Reddit reviews Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra
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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"
It just comes from the way we define sums of infinite sums, aka series. .999... is just shorthand for (.9+.09+.09+.009...), which is an infinite sum. We define the sum of a series to be equal to the limit of the partial sums. The limit is rigorously defined, and you can read the definition on wikipedia if you google "epsilon delta". The limit of an infinite sum, if it exists, is unique. For this infinite sum, that limit is exactly 1. By the way we define infinite sums, .999... is therefore exactly equal to 1.
It's not so bad when you remember that all real numbers have a representation as a non-terminating decimal. 0.5 can be written as 0.4999... and 1/3 can be written as 0.333... and pi can be written as 3.14159... for example.
And lastly, if .999... and 1 are different real numbers, then there must exist a number between them. This is because of an axiom we have called trichotomy: for any two real numbers a and b, exactly one of the following is true: a<b, a=b, a>b. If a=/=b, then there exists a real number between them, because the real numbers have a property called "dense". It is easy to prove that here is no such number between .999... and 1, real or otherwise. Therefore .999... is exactly equal to 1.
e: The sum (.9+.09+.009...) is bigger than every real number less than 1. You can check if you want. The smallest number that is greater than every real number less than 1 is 1 itself. We get this from an axiom called the "least upper bound property". Therefore .999... is at least 1. Using our rigorous definition of a limit, we find that it is exactly 1.
e2: Apostol's Calculus vol 1 is a fantastic place to start learning about rigorous math shit. Chapter one starts you out with axioms for real numbers, and about half way through chapter 1 you prove the whole thing about repeating decimals corresponding to rational numbers. It is slow and easy to follow. Other people recommend Spivak but I haven't seen it so idk.
If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.
If your degree is anything else, you may have no formal requirement to learn the why.
For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.
I speak of this famous calculus book: https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Which is a "theoretical" approach to Calculus rather than a mechanical approach.
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.
Apostol and Spivak are the best calculus texts I know; paperback versions of each exist.
It's probably not possible to review everything you need, but getting more experience with proofs is a good start. This course might be helpful:
https://www.coursera.org/course/matrix
and these texts are great examples of mathematical thinking in prose:
Grinstead and Snell's Introduction to Probability:
https://math.dartmouth.edu/~prob/prob/prob.pdf
Apostol's Calculus I and II:
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Start With "Foundations Of Analysis" By Edmund Landau
http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X
It's a tiny book, but is very good at explaining basic abstract algebra.
Here is the description from Amazon:
"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."
With the above book you should then have enough knowledge to move on to calculus.
I recommend the two volume series called "Calculus" by Tom M. Apostol.
The first volume is single variable calculus and the second is multivariate calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-4
http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-3
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.
Apostol's classic calculus textbook, used at Caltech and MIT. The Art of Problem Solving textbook for calculus. The Stanford and Harvard-MIT Math Tournaments have calculus subject tests. The college-level Putnam competition has calculus problems, in addition to linear algebra, abstract algebra, etc.
Calculus Made Easy -- Can't get much better as far as bang for the buck. Follow it up something more rigorous. Maybe, Calculus, Vol 1 by Apostol. The problem with Apostol, as most calculus texts, is price.
Apostol's Calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Here you go. Apostol wrote this classic a while back, and it's currently used at MIT. It treats integration before differentiation. It is mathematically more mature than anything most engineers will ever encounter.
If you're looking at it from a mathematical "I want to prove things" standpoint, I'd recommend Apostol. I've also heard good things about Spivak, although I've never read that book.
If you're looking at it from an engineering "Just tell me how to do the damn problem" perspective, I'm no help to you.
The popular opinion by some mathematical elite is that Stewart dumbs down calculus, focuses too much on applications, and not enough on theory, which is important for those moving beyond to real analysis and other upper division courses. You should read the reviews of Spivak's or Apostol's calculus text books to see what I mean.
I'm currently on this journey as well! I'm a programmer teaching my self rigorous maths, so I can definitely help you out.
I find it's best to simultaneously look at several resources on topics such as proofs, so you can get a few perspectives on the same essential topics and have an easier time of finding something.
As a preliminary to proofing, I would suggest a survey of basic logic and Set Theory. I picked up my Set Theory from google searches and the introduction in Apostol's Calculus, and wiki articles on logic and set operations.. It's really easy to learn enough set theory and logic to begin understanding rigorous proofs.
To learn the proofing skills needing for Real Analysis I recommend
a) "Foundations of Analysis" by Edmund Landau.
b) Math 378: Number Systems: An Axiomatic Approach
For an actual book on real analysis, there can be no greater book than Apostol's Calculus.
Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).
Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.
Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.
Spivak or Apostol, for me Spivak is the best.
It's rigorous and easy to understand, if you bite through the whole book you will know Calculus.
But for Intro to Physics, I would suggest going through the MIT videos and do all the assignments and exams.