My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.

Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.

General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.

Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.

Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.

Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.

Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.

Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.

Just finished watching it. Thanks for letting me know.

I enjoyed it, but I'm wondering if there are any others who share my misgivings. On the whole it seems a bit like an elitist morality play, especially with the intermittent text. Incidentally I have a very strong grasp of certain aspects of the film's source material, so I was able to appreciate it as a well-crafted work of historical fiction. However, I fear that someone without my background would mistake some of the very bold liberties taken in the script as fact. There are things which are, to be flattering, horribly anachronistic and speculative. If anyone is interested I can expound more, but as a blanket statement, I will say that we know very little about Hypatia and her discoveries, as well as very little of the actual contents of the library. For those interested in getting their hands dirty, buy and read these books (It will take a lot of time and effort), in the following order. They are all pretty much inaccessible until you've read Euclid's Elements rather thoroughly.

Euclid's Elements, Translated by T.L. Heath (one of my Personal Heros). You might also look into the dover editions of the Elements(in 3 volumes), as they contain Heath's fascinating notes.

Ptolemy's Almagest (or Syntaxis), Talliafero translation. I advise against the Toomer, since he takes liberties in re-organizing the tables, obscuring the methods actually used by Ptolemy (and his slave-scribes) to calculate them. People disparage the "great books" collection with good and bad reasons, but I find the mathematical volumes comfortable to study with. Beware of typographical errors and omissions though, of which there are, sadly, quite a few. Another nice thing about this version is you can immediately read Copernicus and Kepler. If you make it through this volume, you will be able to explain exactly how humanity knows that the world is round, revolves around the sun, and does so in an ellipse, in exactly the way the people who explained them first and/or best did.

Next, for some interesting perspective, here is Heath's magnificent compendium of material on Aristarchos himself. http://www.amazon.com/Aristarchus-Samos-Ancient-Copernicus-Astronomy/dp/0486438864/ref=sr_1_1?s=books&ie=UTF8&qid=1301222044&sr=1-1

If you really want to split your mind-cock in half, try Apollonius' Conics If you can find the "Great Books" version, I think the diagrams are better-drawn, but to each his own.

To really master the mathematics in these books takes about 18 months of dedicated study. But if you are able to get through them, you will stand in awe upon the mount of time, peering back over our collective past like a trillionaire's son on the day of his inheritance. Besides, if you're serious about atheism, you'll probably be conned into spending that much time reading the bible. Which would you rather know by heart?

For more information on Hypatia herself, this is a good start: http://www.polyamory.org/~howard/Hypatia/primary-sources.html

One of my favorite things about the movie was how it constantly zoomed in and out from the surface of the planet, and the cinematography of the final scene was fucking fantastic.

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

1. What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)
   
   The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.
   
   As a general rule, I'd say that working on proof-based contest questions that are just beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.
   
   
2. What's your current mathematical background?
   
   This may be especially true for things like logic and very elementary set theory.
   
   
3. What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?
   
   Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)
   
   
4. What resources are available to you for vetting your work?
   
   Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's orders of magnitude more effective when you have someone who can guide you.
   
   
5. Are you trying to learn the basics of mathematical proofs, or genuinely rigorous mathematical proofs?
   
   Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone else?
   
   
6. What experience have you already had with proofs in particular?
   
   Have you had at least, for example, a geometry class that's proof-based?
   
   
7. How would you characterize your general writing ability?
   
   Proofs are all about communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.
   
   ---
   
   With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.
   
   

• The book How to Prove It: A Structured Approach by Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as is How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  
• Since you've already taken calculus, it would be worth reviewing the topic using a more abstract, proof-centric text like Calculus by Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.
  
  
• In order to learn how to write mathematically sound proofs, it helps to read as many as you can find (at a level appropriate for your background and such). You can find plenty of examples in certain textbooks and other resources, and being able to work from templates of "good" proofs will help you immeasurably.
  
  
• It's like the old joke about how to get to Carnegie Hall: practice, practice, practice.
  
  Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.
  
  ---
  
  How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!

I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.

If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.

If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.

Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.

After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.

After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.

Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).

If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!

TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres

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

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  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
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

The best description of Fractal Dimension that I am presently aware of is the one presented in Mandelbrot's book: The Fractal Geometry of Nature.

You start off some time in the 19th or early 20th century, when cartographers were trying to work out the length of the coastline of Britain. Despite cartography being so mature of a discipline that we can launch rockets and photograph the Earth from space for the first time and find basically zero surprises compared to what we've already mapped by crawling across the surface like microbes on a watermelon, here we are with a dozen survey teams all reporting lengths for the same portions of British coastline off by factors of 2-5. I mean, it's simply preposterous!

Hell, cartographers from Portugal are reporting coastal lengths for their country — with impeccable methodology, mind you — greater than Spanish cartographers find for the entire Iberian peninsula.

Well, somebody did a meta-analysis and found that reported coastal lengths not only correlate directly with what atomic measurement scale the surveyors used (EG: over how short of a distance do you stop trying to count the winding details), but the correlation was exponential and it followed different exponential constants for different coastlines. For example, shrinking how short a measuring stick you use to measure the coastline of West Britain by N will give you a total length that is longer by about N^1.25, regardless the starting value of your yardstick or the value you choose for N.

Mathematically, this means that if you keep shrinking your yardstick and count every bay, every outcropping of rock, every pebble, every molecule dividing a time-perfect snapshot of sea from land, the total length that you measure will not converge onto any attractor representing the "real" length of the coastline.. it will instead predictably diverge to infinity.

But we get the same effect if we try to measure the "length" of a square area, say 1 foot square. You can try splitting it into square inches, by lining them up in a row and seeing that they measure 144 inches long. Or you can divide smaller into square half-inches.. but now they get to be 288 inches long. And splitting more finely by N always nets you a "length" that is N^2 yardsticks "longer".

So, any mathematician would just patiently explain to somebody trying to find such a length that there isn't one because they're trying to measure magnitude in the wrong number of dimensions, and that the exponential constant they are running against is the number of dimensions they should measure with to get a reliable and finite result.

That said, one can theoretically measure the coastline of Britain and converge to a finite result so long as they are constantly considering inch^1.25 's, but of probably more use is the understanding that the 1.25 gives us a reliable measure of how "rough" the coastline is: how much extra length one gets from studying another successive factor of detail. :)

All surfaces that remain "rough" or bumpy no matter how far you zoom in can be said to have fractal dimension. From "dusts" of points like the cantor set (log(2)/log(3) ≈ 0.631) between dimensions 0 and 1 .. infinitely complicated collections of elements each dimension 0 to coastlines like the Koch Snowflake (log(3)/log(4) ≈ 1.2619) or foams like the Seirpinski Triangle (log(3)/log(2) ≈ 1.585) between 1 and 2.. infinitely complicated collections of (or kinks in) elements each dimension 1, to surfaces like any given land area on Earth, or foams like the Menger Sponge (log(20)/log(3) ≈ 2.727) with dimensions between 2 and 3 represented by infinitely varied kinks and folds in 2d elements or continued aspiration of 3d elements until all 3d volume is lost. (obviously cantor set and sierpinski triangle can equally be described as aspiration of larger-dimensional solids as well! ;D)

Fractional dimensionality can obviously be extended farther, and even measurably in our own universe one can posit that the gravitational warping of spacetime around infinitely varied mass distribution gives us slightly greater than 4 space+time dimensions prior to even leaving the bounds of mundane general relativity: EG, any attempted measurement of volume * duration of any portion of the universe is doomed to diverge to infinite values by some constant as your measuring stick to account for smaller and smaller curvatures around smaller and smaller gravity wells keeps shrinking.

But in addition to cylindrical and spherical coordinate systems (themselves just elliptical dimensions combined with euclidean ones) it is fun to consider more exotic additions like hyperbolic dimensions (Yeah, you can cross hyperbolic dimensions with Euclidian ones in the same space) or fractional dimensionality or add more Minkowski dimensions because you did remember that we already have one of those, right? Well heck, we can even take that one away and make it Euclidian instead. xD

But yeah, it's true that "adding more Euclidean spatial dimensions to our 3E+1M reality" is a fun thought exercise, and that the result of adding more E is the same as adding more elements to a vector for our linear algebra formulas to nom upon. And there are a ton of fun alternative to consider as well. :)

> For those who prefer video lectures, Skiena generously provides his online. We also really like Tim Roughgarden’s course, available from Stanford’s MOOC platform Lagunita, or on Coursera. Whether you prefer Skiena’s or Roughgarden’s lecture style will be a matter of personal preference.
>
> For practice, our preferred approach is for students to solve problems on Leetcode. These tend to be interesting problems with decent accompanying solutions and discussions. They also help you test progress against questions that are commonly used in technical interviews at the more competitive software companies. We suggest solving around 100 random leetcode problems as part of your studies.
>
> Finally, we strongly recommend How to Solve It as an excellent and unique guide to general problem solving; it’s as applicable to computer science as it is to mathematics.
>
>
>
> [The Algorithm Design Manual](https://teachyourselfcs.com//skiena.jpg) [How to Solve It](https://teachyourselfcs.com//polya.jpg)> I have only one method that I recommend extensively—it’s called think before you write.
>
> — Richard Hamming
>
>
>
> ### Mathematics for Computer Science
>
> In some ways, computer science is an overgrown branch of applied mathematics. While many software engineers try—and to varying degrees succeed—at ignoring this, we encourage you to embrace it with direct study. Doing so successfully will give you an enormous competitive advantage over those who don’t.
>
> The most relevant area of math for CS is broadly called “discrete mathematics”, where “discrete” is the opposite of “continuous” and is loosely a collection of interesting applied math topics outside of calculus. Given the vague definition, it’s not meaningful to try to cover the entire breadth of “discrete mathematics”. A more realistic goal is to build a working understanding of logic, combinatorics and probability, set theory, graph theory, and a little of the number theory informing cryptography. Linear algebra is an additional worthwhile area of study, given its importance in computer graphics and machine learning.
>
> Our suggested starting point for discrete mathematics is the set of lecture notes by László Lovász. Professor Lovász did a good job of making the content approachable and intuitive, so this serves as a better starting point than more formal texts.
>
> For a more advanced treatment, we suggest Mathematics for Computer Science, the book-length lecture notes for the MIT course of the same name. That course’s video lectures are also freely available, and are our recommended video lectures for discrete math.
>
> For linear algebra, we suggest starting with the Essence of linear algebra video series, followed by Gilbert Strang’s book and video lectures.
>
>
>
> > If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
>
> — John von Neumann
>
>
>
> ### Operating Systems
>
> Operating System Concepts (the “Dinosaur book”) and Modern Operating Systems are the “classic” books on operating systems. Both have attracted criticism for their writing styles, and for being the 1000-page-long type of textbook that gets bits bolted onto it every few years to encourage purchasing of the “latest edition”.
>
> Operating Systems: Three Easy Pieces is a good alternative that’s freely available online. We particularly like the structure of the book and feel that the exercises are well worth doing.
>
> After OSTEP, we encourage you to explore the design decisions of specific operating systems, through “{OS name} Internals” style books such as Lion's commentary on Unix, The Design and Implementation of the FreeBSD Operating System, and Mac OS X Internals.
>
> A great way to consolidate your understanding of operating systems is to read the code of a small kernel and add features. A great choice is xv6, a port of Unix V6 to ANSI C and x86 maintained for a course at MIT. OSTEP has an appendix of potential xv6 labs full of great ideas for potential projects.
>
>
>
> [Operating Systems: Three Easy Pieces](https://teachyourselfcs.com//ostep.jpeg)
>
>
>
> ### Computer Networking
>
> Given that so much of software engineering is on web servers and clients, one of the most immediately valuable areas of computer science is computer networking. Our self-taught students who methodically study networking find that they finally understand terms, concepts and protocols they’d been surrounded by for years.
>
> Our favorite book on the topic is Computer Networking: A Top-Down Approach. The small projects and exercises in the book are well worth doing, and we particularly like the “Wireshark labs”, which they have generously provided online.
>
> For those who prefer video lectures, we suggest Stanford’s Introduction to Computer Networking course available on their MOOC platform Lagunita.
>
> The study of networking benefits more from projects than it does from small exercises. Some possible projects are: an HTTP server, a UDP-based chat app, a mini TCP stack, a proxy or load balancer, and a distributed hash table.
>
>
>
> > You can’t gaze in the crystal ball and see the future. What the Internet is going to be in the future is what society makes it.
>
> — Bob Kahn
>
> [Computer Networking: A Top-Down Approach](https://teachyourselfcs.com//top-down.jpg)
>
>
>
> ### Databases
>
> It takes more work to self-learn about database systems than it does with most other topics. It’s a relatively new (i.e. post 1970s) field of study with strong commercial incentives for ideas to stay behind closed doors. Additionally, many potentially excellent textbook authors have preferred to join or start companies instead.
>
> Given the circumstances, we encourage self-learners to generally avoid textbooks and start with the Spring 2015 recording of CS 186, Joe Hellerstein’s databases course at Berkeley, and to progress to reading papers after.
>
> One paper particularly worth mentioning for new students is “Architecture of a Database System”, which uniquely provides a high-level view of how relational database management systems (RDBMS) work. This will serve as a useful skeleton for further study.
>
> Readings in Database Systems, better known as the databases “Red Book”, is a collection of papers compiled and edited by Peter Bailis, Joe Hellerstein and Michael Stonebreaker. For those who have progressed beyond the level of the CS 186 content, the Red Book should be your next stop.
>
> If you insist on using an introductory textbook, we suggest Database Management Systems by Ramakrishnan and Gehrke. For more advanced students, Jim Gray’s classic Transaction Processing: Concepts and Techniques is worthwhile, but we don’t encourage using this as a first resource.
>

> (continues in next comment)

I'm in a similar boat with you. I went through calculus in high school, graduated university with a B.A. in music, but have recently taken a keen interest in developing an actual understanding of math.

Aside from music, I have a strong background in philosophy, and from philosophy, so do the natural sciences extend and I've taken advantage of that. Math was discovered through raw observation of the world and through the concourse of logic, and so I have designed for myself the study of math through the source works of where the math originated, for practical and ontological purposes. Here's a few books that I've picked up and began reading:


A History of Greek Mathematics, Vol. 1: From Thales to Euclid https://www.amazon.com/dp/0486240738/ref=cm_sw_r_cp_apa_RljGybYRSB723

The Mathematical Principles of Natural Philosophy: The Principia https://www.amazon.com/dp/1512245844/ref=cm_sw_r_cp_apa_AmjGyb14R4B2V

Euclid's Elements https://www.amazon.com/dp/1888009187/ref=cm_sw_r_cp_apa_7mjGybZ97DBR7


Introduction to Mathematical Philosophy https://www.amazon.com/dp/1420938401/ref=cm_sw_r_cp_apa_OnjGybQ0078ZX

The Fractal Geometry of Nature https://www.amazon.com/dp/0716711869/ref=cm_sw_r_cp_apa_lojGybPPY25P4

The study of equations and formulas had been unfulfilling and unengaging until I framed it with the historical context of the natural sciences. I'm still a novice to this approach, but I believe it to be of merit- Ive also see some indication (when researching my own self-study method) that this is more similar to the method which Waldorf schools teach math and science as opposed to the traditional American Public school classroom, which as I grow older and reflect upon the majority of my experiences in classrooms, were uninspired, with the exception of very few memorable educators.

You could even base your study on other, less abstract interests than the interest of learning mathematics, such as an interest in modern physics or economy (or Comp sci, anything that utilizes math). Using that interest as a guide, you would be more clear minded to reverse-engineer your own individually purposed self-study. Such a direction of interest would certainly help for you to be able to design your course and keep you engaged. I hate how I've worded most of this Frankenstein of a comment; it's unnecessarily verbose and unorganized, but it's late and I'm tired to I'm not gonna edit it, nevertheless, hopefully you'll get the point(s).

Anyway, I'm curious what other people have to say about this approach, and especially I am open for people to suggest in response here to additional and essential sourcebooks!

> My lack of confidence, I think, is centered around the idea that complex math is a prerequisite to engineering. So I have a few questions.

The math in engineering is not hard, comparative to actual math. It's low tier within math departments (100/200 level), and most professors aren't expecting you to create some new theorem. You're there to apply theorems that have already been well established within the discipline.

> How hard is EE?

People say it's the hardest of the engineering disciplines. I've taken a few EE courses, and didn't have much trouble, granted, it's only circuits and the second physics course. Difficulty will probably depend on how much you catch on with the subject.

> What kind math will I need to learn?

Algebra, Trigonometry, Analytic Geometry, Calculus (Differential, Integral, Multivariable), and Differential Equations. Most universities merge Algebra, Trig, Analytic Geometry into the Calculus courses.

> Will the stuff from high school matter?

Of course. I am of the opinion that everything matters and is useful. All your science, math, english, and even history can come into play depending on your engineering discipline.

> At what point do you start?

Depends. I did Calculus in high school, but retook it since I added my engineering major in my junior year. Met some kids who came in with Calculus I and II credits and started in Multivariable. Some kids who started with Calc II and some who started with algebra or pre-calc. It really depends on you.

> Do professors assume you know close to nothing?

Depends on the professor. My statics professor was very hand holdy. A Physics professor was very hands off. My Calc III professor would teach you a concept and expect you to be able to apply it to contexts you were never taught. There is not a blanket statement to really apply to this question.

> Will I have to teach myself in order to catch up?

You should be teaching yourself because you want to improve, not to reach some arbitrary point.

If you want to be an engineer and you find deficiencies in your skills, then you need to take some time and think about what you want. I am, unfortunately, in the camp of, you should pursue what you're passionate in, as you are more likely to do that work without being told, hence, getting better. It took me several years to actually figure out that I wanted to be an engineer, and despite not having a math course in three years, I went in quite over prepared because I was already doing the work for fun.

There's no rush to make a decision, but in the event you do decide to work on being an engineer, here's some useful references.

Just-in-Time: http://www.amazon.com/Just--Algebra-Trigonometry-Transcendentals-Calculus/dp/0321671031

Great book to brush up and refine your algebra and trig skills

Schaum's Outlines: http://www.amazon.com/Schaums-Outline-Calculus-6th-Problems/dp/0071795537/ref=sr_1_1?s=books&ie=UTF8&qid=1465145307&sr=1-1&keywords=schaum+outline+calculus

I am in love with this series. I buy a book for every class I take (and they have one for every engineering/math/physics course). It's a great supplemental text, and this particular book covers Calculus I, II, III and a bit of Differential Equations!

Paul's Online Notes: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx

Paul has great lecture notes and tons of practice problems. Invaluable resource.

Using Paul's site, you can also scan what kind of math is in engineering. Bit of warning, some of it will look scary at first. When I first started, the symbol for a partial derivative looked really intimidating. When you reach that point, you'll laugh at the fact you found it intimidating in the first place.

Best of luck OP!

I hope others will chime in here, but I'll answer as well as I can.

Euclidean and Non-Euclidean Geometry

Euclidean and non-Euclidean geometries are interesting and important for various reasons, so I certainly wouldn't say it's a bad idea to study them in depth.

If you want to study these subjects first because you find them interesting and you have plenty of years to spend, then go for it! However, it's not necessary (more on this below).

Multivariable Calculus and Linear Algebra

Before attempting even an elementary treatment of differential geometry, you'll want to have a working knowledge of calculus (single and multivariable) and linear algebra.

Elementary Differential Geometry

You could potentially skip the elementary treatments of differential geometry, but these might be useful for tackling more advanced treatments. Studying elementary differential geometry first is perhaps similar to taking a calculus class (with an emphasis on computation and hopefully on intuition) before taking a class in real analysis (with an emphasis on abstraction and rigorous proofs).

If you do want to work through an elementary treatment, then you have options. One well reviewed book, and the one I learned from as an undergraduate, is Elementary Differential Geometry by Barrett O'Neill.

Note that O'Neill lists calculus and linear algebra as prerequisites, but not Euclidean and Non-Euclidean geometry. Experience with Euclidean geometry is definitely relevant, but if you understand calculus and linear algebra, then you already know enough geometry to get started.

Abstract Algebra, Real Analysis, and Topology

The next step would probably be to study a semester's worth of abstract algebra, a year's worth of real analysis, and optionally, a semester's worth of point-set topology. These are the prerequisites for the introduction to manifolds listed below.

Manifolds

An Introduction to Manifolds by Loring W. Tu will give you the prerequisites to take on graduate-level differential geometry.

Note: the point-set topology is optional, since Tu doesn't assume it; he expects readers to learn it from his appendix, but a course in topology certainly wouldn't hurt.

Differential Geometry

After working through the book by Tu listed above, you'd be ready to tackle Differential Geometry: Connections, Curvature, and Characteristic Classes, also by Loring W. Tu. There may be more you want to learn, but after this second book by Tu, it should be easier to start picking up other books as needed.

Caveat

I myself have a lot left to learn. In case you want to ask me about other subjects, I've studied all the prerequisites (multivariable calculus, linear algebra, abstract algebra, real analysis, and point-set topology) and I've tutored most of that material. I've completed an elementary differential geometry course using O'Neill, another course using Calculus on Manifolds by Spivak, and I've studied some more advanced differential geometry and related topics. However, I haven't worked through Tu's books yet (not much). The plan I've outlined is basically the plan I've set for myself. I hope it helps you too!

There are many but they depend on the logic you want to solve.

Good general problem solving, How to Solve It by Polya is a standard in math departments. Much of that information applies to solving any rational problem. A lot of it is not as relevant to a programmer and it will be well more than most people would need but if you are going to get a wrench why not get fully stocked a toolbox?

Standard binary logic, !(a and b) = (!a or !b) type stuff? I don't have any specific recommendations but look to philosophy sections for books on logic. Philosophy literally wrote the book on that topic before math latched on to it. Most math books on the topic will be particularly unwieldy and overly broad to what a programmer might need.

Any of those books is likely to go well beyond what you actually need. None of them are programming focused, programmers tend to learn this stuff by example or practice debugging is a great cause-effect based teacher and if you practice you will learn. Can you be more specific about what logic you need to improve and what level of skill you feel you have?

Someone else recommended learning math from the pre-calc up. I would second that if it is an option but that can be a very long road and some people just shut down at math. I know some great programmers who failed college freshmen level math classes. I know many other really intelligent and capable people who do not believe 'they can math' so I have tried to offer another path.


Edit: I meant to include with the amazon link, look for older editions on any textbooks and evaluate whether the comments and reviews indicate significant change worth the new edition price. Books on logic don't really change much but sometimes they will reword examples or update them to be better. I didn't remember to note that until the amazon price bot replied to me.

Indeed; you may feel that you are at a disadvantage compared to your peers, and that the amount of work you need to pull off is insurmountable.

However, you have an edge. You realize you need help, and you want to catch up. Motivation and incentive is a powerful thing.

Indeed, being passionate about something makes you much more likely to remember it. Interestingly, the passion does not need to be a loving one.

A common pitfall when learning math is thinking it is like learning history, philosophy, or languages, where it doesn't matter if you miss out a bit; you will still understand everything later, and the missing bits will fall into place eventually. Math is nothing like that. Math is like building a house. A first step for you should therefore be to identify how much of the foundation of math you have, to know where to start from.

Khan Academy is a good resource for this, as it has a good overview of math, and how the different topics in math relate (what requires understanding of what). Khan Academy also has good exercises to solve, and ways to get help. There are also many great books on mathematics, and going through a book cover-to-cover is a satisfying experience. I have heard people speak highly of Serge Lang's "Basic Mathematics".

Finding sparetime activities to train your analytic and critical thinking skills will also help you immeasurably. Here I recommend puzzle books, puzzle games (I recommend Portal, Lolo, Lemmings, and The Incredible Machine), board/card games (try Eclipse, MtG, and Go), and programming (Scheme or Haskell).

It takes effort. But I think you will find your journey through maths to be a truly rewarding experience.

I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.

Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).

Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)

In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.

As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:

• elementary real analysis
  
• linear algebra
  
• differential equations
  
• abstract algebra
  
  And a couple electives:
  
  
• topology
  
• graph theory
  
  And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
  
  
• abstract algebra
  
• topology
  
  Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.

Mathematics and Its History by John Stilwell

This really is a great book. From a review by Richard Wilders, MAA Reviews

>The author’s goal for Mathematics and its History is to provide a “bird’s-eye view of undergraduate mathematics.” (p. vii) In that regard it succeeds admirably. ... Mathematics and its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. ... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics. ... While Stillwell does a wonderful job of tying together seemingly unrelated areas of mathematics, it is possible to read each chapter independently. I would recommend this fine book for anyone who has an interest in the history of mathematics. For those who teach mathematics, it provides lots of information which could easily be used to enrich an opening lecture in most any undergraduate course. It would be an ideal gift for a department’s outstanding major or for the math club president. Pick it up at your peril ― it is hard to put down!

I'm not sure if you are looking for recommendations regarding more pop-math reading or actual textbooks, so I will try and recommend both.

• Love and Math by E. Frenkel is a great high-level view of mathematics with a very interesting autobiography woven in.
  
  
• If you don't have a background in proof-writing, I recommend acquainting yourself with proof writing techniques. A Transition To Advanced Mathematics is the book my university uses for Intro to Proofs. Another book that pops up often is How to Solve It, though I can't personally comment since I haven't used it.
  
  
• If you are interested in computer science, start learning about basic algorithms or graph theory. The YT lectures on graph theory are fantastic and easy to follow (or so I think); I used to use them as a supplement to my graph theory course.
  
  That's probably a really wide variety of resources, so my recommendation is to pick one and see how you like the material! I'm sure if you are really ambitious, you can try working through a topic with guidance from one of your teachers and maybe even work on some sort of project with them.
  
  It's also worth joining - or starting - a math club at your school. And if you are still looking for other activities, look into local math competitions that you can participate in.

We need to make a few definitions.

A group is a set G together with a pair of functions: composition GxG -> G and inverse G -> G, satisfying certain properties, as I'm sure you know.

A topological group is a group G which is also a topological space and such that the composition and inverse functions are continuous. It makes sense to ask if a topological group for example is connected. Every group is a topological group with the discrete topology, but in general there is no way to assign an interesting (whatever that means) topology to a group. The topology is extra information that comes with a topological group.

A Lie group is more than a topological group. A Lie group is a group G that is also a smooth manifold and such that the composition and inverse are smooth functions (between manifolds).

In the same way that O(n) is the set of matrices which fix the standard Euclidean metric on R^n, the Lorentz group O(3,1) is the set of invertible 4x4 matrices which fix the Minkowski metric on R^4. The Lorentz group inherits a natural topology from the set of all 4x4 matrices which is homeomorphic to R^16. It is some more work to show that the Lorentz group in fact smooth, that is, a Lie group.

It is easy to see the Lorentz group is not connected: it contains orientation preserving (det 1) matrices and orientation reversing (det -1) matrices. All elements are invertible (det nonzero), so the preimage of R+ and R- under the determinant are disjoint connected components of the Lorentz group.

There are lots of references. Munkres Topology has a section on topological groups. Stillwell's Naive Lie Theory seems like a great undergraduate introduction to basic Lie groups, although he restricts to matrix Lie groups and does not discuss manifolds. To really make sense of Lie theory, you also need to understand smooth manifolds. Lee's excellent Introduction to Smooth Manifolds is an outstanding introduction to both. There are lots of other good books out there, but this should be enough to get you started.

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

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

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

__

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

Geometry I and II by Kiselev

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

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

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

The first book covers:

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

  Geometry by Lang, Murrow
  

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

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

  Basic Mathematics by Lang
  

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

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

Anti-disclaimer: I do have personal experience with all the below books.

I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.

It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.

If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.

I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .

Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.

Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.

Hi there,

For all intents and purposes, for someone your level the following will be enough material to stick your teeth into for a while.

Mathematics: Its Content, Methods and Meaning https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

This is a monster book written by Kolmogorov, a famous probabilist and educator in maths. It will take you from very basic maths all the way to Topology, Analysis and Group Theory. It is however intended as an overview rather than an exhaustive textbook on all of the theorems, proofs and definitions you need to get to higher math.

For relearning foundations so that they're super strong I can only recommend:

Engineering Mathematics
https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463

Engineering Mathematics is full of problems and each one is explained in detail. For getting your foundational, mechanical tools perfect, I'd recommend doing every problem in this book.

For low level problem solving I'd recommend going through the ENTIRE Art of Problem Solving curriculum (starting from Prealgebra).
https://www.artofproblemsolving.com/store/list/aops-curriculum

You might learn a thing or two about thinking about mathematical objects in new ways (as an example. When Prealgebra teaches you to think about inverses it forces you to consider 1/x as an object in its own right rather than 1 divided by x and to prove things. Same thing with -x. This was eye opening for me when I was making the transition from mechanical to more proof based maths.)


If you just want to know about what's going on in higher math then you can make do with:
The Princeton Companion to Mathematics
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809

I've never read it but as far as I understand it's a wonderful book that cherry picks the coolest ideas from higher maths and presents them in a readable form. May require some base level of math to understand

EDIT: Further down the Napkin Project by Evan Chen was recommended by /u/banksyb00mb00m (http://www.mit.edu/~evanchen/napkin.html) which I think is awesome (it is an introduction to lots of areas of advanced maths for International Mathematics Olympiad competitors or just High School kids that are really interested in maths) but should really be approached post getting a strong foundation.

The links between topology, geometry and classical mechanics are fairly well documented in the other comments. Geometry and topology are fairly important in modern physics, at least what I've seen of it. General Relativity is the main example of where geometric ideas began to enter into physics. A good resource for this is Sean Carroll's GR notes and corresponding book. There are more advanced GR texts as well, like Wald's book.

There are also some books which deal directly with the links between physics and geometry, such as Frankels book, Szekeres, Agricola and Friedrich and Sternberg. Of these I own Szekeres book which is very good, and Frankels looks very good as well. The other two I am not sure about.

Geometric ideas do raise their head in more areas, as an example it is possible to formulate electromagnetism in terms of tensors or the hodge dual (see here). Additionally, and this is a bit beyond my knowledge, a friend of mine is working on topics in quantum field theory involving knot theory. I'm not exactly sure how this works but the links are certainly there.

Sorry if this all has more of a differential geometry flavour to it rather than a topological one, the diff geo side is what I know better. Hope that all helps. :)

I’m not a mathematician, and my mathematical knowledge doesn’t extend beyond the undergraduate level in most fields, so I’m not sure what most professional geometers would recommend. Depends a lot on which specific “later topics” you’re interested in, I expect.

I don’t think Kiselev or the AoPS book is going to be quite what you’re asking for. In terms of content they’re both fairly typical high school geometry books, just a bit more rigorous than some watered down American textbooks I’ve seen, with better problems. You should take a look at the Coxeter and Greitzer book though. One other book you might look at is Felix Klein’s Elementary Mathematics from an Advanced Standpoint.

For use in physics, computer modeling (for graphics, games, robotics, computer vision, cartography, physical simulation, and the like), etc., I think Hestenes style “geometric algebra” (wikipedia) should be taught to bright high school students and most undergraduates in technical fields. Cf. “Reforming the Mathematical Language of Physics”, “Primer on Geometric Algebra”, “Grassmann’s Vision”.

In particular the “conformal model” is really powerful and pleasant to work with, cf. http://www.geometricalgebra.net (or if you don’t want to buy a book, try this Ph.D thesis.

• • *
    
    It sounds like you might be interested in a math history book though. I like Stillwell’s pretty well.

Okay, I'm going to go against the mold here and say Lee's Topological Manifolds is your best bet. This book seemed to be the clearest, most thorough treatment of the topic of the topic at an introductory level. I think it's definitely a lot more verbose than Munkres, but I see it as an advantage as a lot of details are spelled out explicitly (which I something I like to see in textbooks that introduce a new topic to a unfamiliar audience). Plus the extra emphasis on manifolds is a very nice feature of the text.

That is just my recommendation, though. I think Munkres is a great book as well, but Lee's book seemed to present things in a way that clicked with me a bit better.

I could be interested in reading that paper, however I might need a discussion on the Atiyah-Singer Index Theorem first - It's something I haven't really had to use, but something I'd like to know.

My own personal interests lie in manifolds with special holonomy, and I'd be particularly interested in discussing G2 manifolds, if anyone else is.

Another, more basic, option would be Frenkel's 'Geometry of Physics' book, which has a lot of nice physics formulated in the language of differential geometry. This may be a good option for people with physics backgrounds with little formal DG training, as it does all of DG from scratch while being sure to tie all the math to physics (E&M, Lagrangian/Hamiltonian Mechanics, Relativity, Yang-Mills Theory etc.) Check it out here: https://www.amazon.com/Geometry-Physics-Introduction-Theodore-Frankel/dp/1107602602

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

Ghrist's book makes a great overview of not only a bunch of topics in algebraic and differential topology, but also has a bunch of applications. I don't think it would be very good as a first introduction to topology, but it's certainly good for browsing and getting a general idea of things.

For a textbook, you might be best getting Munkres and working through that. Another book I really like that is shorter than Munkres is Armstrong's topology book.

There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.

For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,

By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.

>I didn’t mean to offend anyone.

None taken; I just meant that math is huge, and none of it is inherently boring, but I understand that it can seem that way if you're not used to thinking about it.

It looks like there are already some good suggestions regarding how math relates to other subjects, so let me propose something purely mathematical: knot theory. It may not seem like "math", depending on what you've been exposed to, but this is what might make it a great topic for you. Here's the kind of thing a knot theorist thinks about:

Take a piece of string, tie a knot in it, and then take the two ends and tape them together. There are many ways of doing this. Here's something you might get; and here's another possibility. But maybe secretly, those are actually the same, in the sense that you can get from one to the other by just adding twists or moving bits of the string around. How can you tell if they're different or not? This is what knot theory focuses on.

If this interests you, here's a really great book written for laypeople about knot theory. It has lots and lots of pictures that you can learn a tremendous amount from just by staring for a while. It will probably be a challenge to read through, and a lot of it might go over your head, but that's fine; that's what reading math is like for everyone. There's also a large chance that reading this will feel nothing like math at all, and in this sense there's no need to be afraid.

This has turned out to be a much more interesting question than I had thought it would be. It seems to be unexpectedly hard to find a good, short book on Euclidean geometry. Most of the really good books are advanced treatments that have a lot more to say than what you probably want. Anyway, there is a good discussion of this question on mathoverflow. It appears that Kiselev is a pretty good choice. Hartshorne might be good as a guide to learning straight from Euclid (and lots more besides). I don't know how far you really want to go with this project. It might be enough to just get a taste of how the whole synthetic geometry program is organized.

By the way, you know about libary.nu, right?

I took a course in geometry recently. We used http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/1441931457 The first chapter is the only one to cover Euclid, and it only reviews books 1 - 4 but if you read it and work through the problems it'll give a good foundation to cover the rest of Euclid as you see fit. The real reason I mention this book is that almost all the problems are constructions with straightedge/compass. They give a par (the minimum number of steps a construction can be completed in with a reasonable amount of time spent thinking). You could give out the problems you are interested in without the par so that your students could compare construction methods. When I began the course I had no geometric intuition. I spent a lot of time trying to find the best possible constructions and felt my (euclidean) geometric intuition bloom.

Finally, after chapter 1 the book goes into Hilbert's Axioms to show how we develop modern geometry and develops a number of interesting geometries. I can't speak of most of this as we only covered Euclid, Hilbert's Axioms, and a quick bit of non-Euclidean geometry. But I found it very interesting and think the book can be used to study geometry in a number of ways. Either way good luck.

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

There's a lot of ground to cover in math, but completely doable. I'm going to recommend a dense book, but I truly think it's worth the read.

Let me leave you with this. You understand how number work correct? 1 + 1 = 2. It's a matter of fact. It's not up for debate and to question it would see you insane.

This is all of math. You need to truly understand

1 + 1 = 2

a + a = b everything is a function. There are laws to everything, even if people wish to deny it. If we don't understand it, it's easier to state that there are no laws that govern it, but there are. You just don't know them yet. Math isn't overwhelming when you think of it that way, at least to me. It's whole.

Ask yourself, 'why does 1 + 1 = 2 ?' If you were given 1 + x = 2, how would you solve it? Why exactly would you solve it that way? What governing set of rules are you using to solve the equation? You don't need to memorize the names of the rules, but how to use them. Understand the terminology in math, or any language, and it's easier to grasp that language.

The book Mathematics

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).

>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

When I was in your position I learned some representation theory of finite groups, from this book. It was at the perfect level for somebody who only has one semester's background in group theory. It'll gently introduce you to some things that you'll constantly need when you get further into algebra, like tensor products. Also, it's a topic which doesn't get covered at all in most undergrad abstract algebra courses, so it's a good thing to learn by yourself.

On the other hand, if you liked topology more than you liked group theory, you'd probably like Tu's Introduction to Manifolds.

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'll throw out some of my favorite books from my book shelf when it comes to Computer Science, User Experience, and Mathematics - all will be essential as you begin your journey into app development:

Universal Principles of Design

Dieter Rams: As Little Design as Possible

Rework by 37signals

Clean Code

The Art of Programming

The Mythical Man-Month

The Pragmatic Programmer

Design Patterns - "Gang of Four"

Programming Language Pragmatics

Compilers - "The Dragon Book"

The Language of Mathematics

A Mathematician's Lament

The Joy of x

Mathematics: Its Content, Methods, and Meaning

Introduction to Algorithms (MIT)

If time isn't a factor, and you're not needing to steamroll into this to make money, then I'd highly encourage you to start by using a lower-level programming language like C first - or, start from the database side of things and begin learning SQL and playing around with database development.

I feel like truly understanding data structures from the lowest level is one of the most important things you can do as a budding developer.

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:

1. Algebra Unplugged
   
2. Calculus for Cats
   
3. Calculus Made Easy
   
4. Trigonometry
   
   I wish you all the best!
   
   

I have this version. One of the best features is that if a proof goes onto the backside of a page, they put another diagram on that page. That was you don't have to flip back and forth trying to follow the construction. I have only read the first book, but it was good, and the other reviews on amazon seem to say it was good throughout. As far as other good, understandable books with limited prerequisites, there is Axler's Linear Algebra Done Right. If you've already taken a Linear Algebra class this book will most likely give you a new and hopefully more intuitive perspective. Furthermore Linear Algebra might have more applications in your field.

What you're probably looking for is a book on foundations, like logic or set theory. The thing is though that such books often do assume a certain level of mathematical maturity and experience with proofs. In addition, while the main results should be self-contained, the examples and exercises may make use of things that are supposed to be familiar but which you haven't learned due to your background (or lack thereof).

My advice is to be patient and focus on really learning the elementary topics, up to and including the standard calculus sequence. After that things really start to open up and you can usually find quality undergraduate-level books on just about anything that interests you.

But it probably isn't too soon to pick up a book on writing proofs and proof techniques. The one I used when I was younger was Peter Eccles' An Introduction to Mathematical Reasoning. I often hear Polya's How to Solve It recommended as well. I'm not sure how similar these are to the AoPS series though.

Anyway, good luck! Stick with it, mathematics is very worth it.

Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.

I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.

In addition to the ones mentioned already, another excellent book is Topology by James Dugundji. I know a lot of older mathematicians who prefer that book over both Munkres' and Kelley's, because it covers more material and has very clear and concise explanations, plus some more challenging exercises (including some esoteric material not normally found at a book at this level).

For a simpler introduction, I think that Basic Topology by M.A. Armstrong is pretty good. It starts out with point-set topology then goes into algebraic topology. It takes an intuitive and geometric approach, and has a good conversational style that's well-suited for an elementary course at the undergrad level.

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

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

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

The only possible issue I see is your selection of textbook: Principles of Mathematical Economics - I've honestly never heard of this book.

The graduate school go-to textbook is Mathematics for Economists by Simon and Blume. However, I think this book would be overkill for you, as it is geared towards pure, PhD level, economics. Also, I was in a similar place to you, with respect to mathematical training at one point, and Simon & Blume proved to be too large a leap.

My advice would be to use one of the following books (in order of my preference):
1. Essential Mathematics for Economic Analysis by Sydsaeter
2. Mathematics for Economics
by Hoy
3. Fundamental Methods of Mathematical Economics
by Chiang

They'll bring your basic command, of the basic required mathematics up to scratch AND these books cover linear algebra. You will also then be in a good place to tackle Simon & Blume if you ever need to in the future. Another piece of advice: PRACTISE PRACTISE PRACTISE. For what you are doing, you don't need to have a deep understanding of the mathematics you are using BUT, you do need to be very comfortable with applying the techniques.

So, as you are working through (for instance) Sydsaeter, I would be attempting the related practice questions you find in:

1. Schaum's Outline of Calculus
   
2. Schaum's Outline of Linear Algebra
   
3. Schaum's Outline of Introduction to Mathematical Economics
   
   Hope this helps.
   
   P.S. Almost all of these books are available for 'free' on Library Gensis

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

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.

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.

I've become greatly interested in geometric concepts in physics. I would like some opinions on these text for self study. If there are better options, please share.

For a differential geometry approach for Classical Mechanics:
Saletan?

For a General self study or reference book:
Frankel or Nakahara?

For applications in differential geometry:
Fecko or Burke?



Also, what are good texts for Geometric Electrodynamics that includes spin geometry?

Hi. The book Basic Mathematics by Serge Lang covers high school math in a way that is similar to most texts on higher mathematics, with theorems and proofs. As such, I think it would make a great stepping stone to higher maths, and some reviewers on Amazon agree. It gives you a solid foundation, and a little bit of an idea what's in store for you if you choose to pursue math. I think it would be a great place to start.

Send me a PM if you need help obtaining (a digital version of) the book.

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

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

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

I'm going to suggest what I suggest to most people your age with promise. Work through Gelfand's set of high school textbooks:

Algebra

Trigonometry

Functions and Graphs

They're written for self-study, so they're structured to help the reader figure things out for themselves. They're challenging at times, but with hard work I'm sure you'll do fine.

You might also pick up a book about proof, like the free Book of Proof and teach yourself how to prove things which is, after all, the basis for math.

Knot Theory

It's simple and accessible great for casual learning.
If you've already had much topology, you'll want a more in-depth book after you finish this one. This book is accessible to undergrad and even high school learners. But it's a great jumping off point even if you're more mathematically sophisticated.

Absolutely!

Math is everywhere and it's just about seeing the patterns emerge from simplicity. My knowledge on this topic has mainly been from my own work in Artificial Life and encoding AI genetic knowledge combined with my general interest in biological patterns (which are everywhere in nature) but the first thing that got many things to click for me was playing around with Turtle Logo in high school that is all about using simple constructs to create amazingly complex structures (i.e. one, two - look familiar?).

Sadly I don't work on my AI research anymore due to ethical concerns so I'm a bit out of date but I'd highly recommend the following that weren't mentioned in the original post though:

• "Godel, Escher, Bach: An Eternal Golden Braid" Book
  
• "The Fractal Geometry Of Nature" Book
  
• Procedural Generation
  
• Gene Expression Programming

The BEST way to study history of math is to read classic math texts from history such as the Elements, the Conics, the Principia, etc.

Original texts aside, I recommend The World of Mathematics by James Newman.

I also recommend Newton's Principia: The Central Argument by Densmore.

I recommend this:

https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

Unlike most professional mathematical literature, it is aimed at novices and attempts to communicate ideas, not details. Unlike most popular treatments of mathematics, and in particular unlike the YouTubers you mention, it is written by expert mathematicians and is about advanced mathematical topics. I got a hardcover set from a used bookstore when I was young and enjoyed it very much. It's well worth your time.

Take my recommendation as a grain of salt as i didn't take my formal math education further than where you're currently at, but I felt the same way after similar classes learning the mechanics but not the motivations. Mathematics: Its Content, Methods and Meaning was recommended to me by a friend and I think it help fills the gaps in motivation and historical context/connecting different fields not covered in classes.

Since you hope to study mathematics more seriously, I would look into this book link.

It's an excellent book that treats high school/basic college mathematics in an "adult" way. By adult I mean in the way that mathematicians think about it.
(The fun thing about Lang is that you can read only his books and get pretty much a high school through advanced graduate education).

the concept you're looking for is fractal.

an important book on the mathematical description of nature called "The Fractal Geometry of Nature" was written about 40 years ago by a guy named Benoit Mandelbrot. in it, he described how iterative natural processes could be described mathematically to model natural phenomena. it's an amazing book, a work of true genius, but heavy reading.

the Fibonacci sequence is not fractal -- that is, self-similar over a broad domain of scales. but some sequence sets are.

in any case, the self-similarity you are observing in this -- how the small branches look just like the big branches but in miniature -- is definitely fractal and just one of the many ways in which human systems represent our nature.

https://www.amazon.com/gp/product/0486409163/ref=ya_st_dp_summary

This does not specifically target game programmers. However, it's not just specific categories of math that is important to game programmers. It's EVERYTHING math related. And knowing the meaning of it and understanding is more important than just a formula.

The book I just linked is an amazing book. It is well written, and avoids academia where possible. It's balance between math and explination is just right where it can effectively get the point across, and even help you understand more complex explinations.

This book features three volumes, and each volume goes over a wide array of topics in depth.

Books on Fractal Geometry tend to have pretty pictures:

Indra's Pearls: The Vision of Felix Klein by David Mumford et al.

Beauty of Fractals by Heinz-Otto Peitgen et al

Fractal Geometry of Nature by Benoit Mandelbrot

For what it's worth New Kind of Science by Stepeh Wolfram has tons of pretty pictures, even if the content is dubious.



you might also want to checkout the Non-Euclidean Geometry for babies and other similar titles.

These are, in my opinion, some of the best books for learning high school level math:

• I.M Gelfand Algebra {[.pdf] (http://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf) | Amazon}
  
• I.M. Gelfand The Method of Coordinates {Amazon}
  
• I.M. Gelfand Functions and Graphs {.pdf | Amazon}
  
  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.

My two cents

1. Maths is difficult. There isn't one of us who at some point has not struggled with it
   
2. Maths should be difficult. The moment you find it easy you are not pushing yourself!
   
   If you want to improve your skills you can do two things in the short term -- read and practice.
   
   I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
   
   The great thing is that both include exercises.

Geometry, Topology, and Physics isn't a complete overview of math (as suggested by the title, it focuses on, well, geometry and topology), but if you're interested in learning about those specific subfields and their application to physics, I'd definitely recommend it.

Geometry: Euclid and Beyond by Hartshorne is a really good book. It starts out by walking you through Euclid's elements (you need a separate copy of that). Then it goes on to Hilberts's axiomatization of geometry, while discussing the ways in which Euclid's was lacking. Then it moves on to other things, like the relationship between algebra and geometry, and non-Euclidean geometries. It's not an easy book, but it does not really have any prerequisites, and it's a lot of fun. It makes Euclid a lot more clear.

The answer is "virtually all of mathematics." :D

Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:

1. Analysis
   
2. Algebra
   
3. Topology
   
   You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
   
   Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
   
   Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
   
   There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
   
   Here are my recommendations.
   
   Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
   
   Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
   
   Topology There's really only one thing to recommend here and that's Topology by Munkres.
   
   If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
   
   I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.

> No Fap

Probably a good idea if you want to start playing the field or find someone special. I have no idea how online porn can effect someone who grew up with it.

> Study Mathematics >4 hours/day

Have you gone through Euclid's Elements yet? It was the standard textbook for 2000 years. Everyone from Newton to Lincoln praised it and kept by them throughout their lives. Give it a shot. It covers everything from geometry to number theory, and all you need is a straight edge and a compass to start from first principles. I recommend this version: Euclid's Elements

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

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  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"
  

Thanks for the detailed reply. I think it's probably best to give Abbott's a shot then. Right now I'm working through Hubbard's multivariate text alongside Spivak's Calculus on Manifolds. I'm having a lot of difficulty with Spivak because I just haven't done enough work with single variable analysis to be comfortable doing it all in n dimensions, as you've said. It took my until the end of chapt. 2 to realize I'm not really getting it and I need to take a step back and figure out the simpler stuff first.

As a side question, what do you think about a side-text for topology/metric spaces like: https://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523 ? The only exposure I have to it is from a linear algebra text and the beginning of Spivak (in other words, not much at all).

If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.

Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.

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

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

Not quite encyclopaedic, but this gives a good overview of most topics you might encounter in an undergraduate course. The first section also gives a very good defense of the need for basic research into mathematics.

If there is something close to an Encyclopaedia Mathematica, but you can read it like a novel, it is these three volumes from Aleksandrov/Kolmogorov/Laurentiev. Amazon

Edit: Ahem, but after reading carefully post0, I would recommend you simply to begin with the textbooks of secondary school or so.

Some of my physics major friends liked Nakahara. If you want to instead just do Riemannian geometry computations like a physicist you can try a general relativity book like Wald or Carroll.

The standard/classic intro undergrad textbook is Munkres.

I actually never took a proper Topology course, I've just been forced to pick up a lot of it along the way. This book has been helpful for that. It's very friendly for reading/self-study.

If you don't want to buy a $60 book, I'm sure you can find it online somewhere, though I learn a lot better when trying to teach myself from a book I can easily flip through rather than a pdf in any form.

Thanks! what do you think about Mathematics: Its Content, Methods and Meaning ? from what I searched it can teach a lot a novice like me and quite the wonderful book.

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

>What books would you recommend?

How about doing your own research?

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



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

Mathematics, a learning map

Edit:

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

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

Though not an intro to topology book in the strictest sense (it doesn't go into too much detail on seperation axioms, metrization theorems, or Tychnonoff's theorem) I found going through Lee's Introduction to Topological Manifolds to be a fantastic book to learn from, especially if you're interested in going into differential geometry afterwards. I find that many a topology book commits the sin of focusing way too much on curious pathological counterexamples and the beginning students find themselves awash in a sea of formalities without any intuition to guide them. Having many geometrically motivated examples will make the subject highly approachable and eases the transition into thinking about abstract topological spaces.

The two books already mentioned sound awesome, but if you ever wanted a textbook with a formal approach to mathematics (written by a well-known and respected mathematician), check out Basic Mathematics by Serge Lang.

This is more for anyone reading who would like to continue on to a math or perhaps a physics major. The book takes you from elementary algebra and geometry all through pre-calculus; basically the only book you should need to prepare you for calculus and elementary linear algebra.

The classic textbook for a first course in topology is Topology by Munkres. It's a very good book.

Michael Starbird offers his topology "book" free of charge on his website. Here's the link. It's really closer to lecture notes for the course, and it's intended for an inquiry-based learning (IBL) course. What this means is that all of the proofs are omitted. The reader is expected to prove each result themselves. This obviously works much better in a group setting.

If you see any book titled "algebraic topology," I would recommend you ignore it for now. Algebraic topology courses assume you've at least had the one semester course in point-set topology (i.e. the books I linked) and one or two semesters in abstract algebra.

I was in a similar position to you, until I had to read Euclid's Elements for my freshman math course in college. The difference between what I'd been taught in school before that and what I came to understand after reading source texts was huge.

While I somewhat understand why the modern American education system focuses almost exclusively on applied math, I can't help but feel that something significant was lost when we moved away from using texts like the Elements as primary textbooks; it lasted in that role for more than 2,000 years for a reason.

If you're looking to really begin understanding math, I'd start there.

http://www.amazon.com/Euclids-Elements-Euclid/dp/1888009195/

Is an excellent edition of Euclid.

If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.

I'm not too sure about it personally, but several friends have taught from Nakahara, and have a lot of good things to say about it. It's graduate level.

Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:

In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).

AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.

Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.

Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:

• math subreddit
  
• math.stackexchange.com
  

• math on irc.freenode.net
  

• the math department of your college (don't forget that!)
  
  
  Here are two possible routes, one minimal, one less-minimal:
  
  Minimal
  
  
• Get good with proofs/math-thinking. Texts: One of Velleman or Houston (followed by Polya if you get a chance).
  
• Elementary real analysis. Texts: One of Spivak (3rd edition is more popular), Ross, Burkill, Abbott. (If you're up for two texts, then Spivak plus one of the other three).
  
  
  Less-minimal:
  
  
• Two algebras (linear, abstract)
  
• Two analyses (real, complex)
  
• One or both of geometry, and topology.
  
  
  NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.

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.

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.

Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?

The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.

In the grand scheme of math: jack shit. But who's to stop you after 2 months of studying?

What do you know so far? Are you comfortable with inequalities and math induction?

Check out the books below for a nice intro to Real Analysis:

How to Think About Analysis by Lara Alcock.

A First Course in Mathematical Analysis by D. A. Brannan.

Numbers and Functions: Steps to Analysis by R. P. Burn.

Inside Calculus by George R. Exner .

Discrete And Continuous Calculus: The Essentials by R. Scott McIntire.

Good Look.

I like to reccommend Basic Mathematics by Serge Lang. It will take you exactly from addition and subtraction to a prepared state for calculus and beyond. Don't let the name fool you though, it is a rigorous study, but with an honest effort you will do well.

I think Basic Mathematics is basically a precalculus text. I can't stand normal textbooks, everything is disconnected and done for you. This is written by one of the best mathematicians and will provoke thought and understanding. He knows his audience too, he's good with kids, check out his book Math! Encounters with High School Students. He's also written a 2-volume calculus text that I know has been used well in high school settings.

Nakamura is ok. I like Bleecker. The classical reference is Kobayashi and Nomizu. Nakamura is advanced undergrad. Bleecker is masters / post grad and K & N is renowned for both it's rigour and difficulty. From taking a brief look over the notes you are currently using any of these books would be fine.

I'm a bit surprised that you've had difficulty finding resources. Maybe it's your search terms? Try looking for principal fibre bundles, differential geometry, geometric analysis... etc...

Oh. Speaking of geometric analysis Josh does an ok job of reviewing fibre bundles / connections. There's a little bit of a connection to physics via Yang-Mills.

https://www.amazon.com/Geometry-Topology-Physics-Graduate-Student/dp/0750306068

https://www.amazon.com/Gauge-Theory-Variational-Principles-Physics/dp/0486445461/ref=sr_1_fkmr0_3?s=books&ie=UTF8&qid=1467948564&sr=1-3-fkmr0&keywords=bleecker+guage

https://www.amazon.com/Foundations-Differential-Geometry-Classics-Library/dp/0471157333/ref=sr_1_1?s=books&ie=UTF8&qid=1467948581&sr=1-1&keywords=kobayashi+and+nomizu

https://www.amazon.com/Riemannian-Geometry-Geometric-Analysis-Universitext/dp/3642212972/ref=sr_1_fkmr1_3?s=books&ie=UTF8&qid=1467948599&sr=1-3-fkmr1&keywords=Josh+geometric+analysis

I would strongly encourage you to pick up Mandelbrot's book on fractals as it shows the intersection of real-world problems with fractal theory. There are now better introductions now but this is THE CLASSIC reference (and a good read).

Here is the amazon link, but you can often grab it in used bookstores.

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

​

It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

​

Tu looks good, but I haven't read it carefully.

There's a lot of category theory, but this only uses the basics. Galois theory is deeper than anything used so far. Category, functor, natural transformation (co)limit, and maybe adjoint should be plenty (it looks like gibbon's tries to explain everything about adjoints that he uses). That's all in the first four or five (short) chapters of MacLane.

Thanks for your reply. I read positive reviews about this , what do you think?

For those interested in the "abstractness" of non-natural numbers, there's a phenomenal brief introduction in one of my favorite math texts, Mathematics: Its content, methods and meaning. A cold war Russian standard that covers a helluva lot of ground in applied math.

They make the point that the number "1" seems pretty intuitive to humans... you can have "1" of something, or "2" of something. But having "" of something doesn't really make any sense, and for a long time it was argued whether or not "" was even a "number". You certainly can't have "1/2" of a thing. If you cut an object in half, you just have 2 things now. And to have negative something is just absurd. There's a blurb about some primitive isolated tribes that have words for the number "1", "2", and "many". The number 1,237,298 is still pretty abstract to a human, because it's not like you can count that or really visualize that many things, but we acknowledge such a quantity can be useful.

Your post belongs to this sub: https://www.reddit.com/r/badmathematics/

It is good to have interest in mathematics, but you have serious gaps in the foundations. Get good books on logic, abstract algebra and general mathematics.

This one is decent for general mathematics introduction: https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev.

Personally read only the first chapter, but the book is praised by lots of people. I bet Mr. Nikolaevich has read it.

You can find it on Amazon https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

Okay. So..


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

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

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

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



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

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

I think Lang's basic mathematics is pretty good ( http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877 )

Patterns are everywhere in nature.

Once I was eating grapes while watching a video about the inter-filamental structure of the universe when my mind suddenly exploded.

When we see the endless variety in nature it's amazing to think of how all life is encoded in DNA. One of the most efficient ways to express a potentially endless variety is using fractals. It's no wonder then that much of the behavior and form of life exhibits fractal patterns.

Jason Silva did a short on the awe of patterns.

Our understanding of much of this came from Mandelbrot's 'The Fractal Geometry of Nature' book (who passed away just a few years ago).

You will likely get a lot out of How to Solve It. It teaches you how to break down problems and how to structure your thinking.

For the math itself, your examples are all covered by pre-algebra and algebra 1 (algebra is a pretty broadly defined area of math, but you want the beginner algebra books that most students would start in 8th-10th grade). I don't have specific book recommendations for that level of math, but taking the Khan Academy classes that the other poster linked would likely be a good start.

'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:

• Chapters 4-7 of Pre-Algebra (pdf) by Hung-Hsi Wu.
  
  
• Trigonometry by I. Gelfand and M. Saul.
  
  
• Continuous Symmetry: From Euclid to Klein by W. Barker and R. Howe [MAA review].

No matter what his interests may be, this wonderful survey will cover it, Mathematics: Its Contents, Methods, and Meaning. It was written by a team of prominent Russian mathemations, and became a classic. It's now a single Dover edition, but if possible, find it used in the original MIT 3-volume hardcover edition -- it demands that kind of respect!

"Mathematics: Its Content, Methods and Meaning"
http://www.amazon.com/Mathematics-Its-Content-Methods-Meaning/dp/0486409163
I thought this was a pretty good read. Its not programming oriented, but it does help for showing a brief dip into different maths fields and some of the concepts therein.

I love Jack Lee's series on manifolds:

Introduction to Topological Manifolds

Introduction to Smooth Manifolds

I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.

I've heard good things about Adam's book The Knot Book if knot theory is your thing.

One thing that you run into in math is a distinct lack of books and articles that explore ideas without proving them. They certainly exist, but are not the norm. We also suffer from a wealth of very poorly written books.

Survey articles and introductory tracks aimed at introducing young mathematicians to advanced topics do exist in some volume and they tend to be better written and in a more conversational tone. They want to familiarize you with a body of work that proving all in one place would be a monumental task so they just don't do it. If I were you I'd look in that direction, but you wont learn very much about the nuts and bolts of what's going on.

John Stillwell, Mathematics and its History. https://www.amazon.com/Mathematics-Its-History-Undergraduate-Texts/dp/144196052X

One of the best math books I've read.

I've heard good things about Serge Lang's Basic Mathematics. It's pre-calculus geometry and algebra mainly I think, but it treats you like a grown-up.

I found it interesting because of the history behind the fractal and how it entered into maths. If you still feel like not watching it you can look into Mandlebrot book

http://www.amazon.co.uk/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869

Well, I quite like this book.

http://www.amazon.co.uk/Mathematics-Its-History-Undergraduate-Texts/dp/144196052X

Mathematics, its Content, Methods, and Meaning - an amazing survey of analytic geometry, algebra, ordinary and partial differential equations, the calculus of variations, functions of a complex variable, prime numbers, theories of probability and functions, linear and non-Euclidean geometry, topology, functional analysis, and more.

For your situation I would highly recommend Mathematics: Its Content, Methods and Meaning, which is ~1000 page survey of mathematics topics.

I would also highly suggest the 3 volume set, Mathematical Thought from Ancient to Modern Times by Morris Kline. I'm not finding the words for why I think anyone, but particularly teachers, to have a historical context for mathematics, but I strongly believe it.

It also helps to read about what sort of problems people were interested in when they came up with things such as groups, or sqrt(-1), etc.

You might like a book like Serge Lang's or the Art of Problem Solving books.

These are a couple of nice old books about mathematical thinking:

• What is Mathematics? by Courant, Robbins, and Stewart
  
• How to Solve It by Polya

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

Free online:

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

Try Stillwell. It covers a lot, and some may not be accessible to those with less than an undergraduate math education, but there is enough elementary material to make it interesting for anyone of any background.

Pretty sure Xfocus was being sarcastic.



See "The Fractal Geometry of Nature" by Benoit B. Mandelbrot (the Mandelbrot - the man that coined the word).

Remember the big thing is not to just watch lectures, but to spend a lot of time solving a lot of problems. Make sure you find text books with problems and solution manuals (easy to do if you look) and solve the problems. The biggest thing about learning to be an EE or subset of EE is that it takes a crap load of time and effort. It will be frustrating. Plow through it. There are big rewards at the end for those who truly want to know how and why electricity do what they do.

Schaums outlines are great:

http://www.amazon.com/Schaums-Outline-Calculus-6th-Edition/dp/0071795537

So many subjects with so many solved problems.

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.

I'm really glad you posted this, I'm buying Mandelbrot's book now.

If you’d like a physical textbook, I’d recommend Basic Mathematics by Serge Lang, a celebrated mathematician and teacher. It’s an oldie but a goodie. https://www.amazon.com/dp/0387967877/

If you progress past that and want to refresh your calculus, it’s hard to go wrong with James Stewart’s Calculus. https://www.amazon.com/dp/B00YHKU50E/

Let me point you to my friend Nakahara.

Thanks a lot! I'm currently trying to get through Bert Mendelson's Intro to Topology book. Would you say this is a good enough start in your opinion?

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

Aside from The Princeton Companion to Mathematics, you might like to check out What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant and Robbins, and Mathematics: Its Content, Methods and Meaning by three Russian authors including Kolmogorov.

Saying literally anything is better than saying nothing. Just keep your mouth moving. Verbalize your thought process. Something I think a lot of people misunderstand about this sort of question is that there usually isn't a specific right answer: they're mainly trying to evaluate how you think about problem solving. It's probably fine if your response is along the lines of "I'm not entirely sure how I'd go about that, but here's how I'd start thinking about the problem." A good way to tackle this sort of thing is to identify ambiguities in the problem statement and start suggesting assumptions you could make to concretize them, and look for ways to break large problems down into subproblems. Even if you can't solve the problem, at least show that you have some idea how to get started.

EDIT: Another book you might find useful is George Polya - How to Solve It. Classic book on general strategies for problem solving. Just google "Polya's Method" for an overview.

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.

What do you mean by advanced Calculus? Multivariate Calculus without proofs?

Anyway,

Mathematical Analysis and Proof by Stirling

A First Course in Mathematical Analysis by Brannan

Yet Another Introduction to Analysis by Bryant

The standard textbook, which doesn't require much background (just calculus and a bit of set theory) is Topology by James R. Munkres.
Topology stands at the base of many mathematical subjects, but I don't know of many real world applications of general topology per se. Algebraic topology and knot theory have applications in biology, astronomy and I'm sure plenty else.

Here is a good book on trigonometry.

Here is one for algebra.

Here's another

Introduction to Topology by Mendelson is good.