class? I'd worry that as an already-busy grad student, this might be biting off more than you can chew.
One additional question: is "grad analysis" a graduate-level class in analysis beyond an undergraduate-level class also offered at your school? I ask because typically, such a graduate-level class would assume considerable familiarity with undergrad-level analysis as a prerequisite. If you're in a situation where understanding the rigorous ε-δ definition of limit isn't something you've already internalized intuitively, then you'll likely find a grad-level introduction to something like measure theory to have a very steep learning curve.
---
I second /u/Gwinbar's recommendation above of Stephen Abbott's Understanding Analysis as a textbook for self-directed learning. But even that might be premature if you don't first develop sufficient background in the basics of set theory and mathematical logic. In particular, lots of concepts in analysis involve logical quantifiers, meaning that you'll need to be comfortable with both the meaning of a statement like
and how you would take the logical negation of the above statement. If none of this is familiar or transparently clear to you, then you might be better served by taking an undergraduate class in real analysis. Another option, of course, would be to audit a class, though that would be less advantageous in the context of buttressing your CV.
---
I think the best advice I can give you at this point would be to talk to someone at your school. Someone in the economics department would have the best sense of how valuable having a graduate-level analysis class could be for your pursuit of a doctorate—as well as how damaging flaming out from such a class might be. I'd recommend talking to someone at your school's math department, too, since the best way to evaluate your background would be through a conversation by someone who's familiar with your school's analysis curriculum. They're in the best position to make the recommendation that best fits your current background level in mathematics, given what your school's academic standards are for such analysis classes. They can also provide final exams from past iterations of the undergrad- and grad-level analysis courses, respectively. That might give you some additional data to illuminate what such classes entail.
I hope you can find more concrete information that's more custom-tailored to your specific circumstances. Good luck, whatever you decide!
In addition to Baby Rudin, I really liked this book when I first start learning analysis.
The Nature of Computation
(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)
Concrete Mathematics
Understanding Analysis
An Introduction to Statistical Learning
Numerical Linear Algebra
Introduction to Probability
>I mean its to late now to enroll...
Why wait? Pick up a few books on math and use your Google Fu to get yourself started.
I really like this book.
And instead of studying geometry (which I doubt you'd be using in college), study Logic instead. The way problems are constructed is similar to geometry. In geometry you have theorem and postulates, in logic you make proofs. You start out with two or three opening statements, and by using different combinations of OR, AND & IF-THEN statements, you can prove the final statement.
I'll give you this link about it but I'm hesitant to because it has lots of scary symbols and letters. Here. But save that for later. If you want to get started, take a look at truth tables .
Logic is so much more interesting than geometry because it'll help your Google Fu get even better. You can make Boolean statements when you enter a Google query. It also gets you on the path to learning SQL (which your brother may also be able to help you with). SQL is all about sets - sets of records, and how you can join them and select those that have certain values, etc.
You may even find this book a nice, gentle introduction to logic that doesn't require much math.
Basically, what I'm saying to you is this: you live in the most incredible time to be alive ever. The Internet is a super-powerful tool you can use to educate yourself and you should make full use of it.
I also want you to know that if you don't have a specialized skill, you're going to be treated like a virtual slave for the rest of your life. Working at WalMart is not a good career choice. That's just choosing a life of victimhood. Make full use of the Internet, and your lack of a car will seem less problematic.
I've been working my way through the Humongous Book of Algebra Problems. It's about a thousand math problems with complete (and very good explanations). The only way to get good is to get out the paper and plow through problems. I supplement that with videos from Khan Academy (which has it's own math quiz system that is also excellent). I try to do every problem, even if I hate it (looking at you matrices!). And if I get it wrong, no matter the mistake, I re-do the whole problem.
After I do that one, it's on to the Humongous book on Trig. Then calculus. All for the randy hell of it (I grew up with an interest in science and bad math teachers).
http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229
Maybe? I'm thinking about picking this up when I finish Khan academy algebra.
No worries.
There's also a book called Humongous Book of Algebra Problems
Linear Algebra (Fourth Edition) by Stephen H. Friedberg
EDIT: I just realized that you already mentioned this book in your comments. I used this book in my upper level course too and it was a real treat.
Almost forgot to reply. Linear Algebra by Friedberg is one of the more mathematically rigorous texts I've seen for undergraduates. My school used it in the honors linear algebra course. I think you'll find that it covers most of what you need. Hope it helps (if you can find it at the library or something).
Personally, I would take the time to read them both. A strong linear algebra background will be very helpful in ML. Its especially useful if you want to expand out a little bit more into other areas of signal processing. Make sure you also spend some time getting a good background in probability and statistics.
EDIT: I haven't actually read Axler's book but me and some of my friends are partial to this book.
For Geometry
https://www.amazon.com/Mathematical-History-Golden-Number-Mathematics/dp/0486400077
https://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859
For trig
https://www.amazon.com/Mathematics-Heavens-Earth-History-Trigonometry/dp/0691129738
Edit:
Boyer also has a history of calc only
https://www.amazon.com/History-Calculus-Conceptual-Development-Mathematics/dp/0486605094
And analytic geometry
https://www.amazon.com/History-Analytic-Geometry-Dover-Mathematics/dp/0486438325/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1540239797&amp;sr=1-1&amp;keywords=Boyer+geometry
And more practical for learning how to problem solve for a program is
https://www.amazon.com/Solve-Mathematical-Problems-Dover-Mathematics/dp/0486284336
There were two books that got me completely involved in the world of mathematics.
History of Pi
Golden Ration, Phi
These two books were great when I read them when I was 16 and they got me completely wrapped up in mathematics (currently I am a Physics Grad student working on my Ph.d). Well worth reading.
Try Mathematics: From the Birth of Numbers by Jan Gullberg.
An RCA cable modem/router combo unit from our ISP.
Book is called: "Mathematics from the birth of numbers"
http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_1?ie=UTF8&amp;qid=1321966815&amp;sr=8-1
There are a fair number of popular level books about mathematics that are definitely interesting and generally not too challenging mathematically. William Dunham is fantastic. His Journey through Genius goes over some of the most important and interesting theorems in the history of mathematics and does a great job of providing context, so you get a feel for the mathematicians involved as well as how the field advanced. His book on Euler is also interesting - though largely because the man is astounding.
The Man who Loved only Numbers is about Erdos, another character from recent history.
Recently I was looking for something that would give me a better perspective on what mathematics was all about and its various parts, and I stumbled on Mathematics by Jan Gullberg. Just got it in the mail today. Looks to be good so far.
For history:https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1538960578&amp;sr=1-1
Jan Gullberg's Mathematics: From the birth of numbers is a great book I'd recommend: https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X
It introduces a lot of mathematical topics starting from the "simplest" (numbers you asked about) and advances to common stuff found in university studies (although not going extremely far), but what might be the biggest feat and useful to your case is that tells as a non-fictional story while at it, explaining mathematical tools, their history and how they relate to each other extremely well in a way a normal college textbook doesn't, and it doesn't assume you already know everything from school.
You might try checking out the book, "Mathematics: From the Birth of Numbers"
Well, if you want something light and accessible and suitable for the layperson, I'm quite fond of Jan Gullberg's Mathematics from the Birth of Numbers. It goes over basically everything you would typically learn in primary and secondary school, and it presents everything with historical background. But it doesn't go into tremendous detail on each topic, and it doesn't provide the most rigorous development. It's more of a high-level overview.
But if you really want to learn some mathematics, on a deep and serious level, be prepared to read and study a lot. It's a rewarding journey, and we can give you book recommendations for specific topics, but it does take a lot of discipline and a lot of time. If you want to go that route, I would recommend starting with an intro to proofs book. I like Peter J. Eccles's An Introduction to Mathematical Reasoning, but there are many other popular books along the same line. And you can supplement it with a book on the history of mathematics (or just read Gullberg alongside the more serious texts).
I read Mathematics: From the Birth of Numbers in high school / early college. It's a long book, but it's definitely worth checking out.
I have Mathematics:From the Birth of Numbers and it’s excellent.
Highly recommend
> This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into mathematical logic, set theory, topology, fractals, probability, and assorted other mathematical byways. The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.
In a quite literal sense:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.
It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.
Also, knowing some Linear Algebra is great for Multivariate Calculus.
Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
A Friendly Introduction to Number Theory Joseph Silverman
A First Course in Mathematical Analysis by David Brannan
The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore
The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore
Introductory Modern Algebra: A Historical Approach by Saul Stahl
An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall
There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.
One more thing:
Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
Category theory is an overkill. If you think you're gonna have an easier time with it, you're mistaken. Category Theory is an extreme generalization of abstract math. Although, there's a very nice intro that you can get started with: Conceptual Mathematics: A First Introduction to Categories by Schanuel and Lawvere. It's accessible to most high school students.
What you are trying to understand is trivial. Most any intro to proofs/higher math book has an explanation of the subject.
In general, you need to learn how to think logically because the way you're going right now won't get you anywhere.
Again, read a book on the very basics of logic and sets. It would contain everything you need to know. For example,
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
> ... relation between finite and infinite.
...relation between finite sets and infinite sets. Just about everything in math is a set. There are many different types of relations. Some are functions, some are equivalence relations, some are isomorphisms.
> Just because something is an adjective or property does not mean it can't be negated.
Ok. Opposite of infinite is finite. In fact, we can say that a set is finite if it is not infinite. But limit is a number and infinity is not. You can't compare apples to oranges.
> In fact almost everything has an inverse.
Relations and special kind of relations called functions have an inverse. Also, operations can be inverse.
I really like Nancy Rodgers' "Learning to Reason".
Keith Devlin has a course called Introduction to Mathematical Thinking that covers a subset of the material in Nancy's book.
For algebra, I'd recommend Mac Lane/Birkhoff. They may not be as comprehensive as some other texts but to me, they are more motivating, and will probably provide a better introduction to categorical thinking.
For linear algebra, I'm going to suggest something slightly unusual: Kreyszig's Introductory Functional analysis with applications. Functional analysis is essentially linear algebra on infinite dimensional spaces, and it generalizes a lot of the results in finite dimensions. Kreyszig does a good job motivating the reader. I can definitely sit down and read it for hours, much longer than I can for other books, and I definitely don't consider myself an analyst. However, it could be difficult if you are not familiar with basic topology and never seen linear algebra before.
hey nerdinthearena,
i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.
helpful for intuition and basic understanding
more advanced but still intuitive
hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
three general analysis favorites
Yes. However, you should probably read something that introduces you to proofs. My Intro to Higher Math classes (commonly called Intro to Proof-Writing or Intro to Analysis, the class or series of classes that introduce you to higher math and proofwriting skills) used this book alongside a prepackaged set of detailed lecture notes. I'd say that'd be a good place to start before reading about Abstract Algebra, plus the book is dirt cheap.
This is a pretty good book too. http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&amp;qid=1323212337&amp;sr=8-1
I don't know why more people on here don't recommend it, especially considering how cheap it is.
I am not a big fan of Rudin. The tone is incredibly stuffy and his style is fairly loose.
I would recommend the small Dover book Introduction to Analysis by Rosenlicht. It's a very small book, hardly 200 pages, but the style is much nicer. It doesn't cover nearly as much (there is no introduction to Fourier Analysis, differential forms, or the gamma function), but that's a good thing for an introductory book, since you can expect to master everything in it.
We used Abbott in a class I audited. I skimmed bits of it, and it seemed pretty nice. Very expository, which is always nice to have when self-studying.
I would eventually pick up a copy of Rudin, just because it's a cultural icon. But it's just very brutal for an introduction to the subject.
Introduction to Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486650383/ref=cm_sw_r_cp_api_i_WTGPCbM6P2N4H
It’s actually a pretty decent book for a first look at Real Analysis.
Apologies for the serious comment on /r/funny.
What are you majoring in?
What you're describing could just be a personality issue that's unrelated to maths, that maths is just be an example of. That being said, I find the way people are taught maths to be a form of abuse. It's like the way someone who was molested as a child might have weird issues with sex, so do most people have issues with maths who have had to go through maths in high school.
Just so that you know, what you think maths is, is actually almost not at all what maths really is. I would recommend, after you finish your exams and have nothing better to do, read this book about graph theory. It's $4 + shipping from amazon, or you may have it in the library wherever you're studying. It's kind of pointless, but there are a few nice bits about the philosophy of maths.
Since last we spoke, I have mostly been reading:
Today I purchased:
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
To pedal off of this, graph theory is pretty much everywhere and it's really straightforward to learn. This is a really good intro book and it's really cheap.
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
Yea John Green certainly isn't for everyone, particularly outside of the YA target audience. I wouldn't say it's his strongest book either, but it might be useful to check out.
In terms of mathematical directions you could go, graph theory is actually a pretty solid field to work in. It's basics are easy to grasp, the open problems are easy to understand and explain, and there are many obscure open ones that are easily within reach of a talented high schooler. In fact a lot of combinatorics is like that as well. I would recommend the book Introduction to Graph theory by Trudeau (which was originally titled Dot's and Lines). It's a great introduction to mathematical proof while leading the reader to the forefront of graph theory.
> I'd like to know, how did you learn to use R?
My batshit crazy lovable thesis advisor was teaching intro datascience in R.
He can't really lecture and he have high expectation. The class was for everybody including people that don't know how to program. The class book was advance R http://adv-r.had.co.nz/... (red flag).
We only survived this class because I had a cs undergrad background and I gave the class a crash course once. Our whole class was more about how to implement his version of random forest.
I learned R because we had to implement a version of Random forest with Rpart package and then create a package for it.
Before this a dabble in R for summer research. It was mostly cleaning data.
So my advice would be to have a project and use R.
>how did you learn statistics?
Master program using the wackerly book and chegg/slader. (https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817)
It's a real grind. You need to learn probability first before even going into stat. Wackerly was the only real book that break down the 3 possible transformations (pdf,cdf, mgf).
I learned from Wackerly which is decent, though I think Devore's presentation is better, but not as deep. Both have plenty of exercises to work with.
Casella and Berger is the modern classic, which is pretty much standard in most graduate stats programs, and I've heard good things about Stat Labs, which uses hands-on projects to illuminate the topics.
An Introduction to Mathematical Reason - Peter Eccles. Very good book.
http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188
take the most advance math courses you can. do undergraduate research...summer programs, independent studies. make sure to write a math research paper. it doesn't have to be published, but a published paper would look great. give a talk about your research at an undergraduate math conference. go to many math conferences. many schools require the math subject test gre, which is difficult and requires a fair amount of study outside of coursework.
that being said, since you are still a beginner, be warned that upper level math is very different than high school math. after a certain point, computations are no longer of use and all math is theoretical and abstract. you will be focusing on "proofs" and generally these are much more logic based and theoretical than any math you do before university. any proofs you did proofs in a highschool geometry class are also not relevant. to get a better idea, look at an elementary proof-writing book. for example http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1320289226&amp;sr=1-2#reader_0521597188
more specifically, once you are enrolled in a phd program, you will have to take at least 2 years of coursework. you will also need to pass one or two sets of "qualifying exams", the number and style of testing is based on the university. these test you on your basic knowledge of math, and also on the subject of your research. to obtain a phd you have to do NEW mathematical research and then write a dissertation about it. the research part of the phd can take 2-4 years on average.
It depends a bit on what your college offers and what you think you would benefit from.
On the math side an intro to proofs course would be helpful because it teaches you a lot of math and reasoning that undergirds a lot of cs. A book like this is good for self study: https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188
If your school has a good philosophy dept, then check out the classes on critical reasoning and logic too. Herrick's logic book is not bad for a first course.
This was the class as it was last quarter (Spring 2012), they used this textbook. I live off-campus and only go to the UW to drop off homework and almost never talk to anyone, so you almost definitely don't know me, but perhaps we walked right by each other one time without ever knowing it. EXCITING!
John Allen Paulos' Innumeracy goes into a similar subject. He says that logic puzzles, analytical and inductive skills, and more importantly probability and statistical analysis should be taught alongside regular mathematics.
It's a short read and the man is a genius.
Argh. Numerology. It's like every logical fallacy for numbers rolled into one.
I highly recommend you pick up a copy of Innumeracy.
Innumeracy by Paulos
A great read that deal in part with the general acceptability math incompetence has compared to other subjects. Also a fun book as a "math person" just in the way he speaks and confides in the reader.
http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012
By the way, there's a book called Innumeracy which tackles the problem of the consequences of not knowing math, how come it became OK, or even fashionable to not know math in the society, places where you can apply the knowledge etc. You should give it a read, it's enjoyable and a short read. You can notice that most of the arguments you are making are similar to those that were against literacy ("I'm plantin' seeds all day and then weedin for some, what are books gonna do me for?")
Completely off-topic, but stick with it. I had the same problem. Then everything will start to fit together. In the meantime, might I suggest Adrian Banner's The Calculus Lifesaver as a really approachable second textbook/help guide/reference?
/tangent
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.
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
I am doing this very thing. I found some fantastic books that might help get you (re)started. They certainly helped me get back into math in my 30s. Be warned, a couple of these books are "cute-ish", but sometimes a little sugar helps the medicine go down:
I wish you all the best!
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
http://www.amazon.com/Trigonometry/dp/0817639144
EDIT: I don't know what ACT is, so I don't know how well it will prepare you for that.
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
I love Aluffi! It's a fun read, and more "modern" than texts like Dummit and Foote (in that it uses basic category theory freely). I like category theory, so I really enjoy Aluffi's approach.
this is the book: http://www.amazon.com/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
There is no "one fastest" method to solving them.
Systems of equations are systematic, and it really depends on the problem. The only real way to learn about this is to take a course in Linear Algebra. That is all about systems of linear equations.
But these show up all of the time, here is what I usually do:
If I just need one of the 2-3 variables, Cramer's Rule is a good way to test solvability and extract a single value.
On normal 2x2 systems, I usually do a quick determinant/matrix inverse. Checks the rank as well as the det, and it is always going to work.
On 3x3 or higher systems, it depends. This is why Linear Algebra is important.
Supposedly Linear Algebra Done Right is a good book on the subject, so if you're interested there is one way. The book I used was A custom edition of this one. I thought it was very good as well.
Working through Griffiths is a good idea, but I strongly suggest working through an abstract linear algebra book before you do anything else. It will make your life much better. Doing some of Griffiths in advance might make your homework a bit easier, but you'll be repeating material when you could be learning new things. And learning real linear algebra will benefit you in pretty much every class.
I recommend this book as your primary text and this one for extra problems and and a second opinion.
I've been reviewing linear algebra recently and found that I like my old textbook much more now than when I took the course.
https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514
Its not very good on visual intuition but there are a lot of examples. You could supplement it with the 3blue1brown series for that.
It covers a lot of the topics i needed to review for group theory. For example, it covers dual spaces and the transpose in the second chapter (it stresses invariant subspaces, projection operators, bilinear forms- essentials for group theory.). It's clear, concise and seems popular. One of the prof.s featured on Numberphile said he used it for his course. It might not be a good first linear algebra book for some people. But check it out.
I'm a huge fan of linear algebra. My favorite book for a theoretical understanding is this book. A pdf copy of the solutions manual can be found here.
Linear algebra by Friedberg, Insel and Spence
http://www.amazon.ca/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
I can post a few links from some books about numbers. I haven't read a few of them, but the history of some numbers like phi, pi, zero... all of them are fascinating.
Those six are all the history of the five most important constants in mathematics. If you're looking for the history of some of the most brilliant minds in mathematics, I'm afraid I haven't the resources or expertise to help you out.
There's an excellent book, A History of Pi by Petr Beckmann, ^^Check ^^it ^^out ^^from ^^your ^^local ^^library that explains the cross-cultural interest in this specific ratio from the ancient pre-base 10 people to modern times.
I'm a lot like you, very self directed learning (I spent as little time in HS as I possibly could, and nearly flunked out because of it). This book really sparked my interest in math. Everything from why zero is so exceptional and how hard it was for our species to realize it, to how to figure out a square root by hand (maybe boring, but I was interested in the method, since just pushing the square root button on a calc was dissatisfying).
Calculus is something that is damn near impossible to get without help (you can do it, but you probably won't understand it). Finally, it's pretty important to talk to people to see what's worth learning and what you haven't considered yet. Speaking of programming, if you fail to get yourself out in there and talk to other people (people that are better than you at something) you are liable to feel proud of inventing something like bubble-sort.
Mathematics: From the Birth of Numbers
It's gigantic, but really entertaining to flip around in.
I'm reading Mathematics: From the Birth of Numbers right now, I recommend it for your needs.
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.
No worries for the timeliness!
For Measure and Integration Theory I recommend Elements of Integration and Measure by Bartle.
For Functional Analysis I recommend Introductory Functional Analysis with Applications by Kreyszig.
And for Topology, I think it depends on what flavor you're looking for. For General Topology, I recommend Munkres. For Algebraic Topology, I suggest Hatcher.
Most of these are free pdf's, but expensive ([;\approx \$200;]) to buy a physical copy. There are some good Dover books that work the same. Some good ones are this, this, and this.
I hate to disappoint you OP, but here are some books just in my wish list on Amazon that outdo that:
This has 14, 5 star reviews.
This has 20, 5 star reviews, and 1, 4 star review.
Kreyszig is the best first book on functional analysis IMO. For measure theory I liked Royden, specifically the 3rd edition.
If you're coming from a more applied background (or physics / engineering) https://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599 is pretty easy to follow. Obviously it goes into the infinite dim too but it covers all the finite stuff first.
I have a variety of books to recommend.
Brushing up on your foundations:
http://www.amazon.com/Beginning-Functional-Analysis-Karen-Saxe/dp/0387952241
If you get this from your library or browse inside of it and it seems easy there are then three books to look at:
More advanced level:
(An awesome book with exercise solutions that will really get you thinking)
Working on this book and Rudin's (which has many exercise solutions available online is very helpful) would be a very strong advanced treatment before you go into the more specialized topics.
The key to learning this sort of subject is to not delude yourself into thinking you understand things that you really don't. Leave your pride at the door and accept that the SUMS book may be the best starting point. Also remember to use the library at your institution, don't just buy all these books.
If you really want to understand probability then you'll need to learn measure theory, which will require some background knowledge in real analysis. This is the book I used, which I highly recommend (and it's cheap!): http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&amp;qid=1414974523&amp;sr=8-1&amp;keywords=introduction+to+analysis
As for an actual book on probability, I'm not too sure since my probability course was based on lecture notes provided by the professor, although I just ordered this book because it looked decent: http://www.amazon.com/Graduate-Course-Probability-Dover-Mathematics-ebook/dp/B00I17XTXY/ref=sr_1_1?ie=UTF8&amp;qid=1414974533&amp;sr=8-1&amp;keywords=graduate+book+on+probability
https://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383
For real analysis, I would avoid Rudin. I think it's overrated as a good book to learn from, especially for people who aren't math majors. I'd go with Introduction to Analysis by Rosenlicht. It's basically a friendlier version of Rudin, and a heck of a lot cheaper.
Here is the mobile version of your link
I know this is removed, so I can recommend my tool which builds a graph of products that are often bought together at Amazon.
http://www.yasiv.com/#/Search?q=graph%20theory&amp;category=Books&amp;lang=US - this is a network of books related to graph theory. Finding the most connected product usually yields a good recommendation. In this case it recommends to take a deeper look at https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709
>I like how you came here to make a distinction without a difference
That you think these sets are equivalent is the problem with "STEM" in this country. I'm not blaming you, it's not your fault. For whatever reason, set theory is barely discussed. Even in multivariate calculus, the most you care about sets is with domain and range, just like in algebra. Here are a few topics that are mathematics, and not arithmetic:
-Set Theory
-Topology (Better than Munkres)
-Graph Theory
-Abstract Algebra (Groups/Rings/Fields)
Basic quantifiers pop up first in set theory, which as far as I can tell is only recommended after integral calculus. Things like ∀, and ∃ have a particular meaning, and their orders and quantities are very specific.
If you would like to know more about the difference between mathematics and arithmetic (which is a subset), then start with set theory. You'll need that to do anything else. I can try to answer any other questions you may have.
If anyone is interested in learning more about graph theory, this is a great (and brief) book that requires very little mathematical background. I highly recommend it.
Probability and Random Processes by Grimmett is a good introduction to probability.
Mathematical Statistics by Wackerly is a comprehensive introduction to basic statistics.
Probability and Statistical Inference by Nitis goes into the statistical theory from heavier probability background.
The first two are fairly basic and the last is more involved but probably contains very few applied techniques.
By introductory, do you mean undergrad level and advanced do you grad level?
If that is the case: The most widely used undergrad book is Wackerly et al. I also taught out of Devore before and it is not bad.
Wackerly covers more topics, but does so in a much more terse manner. Devore covers things better, but covers less things (some of which are pretty important).
Grad: Casella and Berger. People might have their qualms with this book but there is really no better book out there.
If you want to do statistics in a rigorous way you should start with calculus and linear algebra.
For calculus I recommend Paul's notes -> http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
They are really clearly written with good examples and provide good intuition.
As supplement go through 3blue1borwn Essence of calculus. I think it's an excellent resource for providing the right intuition.
For linear algebra - linear algebra - Linear algebra done right as already recommended. Additionally, again 3blue1brown series on linear algebra are top notch addition for providing visual intuition and understanding for what is going on and what it's all about.
Finally, for statistics - I would recommend starting with probability calculus - that way you'll be able to do mathematical statistics and will have a solid understanding of what is going on. Mathematical statistics with applications is self-contained with probability calculus included. https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817
Sadly the only university in my city lost their accreditation since they couldn't pay a competitive salary.
I lucked out because my Statistics professor is insanely qualified. (Ph.D in Mathematics and Ph.D in Statistics) So our Stats course covers MGFs and the derivations of all the theorems. Pretty much every question in this book: http://www.amazon.ca/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817
Thanks a lot for the response. The thought of taking on something of this magnitude with no real life mentor-ship is really daunting.
Just completed Probability this semester, and moving on to Statistical Inference next semester. Calc. B is a prerequisite, and wound up seeing plenty of it along with a little Calc C (just double integrals). I'm an Applied Mathematics undergrad major btw and former Physics major from some years ago. I wound up enjoying it despite my bad attitude in the beginning. I keep hearing from fellow math majors that Statistical Inference is really difficult. Funny thing is I heard the same about Linear Algebra and didn't find it overwhelming. I'll shall soon find out. We used Wackerly's Mathematical Statistics with Applications. I liked the book more than most in my class. Some thought it was overly complicated and didn't explain the content well. Seems I'm always hearing some kind of complaint about textbooks every semester. Good luck.
This was mine.
This book comes to mind.
I would try Mathematical Statistics and Data Analysis by Rice. The standard intro text for Mathematical Statistics (this is where you get the proofs) is Wackerly, Mendenhall, and Schaeffer but I find this book to be a bit too dry and theoretical (and I'm in math). Calculus is less important than a thorough understanding of how random variables work. Rice has a couple of pretty good chapters on this, but it will require some mathematical maturity to read this book. Good luck!
There is actually a book called An Introduction to Mathematical Reasoning.
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.
I used this book in my first mathematical reasoning class at my university.
I'm currently in my first year of undergraduate Maths and our course uses the book 'An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions' by Peter J Eccles. It's such a helpful book aimed at introducing first year university students to pure mathematics, the book has definitely helped me feel confident in my pure module.
It states propositions and theorems and proves them and gives problems for you to solve or prove with the solutions at the back.
Only tangentially relevant, but a really good read!
Innumeracy
Innumeracy: Mathematical Illiteracy and Its Consequences by John Allen Paulos, and its sequel Beyond Numeracy are two of my favorites.
Well is not exactly statistics, rather a bunch of anecdote on common mistakes and misconception about mathematics, but there is this book:
"Innumeracy: Mathematical Illiteracy and Its Consequences" by John Allen
(http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012)
and it's topic is vaguely related to OP's concern.
I haven't read it all but so far it was quite fun. Again is more anecdotal than scientific and the author might be a little condescending, but is worth reading.
Some great stuff here.
However, these are MUST READS.
First, for a good introduction to numbers, read:
Innumeracy
http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012
It explains how numbers work very, very well, in a non-technical fashion.
Second, read,
The Structure of Scientific Revolutions by Thomas Kuhn
This excellent, excellent easy to read book is simply THE BEST EXPLANATION OF HOW SCIENCE WORKS.
Next, The Way Things Work by David Macaulay. It is not a 'science' book, per se, more of an engineering book, but it is brilliantly written and beautifully illustrated.
Then, dive into Asimov. "Please Explain" is fantastic. Though dated, so is his Guide to Science.
The great thing about these books is that they are all very short and aimed at people who are not technically educated. From there I am sure you will be able to start conquering more material.
Honestly, Innumeracy and The Structure of Scientific Revolutions, alone, will fundamentally change the way you look at absolutely everything around you. Genuinely eye opening.
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!
Hello!
It's been a while since I last suggested a resource for calculus - so far, I've been finding the following two books extremely helpful and thought it would be good to share them:
http://www.amazon.com/The-Calculus-Lifesaver-Tools-Princeton/dp/0691130884/ref=sr_1_1?ie=UTF8&amp;qid=1398747841&amp;sr=8-1&amp;keywords=the+calculus+lifesaver
I have mostly been using this as my main source of calculus lessons. You can find the corresponding lectures on youtube - the ones on his site do not work for whatever reason. The material is quite good, but still slightly challenging to ingest (though still much better than other courses out there!).
When I first saw this book, I thought it was going to be dumb, but I've been finding it extremely helpful. This is the book I'm using to understand some of the concepts in Calculus that are taken for granted (but that I need explained more in detail). It actually is somewhat entertaining while doing an excellent job of teaching calculus.
The previous website I recommended to you is quite good at giving you an alternative perspective of calculus, but is not enough to actually teach you how to derive or integrate functions on your own. Hope your journey in math is going well!
These are the two things that saved my ass in calc 2:
this book, the calculus lifesaver
and this guy, Mr. McKeague from MathTV
Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.
Free:
www.khanacademy.org
free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.
http://ocw.mit.edu/index.htm
free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.
Cheap $$
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&amp;qid=1331675661&amp;sr=8-1
$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.
http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1
$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.
This calculus book has the highest reviews I've ever seen on Amazon.
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.
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
This book is golden
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:
Dead tree books that cost money:
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.
Here is a good book on trigonometry.
Here is one for algebra.
Here's another
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook
That's sounds like a horrible way to try to learn. If you think this problem is not representative of the school itself, complain (politely) to the department or dean.
I normally do not recommend Khan Academy because his methods are inefficient and boring at best, but that might actually be a step up for you.
Meanwhile, try to find a book to read out of. Unfortunately, textbook writing is a tough thing to be good at, and then a lot of publishers will get in the way of half of those.
Here are some to try though: http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144
http://www.amazon.com/Precalculus-Mathematics-Nutshell-Geometry-Trigonometry/dp/1592441300
And they're on the cheaper side
I think category theory is best learned when taught with a given context. The first time I saw category theory was in my first abstract algebra course (rings, modules, etc.), where the notion of a category seemed like a necessary formalism. Given you already know some algebra, I'd suggest glancing through Paolo Aluffi's Algebra: Chapter 0. It is NOT a book on category theory, but rather an abstract algebra book that works with categories from the ground level. Perhaps it could be a good exercise to prove some statements about modules and rings that you already know, but using the language of category theory. For example, I'd get familiar with the idea of Hom(X,-) as a "functor"from the category of R-modules to the category of abelian groups, which maps Y \to Hom(X,Y). We can similarly define Hom(-,X). How do these act on morphisms (R-module homomorphisms)? Which one is covariant and which one is contravariant? If one of these functors preserves short exact sequences (i.e. is exact), what does that tell you about X?
Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions
The Haskell Road to Logic, Maths, and Programming.
The Haskell Road to Logic, Maths and Programming takes you through a lot of the basic “essential” math for CS, much of what would be covered in a typical discrete math course, but taught along side Haskell which is fun!
Have you seen The Haskell Road to Logic, Maths, and Programming? It's a pretty decent intro to higher math, and each chapter has a Haskell module.
I posted this in /learnmath but didn't get any response so I'll give it a try here.
I'm a senior high school student and I'm learning linear algebra using Pavel Grinfeld's videos and programming in Haskell with this book.
What can I do to practice and apply concepts of linear algebra and programming?
Any recommended textbooks to complement the LA course?
Is it a good idea to solve project Euler problems in order to acquire programming/math skills?
The Haskell Road to Logic, Maths and Programming. I had already fallen in love with programming, and with Haskell, and this book showed me how well math, logic, and computer science play together. Shoutout to my aunt Trisha for giving me this book as a Christmas present in my junior year of high school
You might be interested in The Haskell Road to Logic, Maths and Programming or just google haskell+math. Formal work seems to be navigating towards haskell now. My background is in power engineering, so I'm very familiar with numerical stuff, but lacking in discrete math. That's what I'm trying to patch.
First, to get a sense as to the world of math and what it encompasses, and what different sub-subjects are about, watch this: https://www.youtube.com/watch?v=OmJ-4B-mS-Y
Ok, now that's out of the way -- I'd recommend doing some grunt work, and have a basic working knowledge of algebra + calculus. My wife found this book useful to do just that after having been out of university for a while: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005
At this point, you can tackle most subjects brought up from first video without issue -- just find a good introductory book! One that I recommend that is more on computer science end of things is a discrete math
book.
https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025
And understanding proofs is important: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108
There's this, The Book of Proof https://smile.amazon.com/dp/0989472108/ref=cm_sw_r_cp_awdb_t1_jW04BbNE8ZA19
Here ya go: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1500351882&amp;sr=8-1&amp;keywords=the+book+of+proof
Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.
It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).
You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1537570440&amp;sr=8-1&amp;keywords=book+of+proof
But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.
Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become very apparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.
The book will surely benefit you in time. Both in better understanding linear algebra and computer science classics like isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.
I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.
You could try Book of Proof by Richard Hammack. I've never read Velleman so I can't directly compare, but it's free for pdf (link to author's site above) and quite cheap in paperback (~$15). I found the explanations quite clear, the examples well worked and the exercises plentiful and helpful. Amazon reviewers seem to like it as well.
u/emiliaslarke My undergrad used “The Book of Proof” in our transition to upper level math class Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_i_YBEUDbF7NREFN
Its not too hard, is cheap, and gives a good preview
Well, skinnypenis420,
It's hard to study math on your own--it's not as fun as programming or whatnot, and especially not so when you're doing not-so-interesting high school/freshman-level mathematics. And especially when you're not really sure how to even study mathematics on your own--a skill that takes a long time to develop!
I'd say doing something like https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ would be a summer well spent. It's a short book and not very difficult. But it will probably be entirely new to you and it's a very important skill set and, most of all, it's realistic for you to study it yourself.
The Book of Proof was such a great book, I bought a copy that I regularly refer back to. It's full of worked examples, exercises, and explanations. This should be on the bookshelf of every undergrad.
Stewart, Larson, Rogawski
For calc MATH 1300/1014 and 1310/1014 you need , buy it new from the bookstore cause you will need the online code for assignments also it’s useful for calc 3 if you wanna take that. Man Wong is a good prof I had him for both 1300 and 13010
For EECS 1019 you need it’s not that useful and PDF can be found online for free and no online assignments so no need to buy it new. I had Zhihua Chang he’s a new prof but really nice but his lectures are boring. Trev tutor on YouTube is really helping with the course.
For Math 1025/1021 you need I found the book helpful but unlike calc some profs tend not to use this book so I’d hold out of buying it but most profs use lyryz which is an online assignment program so you will need to buy that. I had Paul Skoufranis, amazing prof but had tests. The book is also useful for linear 2 but again depend if the prof uses it
For EECS 1022 you need
It’s a good book and the guy you wrote it teaches the class.
PM if you have any other questions
Textbooks in the US are priced for what students will pay, not for their actual cost, because the textbook market isn't a free market for students. You either buy the course's reccomended textbook, or find some other way to access the material. You can't shop between different publishers of the same book, unless you start looking at international editions.
>Paying for content btw
you do know that a digital copy of the text book isn't free. And no you can't use the price for a digital copy that you can buy for personal use. There would be a rental charge. The calc book I used for 3 semester of calculus in College is $32/semester to rent. so that means schools are probably paying round $50/year for each digital copy of a text book.
So if you think a school book costs $250 it becomes cheaper than rental after the 5th year (not even including the increased cost of the chromebook and "insurance" required by the student.
Rental
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart-ebook/dp/B00T9X7THG/ref=sr_1_6?s=digital-text&amp;ie=UTF8&amp;qid=1536864274&amp;sr=1-6
physical copy
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=mt_hardcover?_encoding=UTF8&amp;me=&amp;qid=1536864274
https://www.amazon.ca/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552
Pretty sure it's this one. You should be able to find a pdf online.
I strongly suggest you take your time learning calculus, because anything you don't grasp completely will come back to haunt you.
But the good news is that there are lots of great resources you can use. MIT OCW has a full course with lectures, notes, and exams. Here are three free online books. If you're looking to buy a textbook, some good choices are Thomas, Stewart, and Spivak. (You can find dirt-cheap copies of older editions at abebooks.com.)
If you want more guidance, another great place to find it is at /r/learnmath.
That depends on your own level, your goals and your ambition. For example, OP wants to learn machine learning. Assuming OP's highschool math is solid, it might be possible for OP to simply download pytorch and immediately start programming neural networks without worrying too much about the hardcore math in the background.
On the other hand, if OP is more serious about improving as a mathematician, and assuming nothing but highschool math, I would start with linear algebra and differential and integral calculus. The famous professor Gil Strang has an excellent book on linear algebra, which is strangely available online. For differential and integral calculus, probably the standard reference is Stewart's book. At this point, OP would have all the basic things needed to start with machine learning. I'm not aware of the literature for machine learning so I can't recommend any specific books.
If OP wanted to get sidetracked learning more things before plunging into machine learning then the obvious choice would be Scientific Computing (my friends wrote an excellent book on the subject). Scientific Computing is the science of calculating things using computers and supercomputers. In addition, the area of Mathematical Optimization is good to know because Stochastic Gradient Descent is omnipresent in machine learning, but I don't know enough about optimization to recommend a book. There is Boyd and Vandenberghe but that is only for convex optimization. Some more areas that are related and useful are Probability and Statistics.
Sometimes you can find what textbook your school uses before the semester starts (I'm also the weird kid that emails the professor asking about books if I cant find it >.>). Some of my professors have what material they use for each class on their personal web pages though. For calculus, you'll most likely use this book. My brother used it at his Uni my friend at another and I myself used it at mine. Not sure if you're registered yet though. I had a weird case going into my Uni because I did community college then took summer courses so I was enrolled earlier than students who transfer and probably the freshman. YouTube videos will also be your best friend. People I liked for my math classes are TrevTutor (I don't think he ever finished his Calc 2 series) and PatrickJMT. Hope this helps a bit if you have any other questions or need more clarifications don't hesitate to ask.
I recommend this book.
If you want an extremely practical book to complement BDA3, try Statistical Rethinking.
It's got some of the clearest writing I've seen in a stats book, and there are some good R and STAN code examples.
I found 'Understanding Analysis' by Stephen Abbott ( https://www.amazon.co.uk/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ) to be super helpful/enlightening post Real Analysis insofar that it helped me build an intuition and understanding for some of the key ideas. Earlier today someone highly recommended this book as well: 'A Story of Real Analysis'
http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ (download link on the right). I had a quick glance through it and it seems pretty good.
This free pdf book should help you: Proof, Logic, and Conjecture - The Mathematician's Toolbox
It's really well written (I like it better than Velleman's How to Prove It.) After this you should go through something easier than Rudin, like Spivak Calculus. Then you can try a real analysis book, but try using Abbott or Pugh instead; I hear those books are much better than Rudin.
Your question is pretty vague because studying "mathematics" could mean a lot of things. And yes, your observation is correct: "There are a lot of Mathematical problems which are extremely difficult". In fact, that's true for a lot of people as well. So I suggest that you choose a certain field and delve into that.
For proof based subjects, the most basic to start with is Real Analysis. I recommend Stephen Abbott's Understanding Analysis as it is a pretty well-explained book.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
This may be good for example:
http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ref=pd_sim_b_5?ie=UTF8&amp;refRID=0H1GD8HDQZB58PWTY0F5
You could take a look and see if it suits you.
But don't trust me on this. Others on /r/learnmath or /r/matheducation may be more knowledgeable than me about good algebra workbooks.
I'm late to this party, but as a lot of other people have said, missing a negative sign somewhere is not an indication that you're bad at math. What is important in math is understanding why things are the way that they are. If you can look at the spot where you missed a negative sign and understand exactly why there should have been a negative sign there then you're doing fine. Being good at math isn't so much about performing the calculations—I mean, computers can find the roots of a quadratic function polynomial pretty reliably, so probably no one's going to hire you to do that by hand—but it's following the chain of reasoning that takes you from problem to solution and understanding it completely.
That said, there are things you can do to make yourself better at performing the calculations. Go back to basics, and I mean wayyyy back to like grade 5. A lot of students are seriously lacking skills that they should have mastered in around grade 5, and that will really screw up your ability to do algebra well. For instance, know your times tables. Know, and I mean really know and understand, how arithmetic involving fractions works: how and why and when do we put two fractions over a common denominator, what does it mean to multiply and divide by a fraction, and so on. It's elementary stuff but if you can't do it with numbers then you'll have an even harder time doing it with x's and y's. Make sure you understand the rules of exponents: Do you know how to simplify (a^(2)b^(3))^(2)? How about (a^(3)b)/(ab^(5)) How about √(3^(4))? What does it mean to raise a number to a negative power? What about a fractional power? These things need to be drilled into you so that you don't even think twice about them, and the only way to make it that way is to go through some examples really carefully and then do as many problems as you can. Try to prove the things to yourself: why do exponents behave the way that they do? Go out and get yourself something like this and just work through it and make sure you understand exactly why everything is the way that it is.
Feel free to PM me if you are stuck on specific stuff.
this is one of the best for self teaching. the examples are very clear so you don't get tripped up on them jumping steps. You will need to get more problems from somewhere like a more formal textbook, but this will help you get the idea of what to do instead of fuming at an impasse.
http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129
there's also trig and precalc versions if he needs the review.
http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ref=pd_bxgy_b_text_z
http://www.amazon.com/The-Humongous-Book-Trigonometry-Problems/dp/1615641823/ref=pd_bxgy_b_text_y
It's a lot of work but with this book I lost my math anxiety and actually started to enjoy math. The author's philosophy is the only way to get better at algebra is to just do a lot of algebra, it starts out with the most basic fundamentals you need to know too, like if you have trouble with negative numbers or fractions (as I did). It's possible you just need a recap on the foundational stuff you forgot in grade school + more practice. By the end of the book you'll be working with functions and logarithms and you'll understand it.
> Turning in assignments should not be locked behind a pay wall. A student should not fail the class just because they didn't buy it.
You're not wrong, but I have a few issues with that.
First, do you really believe that the school does not have a system in place to help those that genuinely cannot afford it? Every class I've had that mentioned Cengage had the teacher explicitly mention that if paying for it was a problem, to get in contact with them.
As I mentioned, I used to go to a different school. $125 per semester, required by every math class I took. It's a good step down for me to pay that much in a year.
Second, we've got 750 students this semester in Calc 3 alone. I've got 3 assignments that were due yesterday, and 2 more due Monday.
If all the assignments only had 6 questions each, that's ~22k questions to be graded this week. Somebody has to do it. UCF is apparently even making their own software/site for this, but regardless of when it gets finished you know we're gonna foot the bill. One way or another we pay for this shit to get done.
> Also, access codes hurt the used textbook market.
You're not wrong but if we can get the textbook + assignments graded for the same price, what's the big deal?
Renting my textbook for Calculus would have cost the same as paying for access, and I covered both Physics classes too (along with whatever else I want to study on)
Beside the point either way. My issue was that the OP was full of shit, not 'Oh poor Cengage.' My bad for expecting people here to read instead of jumping in on another circlejerk.