(Part 3) Reddit mentions: The best mathematics books

This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

math is a funny thing... our culture gets so hung up on 'good at' and 'bad at', but the more I get into neurobiology and ML, the more amazing our general learning abilities seems to be. My partner and I are radically different, she's better at chess than I am in spite of having a poor ability with 'traditional' chess thinking, she relies almost entirely on pattern recognition, so she has to stand over the board looking down so her brain can feed up ideas from the books she's read (since chess layouts are always shown in those books from the top down).

All this is to say... there's a goddamn giant mountain in front of you, and it's easy to think that you're 'bad' at it because of where you're starting, or even because of base talents and interests that might not seem to line up with math at first glance. Just wanted to start out by saying that's horse shit. You're also 'bad' at judo and chinese (presumably), but given a few years of regular practice, you could get those reasonably under your belt as well. Math is a way of thinking and looking at problems, and it's incredibly helpful. It's kind of mind blowing the doors it can open... information theory, statistics, linear algebra, calculus, game theory, graph theory, group theory, representation theory, category theory... every branch opens up mind blowing new insights, tools, and models for looking at new problems. Don't look at it like this 'thing' you have to learn though. You can't learn all of math. You can just slowly learn new tools, get better at understanding what it even 'means' to learn one of those new fields, and how to organize your study to make real progress as you're slowly getting deeper.

So... my recommendation for where to start? Start with the meta learning. What is math? How can you learn it? How should you study? The best glimpse into those questions I've found is how to think about analysis. It takes a complete beginner's perspective (explaining how to read the standard math notation even... the summation symbol, epsilon, etc) slowly builds up an introduction to the guts of what calculus is, basically. You can read it in a week or two, so it's not a huge time investment, and it'll do a lot I think to arm you for the road ahead.

I'm personally a fan of bottom up learning as much as possible, but that's just because I hat trying to play with half a deck. There's plenty of people though that just treat pieces they're working with like 'black boxes'. You can use a decision tree without any fucking clue about information theory, or even what the decision surface actually looks like for the resulting tree. Finding good visualizations when wrapping your head around that stuff can be really helpful... so if you're struggling with one resource, don't be afraid to look for another. Sometimes a git article with some good graphs can make all the difference.

I don't know what road is best for you, but the only barrier in front of you is your patience, and your willingness to spend time every single week, and turn this into a practice instead of just a hobby. I started a year and a half ago after ten years in an unrelated industry, and while I still have a long way to go, I've also covered a ton of ground too. I'd never even had stats before at all, even in high school... now I'm comfortably following some pretty gnarly multivariate derivations in Bishop's pattern recognition and machine learning. You just keep putting one foot in front of the other, pay attention to your goal, follow your curiosity, and before you know it... people start looking at you funny, because you know things most people don't know, and you can build things most people don't even understand. I can't imagine a more exciting thing to be learning, especially at this time in history. If you have the patience and interest for it, whichever road you take I think you'll find it well worth your time.

My own personal suggestion by the way... take a little time for fundamentals on the regular (starting with linear algebra, a proper textbook with a lot of exercises if possible) and practical (actually implementing stuff, doing Kaggle competitions, whatever). Eventually in the distant future, you'll meet in the middle, and find you have the insight to start pursuing your own questions... possibly even questions no one has ever solved before, and you'll have an enormous amount of practical good to bring to whatever field you've been working in, if you choose to continue there. Good luck!

Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

I would say that it would depend on the problem. If you cannot solve the first ten, I would be worried, as they can all be solved by simple brute force methods. I have a degree in Astrophysics, and some of the 300 and 400 problems are giving me pause, so if you are stuck there you are in good company.

There are elegant solutions to each problem, if you want to delve into them, but the first handful, the first ten especially, can be simply solved.

Once you get beyond the first ten or so, the mathematical difficulty ratchets up. There are exceptions to that rule of course, but by and large, it holds.

If you are interested in Number Theory, the best place to start is a number theory course at a local university. Mathematics, especially number theory, is difficult to learn by yourself, and a good instructor can expound, not only on the math, but also on the history of this fascinating subject.

Gauss, quite arguably the finest mathematician to ever live loved number theory; of it, he once said:

> Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

Although my personal favorite quote of his on the subject is:

> The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.

If you are interested in purchasing some books about number theory, here are a handful of recommendations:


Number Theory (Dover Books on Mathematics) by George E. Andrews


Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, Erica Flapan


An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman


Elementary Number Theory (Springer Undergraduate Mathematics Series) by Gareth A. Jones , Josephine M. Jones

and it's companion


A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland, Michael Rosen

and a fun historical book:


Number Theory and Its History (Dover Books on Mathematics) Paperback by Oystein Ore

I would also recommend some books on

Markov Chains

Algebra

Prime number theory

The history of mathematics

and of course, Wikipedia has a good portal to number theory.

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. :)

Earlier this year I finished my PhD in aero (researching computational fluid dynamics). I'll go ahead and reiterate a couple of the other recommendations in this thread, I think they've given you pretty good advice so far.

Numerical Recipes is great, and you can even read their older editions for free online. Don't worry about them being older, their content really hasn't changed much over the years beyond switching around the programming language. A word of warning, though. The code itself in these books come with rather restrictive licenses, and what it ends up meaning for you is you can copy their code and use it yourself, but you aren't allowed to share it (although I don't think this is carefully enforced). If you want to share code, you'll either have to pay for their license, or use their code only as inspiration for writing your own. If you pay close attention to their licensing, they don't even let you store on your computer more than one copy of any of their functions (again, I can't imagine they actually have a way of enforcing this, but it makes me disappointed they do things this way nevertheless), so it can get problematic fast.

If you want more reading material, I've only paged through it myself but Chapra and Canale's book seems like a nice intro text (if it wasn't your textbook already), and uses MATLAB. Reddy has a well-liked intro to finite element methods. Some more graduate level texts are Moin, LeVeque (he has a bunch of good ones), and Trefethen.

Project Euler is indeed great.

I would also recommend you learn some other (any other, really) programming language. MATLAB is a fine tool, but learning something else as well will make you a better programmer and help you be versatile. I don't really recommend you go and learn half a dozen other languages, or even learn every feature available one language--just getting reasonably comfortable with one will do. I'd say pick any of: C, C++, Fortran 90 (or higher), or Python, but there are others as well. Python is probably the easiest to get into and there are lots of packages that will give it a similar "feel" to Matlab, if you like. One nice way of learning (I think) is going through Project Euler in your language of choice.

Slightly more long term, take other numerical/computational courses. As you take them, think about what you like to use computation for (if you don't have a good idea already). If you like to analyze data, develop more or less "simple" simulations to direct design decisions, and don't care so much for heavy simulations, you'll get a better idea of what to look for in industry. If you like physics simulations and solving PDEs, you may lean toward the research end of things and possibly dumping Matlab altogether in favor of more portable and high performance tools.

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

You're pretty good when it comes to linear vs. generalized linear models--and the comparison is the same regardless of whether you use mixed models or not. I don't agree at all with your "Part 3".

My favorite reference on the subject is Gelman & Hill. That book prefers to the terminology of "pooling", and considers models that have "no pooling", "complete pooling", or "partial pooling".

One of the introductory datasets is on Radon levels in houses in Minnesota. The response is the (log) Radon level, the main explanatory variable is the floor of the house the measurement was made: 0 for basement, 1 for first floor, and there's also a grouping variable for the county.

Radon comes out of the ground, so, in general, we expect (and see in the data) basement measurements to have higher Radon levels than ground floor measurements, and based on varied soil conditions, different overall levels in different counties.

We could fit 2 fixed effect linear models. Using R formula psuedocode, they are:

1. radon ~ floor
   
2. radon ~ floor + county (county as a fixed effect)
   
   The first is the "complete pooling" model. Everything is grouped together into one big pool. You estimate two coefficients. The intercept is the mean value for all the basement measurements, and your "slope", the floor coefficient, is the difference between the ground floor mean and the basement mean. This model completely ignores the differences between the counties.
   
   The second is the "no pooling" estimate, where each county is in it's own little pool by itself. If there are k counties, you estimate k + 1 coefficients: one intercept--the mean value in your reference county, one "slope", and k - 1 county adjustments which are the differences between the mean basement measurements in each county to the reference county.
   
   Neither of these models are great. The complete pooling model ignores any information conveyed by the county variable, which is wasteful. A big problem with the second model is that there's a lot of variation in how sure we are about individual counties. Some counties have a lot of measurements, and we feel pretty good about their levels, but some of the counties only have 2 or 3 data points (or even just 1). What we're doing in the "no pooling" model is taking the average of however many measurement there are in each county, even if there are only 2, and declaring that to be the radon level for that county. Maybe Lincoln County has only two measurements, and they both happen to be pretty high, say 1.5 to 2 standard deviations above the grand mean. Do you really think that this is good evidence that Lincoln County has exceptionally high Radon levels? Your model does, it's fitted line goes straight between the two Lincoln county points, 1.75 standard deviations above the grand mean. But maybe you're thinking "that could just be a fluke. Flipping a coin twice and seeing two heads doesn't mean the coin isn't fair, and having only two measurements from Lincoln County and they're both on the high side doesn't mean Radon levels there are twice the state average."
   
   Enter "partial pooling", aka mixed effects. We fit the model radon ~ floor + (1 | county). This means we'll keep the overall fixed effect for the floor difference, but we'll allow the intercept to vary with county as a random effect. We assume that the intercepts are normally distributed, with each county being a draw from that normal distribution. If a county is above the statewide mean and it has lots of data points, we're pretty confident that the county's Radon level is actually high, but if it's high and has only two data points, they won't have the weight to pull up the measurement. In this way, the random effects model is a lot like a Bayesian model, where our prior is the statewide distribution, and our data is each county.
   
   The only parameters that are actually estimated are the floor coefficient, and then the mean and SD of the county-level intercept. Thus, unlike the complete pooling model, the partial pooling model takes the county info into account, but it is far more parsimonious than the no pooling model. If we really care about the effects of each county, this may not be the best model for us to use. But, if we care about general county-level variation, and we just want to control pretty well for county effects, then this is a great model!
   
   Of course, random effects can be extended to more than just intercepts. We could fit models where the floor coefficient varies by county, etc.
   
   Hope this helps! I strongly recommend checking out Gelman and Hill.

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!

FWIW I had no fun with mathematics in school and didn't start studying it til I was in my thirties. I'm no genius, but I now teach the subject and still self-study it. You don't need any mysterious talent to get very competent at university-level maths, just to be interested enough in it to put the hours in.

Self-study is hard and frustrating. Be prepared for that. Reading one page can take a day. You can stare at a definition or theorem for hours and not understand it. Looking things up in multiple books can really help with that -- there are some good resources online as well. Also, some things just take a while to "cook" in the brain; keep at it. Take lots and lots of notes, preferably with pictures. Do plenty of exercises. When you're really stumped, post here.

I'll echo what others have said: add to Spivak a couple of other books so you can change it up. A book on group theory and one on linear algebra would be a nice combination -- maybe one on discrete maths, probability or something similar as well if that interests you. For group theory I think this book is fantastic, though it's expensive.

If you really want to make it through Spivak, make a plan. Break the book down into, say, 50-page chunks and make 50 pages your target for each week (I have no idea whether this is too ambitious for you -- try it and see). Track your progress. Celebrate when you hit milestones.

Good luck!

[EDIT: Also, be aware that maths books aren't really designed to be read like novels. Skim a chapter first looking for the highlights and general ideas, then drill into some of the details. Skip things that seem difficult and see if they become important later, then go back (with more motivation) etc.]

Hi OP,

I found myself in a similar situation to you. To add a bit of context, I wanted to learn optimization for the sake of application to DSP/machine learning and related domains in ECE. However, I also wanted sufficient intuition and awareness to understand and appreciate optimization it for it's own sake. Further, I wanted to know how to numerically implement methods in real-time (embedded platforms) to solve the formulated problems (Since my job involves firmware development). I am assuming from your question that you are interested in some practical implementation/simulations too.

​

< A SAMPLE PIPELINE >

Optimization problem formulation -> Enumerating solution methods to formulated problem -> Algorithm development (on MATLAB for instance) -> Numerical analysis and fixed-point modelling -> Software implementation -> Optimized software implementation.

​

So, building from my coursework during my Masters (Involving the standard LinAlg, S&P, Optimization, Statistical Signal Processing, Pattern Recognition, <some> Real Analysis and Numerical methods), I mapped out a curriculum for myself to achieve the goals I explained in paragraph 1. The Optimization/Numerical sections of the same is as below:

​

OPTIMIZATION MODELS:

1. Optimization Models by Calafiore and El Ghaoui (Excellent and thorough reference book)
   
2. Non-linear Programming by D.Bertsakas ( I agree that nonlinear programming is very relevant and will be very useful in the future too)
   
   

• Lectures from Ohio State: https://www.youtube.com/playlist?list=PL_Nk3YvgORJtKF6B-cRBCKxi9jWgM0pcx
  
  

1. Convex Optimization by S. Boyd and Vandenberghe (Another very good book for basics)
   
   

• Lectures by S. Boyd on the same: https://www.youtube.com/playlist?list=PL3940DD956CDF0622
  
  NUMERICAL METHODS:
  
  

1. Numerical Linear Algebra by L.N.Trefethen and D.Bau III (Very good explanation of concepts and algorithms and you might be able to find a free ebook version online)
   
2. Numerical Optimization by Jorge Nocedal and S.Wright (Both authors are very accomplished and the textbook is well regraded as a sound introduction to this subject)
   
3. Numerical Algorithms by Justin Solomon (He's a very good teacher whose presentation is digestible immediately)
   
   

• His Lectures are here: https://www.youtube.com/playlist?list=PLHrg69yaUAPeiLEsa-1KauSe2HaA0Wf6I
  
  ​
  
  Personally I think this might be a good starting point, and as other posters have mentioned, you will need to tailor it to your use-case. Remember that learning is always iterative and you can re-discover/go deeper once you've finished a first pass. Front-loading all the knowledge at once usually is impractical.
  
  ​
  
  All the best and hope this helped!

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)

As I see it there are four kinds of books that fall into the sub $30 zone:

• Dover books which are generally pretty good and cover a wide range of topics
  
  
• Free textbooks and course notes - two examples I can think of are Hatcher's Algebraic Topology (somewhat advanced material but doable if you know basic point-set topology and group theory) and Wilf's generatingfunctionology
  
  
• Really short books—I don't a good example of this, maybe Stanley's book on catalan numbers?
  
  
• Used books—one that might interest you is Automatic Sequences by Allouche and Shallit
  
  You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
  
  So that's a lot of suggestions, but two of them are free so you can't go wrong with those.

Everyone has to take MATH 150--MATH 152's prerequisite isn't Calculus 12. So after 150, you're at the same level as everyone else.

A tip: make sure when studying, you understand every part of what's being taught. You won't be able to just memorize this stuff. If you don't get something, spend a bit of time trying to figure it out, move forward if the following information doesn't rely on what you're passing, but come back to it later and try again and again till you understand what that thing is, how it works, and why. YouTube the name of what you're having trouble with, cause there are going to be several tutorials from people on there per topic.


You'll have to put in the hours, though, and study smart. Remember: being a student is your job, and 3 courses is full time (equivalent to 9-5 Mon-Fri). SFU uses the "flipped classroom" where you're supposed to read the sections of the textbook before class, the lecture reinforces and clarifies the most important stuff, then you self-study till you understand it 100%.


The rule of thumb for all classes is 2-3 hours of study for every hour in lecture. That means for MATH 150 you should expect to spend 8-12 hours studying on your own outside of class.


Engineering requires 12 credits/semester, so you'd have at least 13 in the semester you take 150--That means 26-39 hours of studying on your own outside class i.e. 6 hours a day 7 days a week, 6.5 hours every day but Sat/Sun, or 8 hours a day Mon-Fri.


Here are a couple useful resources:

• http://kl1p.com/studytips a sort of summary, lots of the below info is covered, but not everything.
  
• Study less study smart overview -- this guy wrote out notes for
  
• http://www.lbcc.edu/LAR/studyskills.cfm (the videos) -- This guy is awesome. It's a full course on how to study. When I get a job after grad I'm definitely getting him a gift.
  
• http://altair.pw/pub/lib/How%20to%20Become%20a%20Straight-A%20Student.pdf
  
• http://www.cse.buffalo.edu/~rapaport/howtostudy.html -- note: has some repeated information from the first 2 links
  
• https://tomato-timer.com -- makes using the pomodoro technique a bit easier.
  
• http://khanacademy.org
  
• http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx -- apparently this guy's got really good, concise, notes. Also applies for MATH 152 and 153.
  
• How to Ace Calculus: The Streetwise Guide (recommended by the math profs, and I've seen tons of people say it helped a lot. It doesn't go in-depth, but gives you a simple overview of the concepts so you get a good idea of how and why they work)
  
  
  Also, FYI for YouTube, if you hit shift+> on YouTube, it speeds up the video 1.25x 1.5x 2.0x -- which has really saved my ass, cause I don't have patience for slow talkers. (Conversely, shift+< slows the video back down)

There are a fair number of popular level books about mathematics that are definitely interesting and generally not too challenging mathematically. William Dunham is fantastic. His Journey through Genius goes over some of the most important and interesting theorems in the history of mathematics and does a great job of providing context, so you get a feel for the mathematicians involved as well as how the field advanced. His book on Euler is also interesting - though largely because the man is astounding.

The Man who Loved only Numbers is about Erdos, another character from recent history.

Recently I was looking for something that would give me a better perspective on what mathematics was all about and its various parts, and I stumbled on Mathematics by Jan Gullberg. Just got it in the mail today. Looks to be good so far.

Hello!

It's been a while since I last suggested a resource for calculus - so far, I've been finding the following two books extremely helpful and thought it would be good to share them:

1. The Calculus Lifesaver
   http://www.amazon.com/The-Calculus-Lifesaver-Tools-Princeton/dp/0691130884/ref=sr_1_1?ie=UTF8&qid=1398747841&sr=8-1&keywords=the+calculus+lifesaver
   
   I have mostly been using this as my main source of calculus lessons. You can find the corresponding lectures on youtube - the ones on his site do not work for whatever reason. The material is quite good, but still slightly challenging to ingest (though still much better than other courses out there!).
   
   
2. How to Ace Calculus: The Street-Wise Guide
   
   When I first saw this book, I thought it was going to be dumb, but I've been finding it extremely helpful. This is the book I'm using to understand some of the concepts in Calculus that are taken for granted (but that I need explained more in detail). It actually is somewhat entertaining while doing an excellent job of teaching calculus.
   
   The previous website I recommended to you is quite good at giving you an alternative perspective of calculus, but is not enough to actually teach you how to derive or integrate functions on your own. Hope your journey in math is going well!
   
   

I'm an autodidact and currently studying computer science. Usually I'm not good at learning with videos or websites, I prefer to study books, so I will give you the first great books in the order that I studied. I'm assuming that you're fairly good at math, if not read this book before since it's a comprehensive survey about the subject.

• Computer Science Illuminated by Nell Dale
  
• Structure and Interpretation of Computer Programs - 2nd Edition
  
• Land Of Lisp (optional but very fun book)
  
• C Programming Language by Brian W. Kernighan
  
• Introduction to Algorithms by Thomas H. Cormen
  
• Concepts, Techniques, and Models of Computer Programming
  
  Now, these 5 books are going to teach the you the basics and you'll learn to program with Lisp and C, which are great languages that will improve your way of thinking about computing; and since all modern languages come from these two, after you learn them it will be easy to you pick up new languages in a matter of days, just buy a good reference book about the particular language and you're on business.
  
  After that the next step is to learn about networks, and learn HTML, CSS and Javascript:
  
  
• Computer Networking: A Top-Down Approach by Keith W. Ross or Computer Networks by Andrew S. Tanenbaum
  
• HTML and CSS: Design and Build Websites by Jon Duckett
  
• JavaScript and JQuery: Interactive Front-End Web Development by Jon Duckett
  
  It's also a good idea to learn Python since is heavily used for all sorts of development and a lot bitcoin apps use it:
  
  
• Learning Python, 5th Edition
  
• Programming Python by Mark Lutz
  
• Flask Web Development: Developing Web Applications with Python
  
• Creating Apps in Kivy
  
• Python Cookbook by David Beazley
  
  Now for cryptography:
  
  
• Understanding Cryptography by Christof Paar
  
• Applied Cryptography by Bruce Schneier
  
• Mastering Bitcoin By Andreas M. Antonopoulos

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

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

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

Good luck!

Not much, the nice thing for upper math courses is they do a good job of building up from bare bones. If you have some linear algebra and a multivariable calc course you should be good. The big requirement is however mathematical maturity. You should be able to read, understand, and write proof.

A very basic intro to proofs course is usually taught to first year math students, this covers set notations, logic, and some basic proof techniques. A common reference is "How to prove it: a structured approach", I learned from Intro to mathematical thinking. The latter isn't as liked, it does seem to cover some material that I think should be taught early. A lot of classical number theory and algebra, for example fundamental theorem of arithmetic, and Fermat's little (not last) theorem are proven. Try to find a reference for that stuff if you can.

It's really important to do a proof based linear algebra class. It helps build the maturity I mentioned and will make life easier with topology. But even more importantly teaching linear algebra in a more abstract way is important for a physics undergrad as it can serve as a foundation for functional analysis, the theory upon which quantum mechanics is built. And in general it is good to stop thinking of vectors as arrows in R^n as soon as possible. A great reference is Axler's LADR.

Again not strictly required, but it helps build maturity and it serves as a good motivation for many of the concepts introduced in a topology class. You will see the practical side of compact sets (namely they are closed and bounded sets in R^(n)), and prove that using the abstract definition (which is the preferred one in topology). You will also prove some facts about continuous functions which will motivate the definition of continuity used in topology, and generally seeing proofs about open sets will show you why open sets are important and why you may wish to look at spaces described only by their open sets (as you will in topology). The reference for real analysis is typically Rudin, but that can be a little tough (I'm sorry, I can't remember the easier book at the moment)

Edit: I will remove this as it doesn't meet the requirements for an /r/askscience question, we usually answer questions about the science rather than learning references. If you feel my answer wasn't comprehensive enough feel free to ask on /r/math or /r/learnmath

Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.

I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.

I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.

In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.

Another good affordable recommendation is How To Think Like A Mathematician, which is aimed at people making the jump from school to university-level mathematics. It explains mathematical terminology and breaks down the process one might go through to read and write proofs.

I wish you good luck but also not to be demanding on yourself: learning to interpret and construct proofs, along with the required vocabulary, is about half (or more) of an undergraduate degree in mathematics, and some students never get the hang of it. The fact that you are actually motivated to understand proofs is a good start, though, and probably sets you apart from those students already. And of course, you have luxury of choosing which proofs interest you.

Feel free to pm me specific questions, I have a bit of free time this month until I'm back to my own studies. Can't promise I'll know the answer but if not can hopefully direct you somewhere useful.

Both time series and regression are not strictly econometric methods per se, and there are a range of wonderful statistics textbooks that detail them. If you're looking for methods more closely aligned with econometrics (e.g. difference in difference, instrumental variables) then the recommendation for Angrist 'Mostly Harmless Econometrics' is a good one. Another oft-prescribed econometric text that goes beyond Angrist is Wooldridge 'Introductory Econometrics: A Modern Approach'.

For a very well considered and basic approach to statistics up to regression including an excellent treatment of probability theory and the basic assumptions of statistical methodology, Andy Field (and co's) books 'Discovering Statistics Using...' (SPSS/SAS/R) are excellent.

Two excellent all-rounders are Cohen and Cohen 'Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences' and Gelman and Hill 'Data Analysis Using Regression and Multilevel/Hierarchical Modelling' although I would suggest both are more advanced than I am guessing you need right now.

For time series I can recommend Rob Hyndman's book/s on forecasting (online copy freely available)

For longitudinal data analysis I really like Judith Singer's book 'Applied Longitudinal Data Analysis'.

It sounds however as if you're looking for a bit of a book to explain why you would want to use one method over another. In my experience I wanted to know this when I was just starting. It really comes down to your own research questions and the available data. For example I had to learn Longitudinal/fixed/random effects modelling because I had to do a project with a longitudinal survey. Only after I put it into practice (and completed my stats training) did I come to understand why the modelling I used was appropriate.

There is Elementary number theory by William Stein, and A Computational Introduction to Number Theory and Algebra. The latter is better if you are also interested in some of the computation They are both available for free online (legally). I think you would prefer Stein's book but skim through both and see which one you like more.

For something more in depth, I looked at some of the books in this list at mathoverflow. Hardy & Wright , and Niven & Zuckerman's books seem best suited to you (from what I looked at, but go through that list yourself). Many of the other books require some background in abstract algebra.

I haven't read either but just looking through their table of contents I would go with Niven and Zuckerman's book. It seems to go into the more useful things more quickly, and it's not as densely packed with information you probably won't be interested in right now.

TLDR: Start here, or here.

For what it's worth, number theory is a fascinating field. I don't think you'll be disappointed going into it. Good luck!

RPCV checking in. This is a good idea... you're going to have a lot of downtime and it's a great opportunity to read all the things you've wanted to but haven't yet found the time for. That could mean math, or languages, or just old novels.

When I was learning functional analysis, if found this book by Bollobas to be incredibly helpful. Of course, the only real analysis reference you need is Baby Rudin, but if you want to learn measure theory you may want his Real & Complex Analysis instead.

For texts on the other subjects, take a look at this list. You should be able to find anything you need there.

If you have any questions about Peace Corps, feel free to PM me. Good luck!

Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.

Free:
www.khanacademy.org

free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.

http://ocw.mit.edu/index.htm

free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.

Cheap $$

http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&qid=1331675661&sr=8-1

$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.

http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1

$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.

Got it!

Well you have a couple of routes. You can choose to go the traditional route of learning mathematics. That is the following:

Trigonometry
Calculus (Derivatives, Integrals, Series, then Multivariable)
Differential Equations
Linear Algebra

After linear algebra you branch off from there.

If you want a non-traditional approach, you can start by understanding logic. This will really help fortify the way you should think about math. Picking up books on logic is great for this. /r/bibliographies is a great place for this. You should find the logic or philosophy section and dip your feet in it (More specifically the symbolic logic section). I would take a look at those books. If anything, I would try to check out the book How To Think Like a Mathematician. It is a pretty good book. What is best about it is that the ideas of logic is explained with set theory.

Set theory is the next math I would learn after logic. It is a building block of mathematics and quite fun (though proof heavy). I wouldn't be surprised if this even turned you away! I would just try this and just try very elementary set theory.

You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Some user friendly books on Real Analysis:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

I recommend starting by pointing people at Nathan Carter’s book Visual Group Theory (site, amzn), which does a great job providing a number of examples of interesting groups, and motivating understanding of quotient groups etc.

If they start with a stable of nice nice concrete group theory examples, then a lot of the drier and more abstract parts of a “definition, theorem, proof, corollary, definition, ...” style presentation afterward will be much less cryptic.

The basic examples for ring theory / field theory are a bit more familiar to most students, so I expect the rest of abstract algebra usually turns out okay, though throwing in more concrete weird examples of rings and fields would be nice as well.

It would be interesting to start an introductory field theory course with a detailed concrete study of a few specific finite fields, and e.g. vector spaces and linear transformations and polynomials etc. worked out for those fields. It would give a very different flavor to the course.

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

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

I have Mathematics:From the Birth of Numbers and it’s excellent.

Highly recommend

> This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into mathematical logic, set theory, topology, fractals, probability, and assorted other mathematical byways. The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that talks about math in more detail. Textbooks are good for problems and for reference, but I find them very hard to read. They use equations where they should be using words.

Go to your local library, and look in the math section until you find something interesting. I found this book when I was struggling with calculus: How to Ace Calculus: The Streetwise guide. It was smart, funny, and really explained topics in ways I could relate to.

That's the kind of thing I would look for if I were you. Good luck! I hope you see post in all the ~430 comments!

Really interested, actually! But I'm curious about a few things:

When exactly will it start in January? And when will it end? Will it be in the evenings? Which days of the week?

Will we need a text book? I have a Dover book on basic analysis already which I haven't cracked open.

Where will the class be held?

I had an incredibly hard time with calculus as a university student. I took it 5 times because I kept dropping it or withdrawing or not getting a passing grade. I almost got kicked out of my program because I pushed the limits of how many times I could repeat the course. There was a general disinterest on my part, but now, almost 10 years later, I am much more fascinated and genuinely interested in math, number theory, and also in many ways, analysis.

I started reading a book recently that finally explained what calculus actually was in simple terms. I feel like it's the first time that was ever done for me and I can say that helped my interest.

Anyway, I'd really hope to attend your class! The reason I'm curious about exact start date is that I'll be away from the HRM until mid-January. And it's a bummer to miss the first few classes of anything!

OK, then let's try this again, this time using more calculus and less topology-specific results. I'm going to be using LaTeX markup here; see the sidebar to /r/math for a free browser plugin that'll translate my code into readable mathematics.

The following is from Michael Spivak's Calculus on Manifolds, and it's pretty close to the result you want, but with more restrictions in terms of differentiability and such:

• Problem 2-37.
  
  (a) Let [; f \colon \mathbf{R}^2} \to \mathbf{R} ;] be a continuously differentiable function. Show that [; f ;] is not 1-1. Hint: If, for example, [; D_1 f(x,y) \neq 0 ;] for all [; (x,y) ;] in some open set [; A, ;] consider [; g \colon A \to \mathbf{R}^2 ;] defined by [; g(x,y) = \left( f(x,y), y \right). ;]
  
  (b) Generalize this result to the case of a continuously differentiable function [; f \colon \mathbf{R}^n \to \mathbf{R}^m ;] with [; m<n. ;]
  
  The basic idea for (a) is that if there were such an continuously differentiable injection [; f \colon \mathbf{R}^2 \to \mathbf{R}, ;] then (1) we can find some subset [; A \subseteq \mathbf{R}^2 ;] such that (depending on your convention for notation)
  
  [; D_1 f(x,y) = \partial_1 f(x,y) = \partial_x f(x,y) = \frac{\partial f}{\partial x} (x,y) \neq 0 ;]
  
  for all [; (x,y) \in A, ;] and (2) the function [; g \colon A \to \mathbf{R}^2 ;] must have a local continuously differentiable inverse. (This is by the Inverse Function Theorem.)
  
  The problem, however, arises when you consider the actual form of a local inverse for [; g, ;] since [; g^{-1} ;] will be independent of the second coordinate. Accordingly, [; g ;] cannot be injective, whence [; f ;] cannot be injective.
  
  I imagine the generalization to part (b) is similar. The important thing here is that given a function
  
  [; f \colon \mathbf{R}^m \times \mathbf{R}^n \to \mathbf{R}^m, \text{ where } m<n, ;]
  
  one can construct the associated function
  
  [; \begin{align*}<br /> g \colon \mathbf{R}^m \times \mathbf{R}^n &amp;amp;\to \mathbf{R}^m \times \mathbf{R}^n\\<br /> (\mathbf{x}, \mathbf{y}) &amp;amp;\mapsto \left( f(\mathbf{x},\mathbf{y}), \mathbf{y} \right).<br /> \end{align*} ;]
  
  In the above example, we're considering the case [; m=n=1, ;] and we're considering the equivalence [; \mathbf{R}^1 \times \mathbf{R}^1 \simeq \mathbf{R}^2. ;]
  
  The advantage is that [; g ;] now maps between two spaces of the same dimension, so one can often apply the Inverse Function Theorem. (In fact, this is a common way to deduce the Implicit Function Theorem from the Inverse Function Theorem, so you see this technique often enough that it's worth your time to remember it.)
  
  These exercises require stronger assumptions—i.e., continuous differentiability rather than mere continuity—but perhaps this'll at least be a bit more accessible because it doesn't invoke quite so much topology. Hope this helps, and good luck!

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

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.

How long ago did you do it? I work with calculus and statistics a lot and I often go back to earlier concepts to make sure my foundations are still strong.

I would recommend looking at this book and just quickly running through the exercises. That will give you a good idea about what you need to focus on. If you feel comfortable with those, something like this might be good to check out since it's made for self-teaching as opposed to being used in conjunction with a class.

Edited to add: math is like any language, in that the more you practice and manipulate numbers the better you'll be at it!

I'm a physicist/mathematician, but I think it could be useful for you. Exterior algebra (differential forms) in particular is worth learning because it makes the theory of multivariable calculus much more elegant and simple. With exterior algebra you can see that the fundamental theorem of calculus, Green's theorem, and the divergence theorem are special cases of a generalized Stoke's theorem. Spivak's Calculus on Manifolds book (which is actually not a manifolds book despite the name) teaches calculus at an undergrad level using exterior algebra and differential forms if you're interested in learning this stuff.

Exterior algebra can be considered as part of geometric algebra, so you could continue on to learn geometric algebra if you enjoy exterior algebra.

ProfRobBob Youtube - This sir has great videos. His playlists are in order and very useful for Calculus. Loved his pre calculus playlist.

Patrick JMT - I could not have passed Calculus 2 without this guy. For the most part, his Calculus section is in order on his website.

KhanAcademy - Nice courses with problems available for you. Really easy to use and navigate. I worked through Algebra and only watched his videos on Trigonometry and Calculus.

Hope you get back on track buddy. Don't give up.


I self taught myself Algebra through Precalculus, here are books I used:

1. Practical Algebra - This helped when doing KhanAcademy Algebra course
   
   
2. Precalculus Demystified - Easy to understand w/o having any knowledge of precalculus.
   
   
3. Precalculus by Larson - The demystified book above helped form a foundation that allowed me to understand this book fairly well
   
   
4. Calculus for Dummies by PatrickJMT - This goes great for soliving problems in PatrickJMT's 1000 problem book.
   

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

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).

Sure! I have a lot of resources on this subject. Before I recommend it, let me very quickly explain why it is useful.

Bayes Rule basically means creating a new hypothesis or belief based on a novel event using prior hypothesis/data. So I am sure you can already see how useful it would be in medicine to think about. The Rule(or technically theorem) is in fact an entire field of statisitcs and basically is one of the core parts of probability theory.

Bayes Rule explains why you shouldn't trust sensitivity and specificity as much as you think. It would take too long to explain here but if you look up Bayes' Theorem on wikipedia one of the first examples is about how despite a drug having 99% sensitivity and specificity, even if a user tests positive for a drug, they are in fact more likely to not be taking the drug at all.

Ok, now book recommendations:

Basic: https://www.amazon.com/Bayes-Theorem-Examples-Introduction-Beginners-ebook/dp/B01LZ1T9IX/ref=sr_1_2?ie=UTF8&qid=1510402907&sr=8-2&keywords=bayesian+statistics

https://www.amazon.com/Bayes-Rule-Tutorial-Introduction-Bayesian/dp/0956372848/ref=sr_1_6?ie=UTF8&qid=1510402907&sr=8-6&keywords=bayesian+statistics

Intermediate/Advanced: Only read if you know calculus and linear algebra, otherwise not worth it. That said, these books are extremely good and are a thorough intro compared to the first ones.

https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=sr_1_1?ie=UTF8&qid=1510402907&sr=8-1&keywords=bayesian+statistics

https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573/ref=sr_1_12?s=books&ie=UTF8&qid=1510403749&sr=1-12&keywords=probability

For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.

I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.

If you're interested in a sequence of books...keep reading.

If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.

Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.

Working through Griffiths is a good idea, but I strongly suggest working through an abstract linear algebra book before you do anything else. It will make your life much better. Doing some of Griffiths in advance might make your homework a bit easier, but you'll be repeating material when you could be learning new things. And learning real linear algebra will benefit you in pretty much every class.

I recommend this book as your primary text and this one for extra problems and and a second opinion.

I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.

Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).

Elements of Statistical Learning covers KDE pretty well. (It does have a pretty heavy linear algebra prereq. If it is getting too hairy, you may want to look at a numerical linear algebra book, like Trefethen and Bau)

Also Computational Statistics covers it well from what I remember. These are both really good books.

But both are really great books.

This book seems silly, but it's honestly great for learning Calculus, especially the second time: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606

(I read it in 1999 when I went from HS -> College, and the college I went into assumed you had already passed calc, and freshmen all had to start with second year calc. The professors recommended all incoming students refresh before the start of class, and I'm glad they did, because that book retaught some things I don't think I learned correctly the first time, made a huge difference).

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. :)

Start With "Foundations Of Analysis" By Edmund Landau

http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X

It's a tiny book, but is very good at explaining basic abstract algebra.

Here is the description from Amazon:

"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."

With the above book you should then have enough knowledge to move on to calculus.

I recommend the two volume series called "Calculus" by Tom M. Apostol.

The first volume is single variable calculus and the second is multivariate calculus

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&s=books&qid=1239384587&sr=1-4

http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3

This one's well-known and highly regarded as a good source.

I'm also going to start learning number theory because it's a pretty fun subject. So far, Hardy's been pretty good (I've only read excerpts of the 1st chapter though).

As for your background, you would only need to know basic facts about numbers (divisibility/primes etc) when starting Hardy so you should be fine I think.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

Yes, they're awesome. Brought up pretty frequently on /r/math, too. I'm pretty sure I have at least 10 Dover books. Two excellent titles that come to mind are Pinter's A Book of Abstract Algebra and Rosenlicht's Introduction to Analysis.

It sounds like you want some kind of regression, especially to answer 2. In a GLM, you are not claiming that the data by itself has a Normal/Poisson/Negative Binomial/Binomial distribution, only that it has such a distribution when conditioned on a number of factors.

In a nutshell: you model the mean of the distribution as a linear combination of the inputs. Then you can read the weighting factors on each input to learn about the relationship.

In other words, it doesn't need to be that your data is Poisson or NB in order to do a Poisson or NB regression. It only has to be that the error, that is, the difference between the expected based on the mean function and the actual, follows such a distribution. In fact, there may be some simple transformations (like taking the log of the outcome) that lets you use a standard linear model, where you can reasonably assume that the error is Normal, even if the outcome is anything but.

If your variance is not dependent on any of your inputs, that's a great sign, since heteroskedasticity is a great annoyance when trying to do regressions.

If you have time, the modern classic in this area is http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X. It starts with a pretty gentle introduction to regression and works its way into the cutting edge by the end.

If you want to do statistics in a rigorous way you should start with calculus and linear algebra.

For calculus I recommend Paul's notes -> http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
They are really clearly written with good examples and provide good intuition.
As supplement go through 3blue1borwn Essence of calculus. I think it's an excellent resource for providing the right intuition.

For linear algebra - linear algebra - Linear algebra done right as already recommended. Additionally, again 3blue1brown series on linear algebra are top notch addition for providing visual intuition and understanding for what is going on and what it's all about.

Finally, for statistics - I would recommend starting with probability calculus - that way you'll be able to do mathematical statistics and will have a solid understanding of what is going on. Mathematical statistics with applications is self-contained with probability calculus included. https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817

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

I 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!
   
   

Depends what you're interested in, but since we're in the ML subreddit it's probably about computation.

Numerical/computational linear algebra studies how to implement the ideas introduced in a 1st LA course on a finite-precision computer.

Linear programming, integer programming, non-linear optimization, and differential equations all heavily rely on linear algebra. The latter two mainly because of Taylor expansions which allow us to approximate functions in terms of linear and quadratic forms.

For ML you're probably best off skimming through the high level ideas in numerical linear algebra, and then diving into linear programming and non-linear optimization.

I wouldn't bother with Apostol's Calculus. For analysis, you should really look at the first two volumes of Stein and Shakarchi's Princeton Lectures in Analysis.

Vol I: Fourier Analysis
Vol II: Complex Analysis

Then, you should pick up:

Munkres, Analysis on Manifolds or something similar, you could try Spivak's book but it's a bit terse. (on a personal note, I tried doing Spivak's book when I was a freshman. It was a big mistake).

In truth, most introductory undergrad analysis texts are actually more invested in trying to teach you the rigorous language of modern analysis than in expositing on ideas and theorems of analysis. For example, Rudin's Principles is basically to acquaint you with the language of modern analysis -- it has no substantial mathematical result. This is where the Stein Shakarchi books really shines. The first book really goes into some actual mathematics (fourier analysis even on finite abelian groups and it even builds enough math to prove Dirichlet's famous theorem in Number Theory), assuming only Riemann Integration (the integration theory taught in Spivak).

For Algebra, I'd suggest you look into Artin's Algebra. This is truly a fantastic textbook by one of the great modern algebraic geometers (Artin was Grothendieck's student and he set up the foundations of etale cohomology).

This should hold you up till you become a sophomore. At that point, talk to someone in the math department.

The most important thing you can do is memorize the definitions. I mean seriously have them down cold. The next thing I would recommend is to get another couple of analysis books (go cheap by getting old books, it isn't like the value of epsilon has changed over the past two hundred years) and look at their explanations, work those problems. Having a different set can be enlightening. Be prepared to spend a lot of time on it all.

Good luck!

EDIT: Back home now and able to put in some specific books. I used Rosenlicht and you wouldn't believe how happy I was to buy a textbook that, combined with a slice of pizza and a coke, was still less than $20. One of my books that I looked at for a different view point was Sprecher.

I also got a great deal of value out of Counterexamples in Analysis because after seeing things go wrong (a function that is continuous everywhere but nowhere differentiable? Huh?) I started to get a better feel for what the definitions really meant.

I hope you're also sensing a theme: Dover math books rock!

I'm planning on relearning calculus also. The books that were recommended to me were:

http://www.amazon.com/gp/aw/d/1592575129?pc_redir=1412262976&robot_redir=1

http://www.amazon.com/gp/aw/d/0716731606/ref=pd_aw_sims_3?pi=SL500_SY115&simLd=1

They're not exactly textbooks, but they appear to be good guides. Best of luck.

Yes!

If you don't know any calculus Stewart Calculus is the typical primer in colleges. Combine this with Khan Academy for easy mode cruise control.

After that, you want to look at The Big Orange Book, which is essentially the bible for undergrad astrophysics and 100% useful beyond that. This book could, alone, tell you everything you need to know.

As for other topics like differential equations and linear algebra you can shop around. I liked Linear Algebra Done Right for linear personally. No recommendations from me on differential equations though, never found a book that I loved.

Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).

Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.

Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.

I'm a lot like you, very self directed learning (I spent as little time in HS as I possibly could, and nearly flunked out because of it). This book really sparked my interest in math. Everything from why zero is so exceptional and how hard it was for our species to realize it, to how to figure out a square root by hand (maybe boring, but I was interested in the method, since just pushing the square root button on a calc was dissatisfying).

Calculus is something that is damn near impossible to get without help (you can do it, but you probably won't understand it). Finally, it's pretty important to talk to people to see what's worth learning and what you haven't considered yet. Speaking of programming, if you fail to get yourself out in there and talk to other people (people that are better than you at something) you are liable to feel proud of inventing something like bubble-sort.

I'm currently on this journey as well! I'm a programmer teaching my self rigorous maths, so I can definitely help you out.

I find it's best to simultaneously look at several resources on topics such as proofs, so you can get a few perspectives on the same essential topics and have an easier time of finding something.

As a preliminary to proofing, I would suggest a survey of basic logic and Set Theory. I picked up my Set Theory from google searches and the introduction in Apostol's Calculus, and wiki articles on logic and set operations.. It's really easy to learn enough set theory and logic to begin understanding rigorous proofs.

To learn the proofing skills needing for Real Analysis I recommend

a) "Foundations of Analysis" by Edmund Landau.

b) Math 378: Number Systems: An Axiomatic Approach

For an actual book on real analysis, there can be no greater book than Apostol's Calculus.

I would try Mathematical Statistics and Data Analysis by Rice. The standard intro text for Mathematical Statistics (this is where you get the proofs) is Wackerly, Mendenhall, and Schaeffer but I find this book to be a bit too dry and theoretical (and I'm in math). Calculus is less important than a thorough understanding of how random variables work. Rice has a couple of pretty good chapters on this, but it will require some mathematical maturity to read this book. Good luck!

It's probably not possible to review everything you need, but getting more experience with proofs is a good start. This course might be helpful:

https://www.coursera.org/course/matrix

and these texts are great examples of mathematical thinking in prose:

Grinstead and Snell's Introduction to Probability:
https://math.dartmouth.edu/~prob/prob/prob.pdf

Apostol's Calculus I and II:
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051

Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.

As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.

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

Awesome! As mentioned, Rudin, Folland, and Royden are the gold standards of measure theory, at least from what I have heard from professors and the internet. I'm sure other people have found other good ones! Another few I somewhat enjoy are Capinski and Kopp and Dudley, as those are more based on developing probability theory. Two of my professors also suggested Billingsley, though I have not really had a good chance to look at it yet. They suggested that one to me after I specifically told them I want to learn measure theory for its own right as well as onto developing probability theory. What is your background in terms of analysis/topology? Also, I am teaching myself basic measure theory (measures, integration, L^p spaces), then I think that should be enough to look into advanced probability. Feel free to PM me if you need some help finding some of these books! I prefer approaching this from the pure math side, so mathematical statistics gets a bit too dense for me, but either way, I would look at probability then try to apply it to statistics, especially at a graduate level. But who am I to be doling out advice?!

*Edit: supplied a bit more context.

A lot of people recommend Khan Academy, but you cannot really learn from the Khan Academy; there is just too much material to cover. I recommend either going into an algebra class at your local community college, and/or get some good algebra/maths books. This one gets a lot of praise on Amazon.com:

http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123/ref=sr_1_fkmr0_1?ie=UTF8&qid=1288684060&sr=8-1-fkmr0

and, this one is the one nobel laureate Richard Feynman taught himself with:

http://www.amazon.com/Algebra-practical-Mathematics-self-study/dp/B0007DZPT6

If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.

It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.

Well, if you want something light and accessible and suitable for the layperson, I'm quite fond of Jan Gullberg's Mathematics from the Birth of Numbers. It goes over basically everything you would typically learn in primary and secondary school, and it presents everything with historical background. But it doesn't go into tremendous detail on each topic, and it doesn't provide the most rigorous development. It's more of a high-level overview.

But if you really want to learn some mathematics, on a deep and serious level, be prepared to read and study a lot. It's a rewarding journey, and we can give you book recommendations for specific topics, but it does take a lot of discipline and a lot of time. If you want to go that route, I would recommend starting with an intro to proofs book. I like Peter J. Eccles's An Introduction to Mathematical Reasoning, but there are many other popular books along the same line. And you can supplement it with a book on the history of mathematics (or just read Gullberg alongside the more serious texts).

Have a look at the courses that you'll be studying in your first year - find some introductory texts on them. Many universities publish course notes online - you might even be able to find some from your own university. Anything you do beforehand will make it easier once you get there.

Depending on your previous experience, there a couple of things to consider that will make the transition to university mathematics easier.

The main thing is proofs - if you're not comfortable setting out a (rigorous) proof, practise beforehand. Our recommended text was 'How to think like a mathematician' (Houston).

Another area is algebra - it helps to be very comfortable with standard algebra. You don't want to have any issues manipulating lots of Greek letters. For applied courses it helps to be sharp with standard derivatives and integrals, too.

R for Data Science is great, especially because it teaches tidyverse.

Another good book is Learning Statistics with R: A tutorial for psychology students and other beginners, which also teaches the implementation of basic statistical techniques, like ANOVA or linear regression.

If you have some time spare, you can follow it by Data Analysis Using Regression and Multilevel/Hierarchical Models, which is also (mostly) based on R.

The Visual Display of Quantitative Information is a good book on the principles of data visualization. It’s theoretical, so no R examples.

Complex Surveys: A Guide to Analysis Using R is great if you work with survey data, especially if you work with complex designs (which nowdays is pretty much all the time).

Personaly, I would also invest some time learning methodology. Sadly, I can’t help you here, because I didn’t used textbook for this, but people seem to like books from Earl Babbie.

I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.

​

• Probability and Statistical Inference by Hogg, Tanis and Zimmerman
  
• Mathematical Statistics with Applications by Wackerly
  
  These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
  
  ​
  
  The other two I've been looking at which seem to be a lot more complex are
  
  
• Introduction to Mathematical Statistics by Hogg as well. I'd think it's the more rigorous version of the book mentioned above by the same author
  
• All of Statistics by Wasserman which seems to require a lot of prior knowledge in statistics, but I think tackles just the perfect topics for machine learning
  
  And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
  
  ​

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.

Yeah, definitely the best book I've read on differential forms was Spivaks Calculus on Manifolds. Its very readable once you have a solid foundational calculus background and is pretty small given what it covers (160pp). If you need to know this stuff then this is definitely the right place to learn it.

Book of proof is a more gentle introduction to proofs then How to Prove it.

​

No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.

​

An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.

I've used a book by Gelman for self study. Great author, very good at using meaningful graphics -- which may be an effective way to convey ideas to students.

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.

My favorite two books for Calc 1,2, and 3 hands down:

How to Ace Calculus

How to Ace the Rest of Calculus

They're short, to the point, and pretty funny honestly.

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.

For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.

For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.

One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.

Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.

Also, Zolv mentioned the book Practical Algebra (A Self Teaching Guide), by Peter Selby and Steve Slavin. I concur. It's cheap, about $11, and has great reviews on Amazon. I found it extremely helpful when I was getting started. Practical Algebra

I think this sample paragraph is something you'd agree with (from page 79 of the second edition):
"We have some good news and some bad news. This chapter and the two that follow [about factoring] introduce some fairly difficult concepts. That's the bad news. The good news is that if you can learn about 75 or 80 percent of this material, you're way ahead of the game....Remember, you're teaching yourself math, and the only thing that's helping you is this book, which is kind of like doing open heart surgery over the phone. So don't get down on yourself if you don't comprehend something the first time - or even the second time. If you get stuck, go on to the next frame..."

I have Abbott's and Charles Pugh's books. Both excellent and probably in your reserve library. There's another book I noticed on Amazon, I've never heard anybody on reddit or math.stackexchange mention, probably worth $20: https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539

Also Spivak, Apostol, other books: https://www.reddit.com/r/math/comments/3drlya/what_mathematical_analysis_book_should_i_read/

There's lots of other threads here and math.SE that're helpful. Maybe looking thru Courant/Robbins What is Math witht he mindset that it's an enjoyable read

Just completed Probability this semester, and moving on to Statistical Inference next semester. Calc. B is a prerequisite, and wound up seeing plenty of it along with a little Calc C (just double integrals). I'm an Applied Mathematics undergrad major btw and former Physics major from some years ago. I wound up enjoying it despite my bad attitude in the beginning. I keep hearing from fellow math majors that Statistical Inference is really difficult. Funny thing is I heard the same about Linear Algebra and didn't find it overwhelming. I'll shall soon find out. We used Wackerly's Mathematical Statistics with Applications. I liked the book more than most in my class. Some thought it was overly complicated and didn't explain the content well. Seems I'm always hearing some kind of complaint about textbooks every semester. Good luck.

Somewhat facetiously, I'd say the probability that an individual who has voted in X/12 of the last elections will vote in the next election is (X+1)/14. That would be my guess if I had no other information.

As the proverb goes: it's difficult to make predictions, especially about the future. We don't have any votes from the next election to try to discern what relationship those votes have to any of the data at hand. Of course that isn't going to stop people who need to make decisions. I'm not well-versed in predictive modeling (being more acquainted with the "make inference about the population from the sample" sort of statistics) but I wonder what would happen if you did logistic regression with the most recent election results as the response and all the other information you have as predictors. See how well you can predict the recent past using the further past, and suppose those patterns will carry forward into the future. Perhaps someone else can propose a more sophisticated solution.

I'm not sure how this data was collected, but keep in mind that a list of people who have voted before is not a complete list of people who might vote now, since there are some first-time voters in every election.

If you want to get serious about data modeling in social science, you might check out this book by statistician/political scientist Andrew Gelman.

If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.

If your degree is anything else, you may have no formal requirement to learn the why.

For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.

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"
  

> I'm hoping for something like what Div, Grad, Curl and All That does for Vector Calculus.

Is that a math text? I am not really familiar with it, but from what I heard it sounds more like a physics/engineering text. Does it have any formal proofs in it?

You won't be able to get too far with a proofless(?) Abstract Algebra text if there exists one to begin with. Even Charles Pinter's A Book of Abstract Algebra presupposes some degree of mathematical maturity.

Anyway, try these and see if you like them:


Visual Group Theory by Nathan Carter


Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem by Al Cuoco, Joseph J. Rotman

I am not a big fan of Rudin. The tone is incredibly stuffy and his style is fairly loose.

I would recommend the small Dover book Introduction to Analysis by Rosenlicht. It's a very small book, hardly 200 pages, but the style is much nicer. It doesn't cover nearly as much (there is no introduction to Fourier Analysis, differential forms, or the gamma function), but that's a good thing for an introductory book, since you can expect to master everything in it.

We used Abbott in a class I audited. I skimmed bits of it, and it seemed pretty nice. Very expository, which is always nice to have when self-studying.

I would eventually pick up a copy of Rudin, just because it's a cultural icon. But it's just very brutal for an introduction to the subject.

You should check out Spivaks Calculus on Manifolds.

http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219

Read the first chapter or 2 and see how you like it, if you feel overwhelmed check some of the other recommendations out.
It is however a good book, and you should read it sooner or later.

Rudins principles of mathematical analysis is also excellent, however it
is not strictly multi-dimensional analysis.
Read at least chapter 2 and 3, they lay a very important groundwork.

By the way, do you know if things like linear/nonlinear regression, ANOVA and multivariate statistics is useful for me? Like stuff from https://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=dp_ob_title_bk or https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=pd_sim_14_8?_encoding=UTF8&pd_rd_i=1439840954&pd_rd_r=c9c0f3c5-f332-11e8-9f0a-0f336f5f387a&pd_rd_w=xHkKH&pd_rd_wg=7lXm5&pf_rd_i=desktop-dp-sims&pf_rd_m=ATVPDKIKX0DER&pf_rd_p=18bb0b78-4200-49b9-ac91-f141d61a1780&pf_rd_r=PFCZ1JM04FMAVAHG6VNP&pf_rd_s=desktop-dp-sims&pf_rd_t=40701&psc=1&refRID=PFCZ1JM04FMAVAHG6VNP

​

You can tell how little I know since I'm kinda shooting topics at the wall hoping something sticks

It is hard to provide a "comprehensive" view, because there's so much disperate material in so many different fields that draw upon probability theory.

Feller is an approachable classic that covers all of the main results in traditional probability theory. It certainly feels a little dated, but it is full of the deep central limit insights that are rarely explained in full in other texts. Feller is rigorous, but keeps applications at the center of the discussion, and doesn't dwell too much on the measure-theoretical / axiomatic side of things. If you are more interested in the modern mathematical theory of probability, try Probability with Martingales.

On the other hand, if you don't care at all about abstract mathematical insights, and just want to be able to use probabilty theory directly for every-day applications, then I would skip both of the above, and look into Bayesian probabilistic modelling. Try Gelman, et. al..

Of course, there's also machine learning. It draws on a lot of probability theory, but often teaches it in a very different way to a traditional probability class. For a start, there is much more emphasis on multivariate models, so linear algebra is much more central. (Bishop is a good text).

Jan Gullberg's Mathematics: From the birth of numbers is a great book I'd recommend: https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X

It introduces a lot of mathematical topics starting from the "simplest" (numbers you asked about) and advances to common stuff found in university studies (although not going extremely far), but what might be the biggest feat and useful to your case is that tells as a non-fictional story while at it, explaining mathematical tools, their history and how they relate to each other extremely well in a way a normal college textbook doesn't, and it doesn't assume you already know everything from school.

I learned from Wackerly which is decent, though I think Devore's presentation is better, but not as deep. Both have plenty of exercises to work with.

Casella and Berger is the modern classic, which is pretty much standard in most graduate stats programs, and I've heard good things about Stat Labs, which uses hands-on projects to illuminate the topics.

The single best resource for 103/104 is The Calculus Lifesaver by Adrian Banner. There's a book and a series of recorded review sessions. I stopped showing up to 104 lectures when I found these because they were so much more thorough than the classes. Banner also did review sessions for 201/202 when you reach that point, which are equally good.

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

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

You might want to consider some kind of numerical linear algebra book like the very readable Trefethen and Bau.

While this topic isn't always included in an undergrad curriculum, it's hugely useful. It's critical for a bunch of more advanced areas like physical simulation, graphics optimization, and machine learning.

I had a nice book called Precalculus Mathematics in a Nutshell but it is not geared to starting from scratch. It's a good book if you remember some of your algebra, geometry, and trigonometry.

I've known some people who had good experiences with Practical Algebra

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

In the lead up to calc first thing you want to do is just make sure you're algebra skills are pretty solid. A lot of people neglect it and then find the course to be harder than it needed to be because you really use algebra throughout.

Beyond that, if you want an extra book to study with and get practice problems from The Calculus Lifesaver is a big book of calculus you can use from now and into a first year college calculus course. If you do get it, don't worry about reading the whole thing from cover to cover, or doing all of the problems in it. It is a big book for a reason, it definitely covers more than you need to know for now, so don't get overwhelmed, it all comes with time.

Best of luck

I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.

After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.

Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.

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

Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.

I had a friend who went through the program. I don't think there was a pre-assessment as Academic Math itself is a prerequisite to other stuff, but don't take my word as law on that. The course resource appears [to be here] (https://www.nscc.ca/learning_programs/programs/PlanDescr.aspx?prg=ACC&pln=ACCONNECT) and doesn't mention pre-assessments. [This PDF] (http://gonssal.ca/documents/AcadMathIVCurr2010.pdf) should cover a fair bit of what the course is about.

As an aside, [this book] (https://www.amazon.ca/Practical-Algebra-Self-Teaching-Peter-Selby/dp/0471530123) is a fantastic way to get yourself up to speed on algebra. I can't recommend it highly enough.

Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley

• How to Lie with Statistics by Darrell Huff is a quick and fun read that will sharpen your ability to identify fraudulent claims.
  
  
• My favorite stats book for people who are not math-geeks is Statistical Methods for Psychology by David Howell
  
  
• The particular technique used in the paper is somewhat beyond beginner level. A third good book to read might be Data Analysis Using Regression and Multilevel/Hierarchical Models by Andrew Gelman

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.

Numerical Linear Algebra by Nick Trefethen is a pretty friendly intro to graduate linear algebra/matrix theory from a numerical analysis angle:

http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617

Introduction to Numerical Analysis is very comprehensive, more advanced, but reads like an encyclopedia in a way. A good reference, though not very good as a lone textbook.

http://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X

How to Ace Calculus: The Streetwise Guide is charming. It does an excellent job scaffolding intuitive understanding without unnecessarily sacrificing rigor. It took me at least three attempts to properly spell the word "unnecessarily" in the previous sentence.

Extremely delayed edit: It also has the marked advantage of being quite cheap.

So I am a political scientist (though my research crosses into sociology).

What I would recommend is starting by learning Generalized Linear Models (GLMs). Logistic regression is one type, but GLMs are just a way of approaching a bunch of other type of dependent variables.

Gelman and Hill's book is probably the best single text book that can cover it all. I think it provides examples in R so you could also work on picking up R. It covers GLMs and multi-level models which are also relatively common in sociology.

If you really need it dumbed down, I would recommend Asimow and Maxwell. This text has a solutions manual. Note that this is specifically tailored toward actuarial exams - i.e., people that have to learn the material quickly but not necessarily for grad school. (And yes, the website is legit. I've done some contract work for them in the past and have ordered books through them.)

If you don't mind something more mathematical, I would recommend Wackerly et al.

Well, there's here, of course. Hilbert spaces are a topic in analysis. I've heard good things about this book, which comes at it from a physics perspective.

If your background in analysis is up for it, they are covered in Rudin. This book is pretty intense.

I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.

The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.

Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.

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.

sounds wonderful! ill add that to the list. i was also thinking of 'how to think like a mathematician: a companion to undergraduate mathematics' - kevin houston. from the amazon 'look inside' i'm following pretty well

don't know if you've seen it before but does it look alright?

Gelman's book is pretty awesome. http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X/ref=pd_sim_b_1

A lot of the recommendations in this thread are good, I'd like to add "Bayesian Data Analysis 3rd edition" by Gelman et al. Useful if you encounter Bayesian models, especially hierarchical/multilevel models.

Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.

I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/

https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful

If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2

And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

I took a long, long break between undergrad and grad school (think decades). I found this GRE math prep book very helpful. (The GRE math section tests high school math knowledge), I'd take the sample tests, see where I fell short, and focus on understanding why. I also found Practical Algebra to be a good review-and-practice guide, for the fundamentals. I boned up on discrete math by buying an old copy of Rosen and the matching solutions guide. And, I watched a bunch of videos of this guy explaining various facets of the math you need for computer science.

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'd like to know, how did you learn to use R?

My batshit crazy lovable thesis advisor was teaching intro datascience in R.

He can't really lecture and he have high expectation. The class was for everybody including people that don't know how to program. The class book was advance R http://adv-r.had.co.nz/... (red flag).

We only survived this class because I had a cs undergrad background and I gave the class a crash course once. Our whole class was more about how to implement his version of random forest.

I learned R because we had to implement a version of Random forest with Rpart package and then create a package for it.

Before this a dabble in R for summer research. It was mostly cleaning data.

So my advice would be to have a project and use R.

>how did you learn statistics?

Master program using the wackerly book and chegg/slader. (https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817)

It's a real grind. You need to learn probability first before even going into stat. Wackerly was the only real book that break down the 3 possible transformations (pdf,cdf, mgf).

How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.

How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).

When I was (approximately) in 8th grade I read https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and I loved it. :)

Hardy and Wright, An Introduction to the Theory of Numbers. Awesome book.

http://www.amazon.com/An-Introduction-Theory-Numbers-Hardy/dp/0199219869

I second Michael Spivak's Calculus if you haven't done a proper analysis course before. It's a 100% rigorous treatment of calculus from first principles and is probably better thought of as "analysis in one dimension." I post on a subreddit of folks working through the book pretty frequently: /r/calculusstudygroup

After that, I like Kolmogorov and Fomin's Introductory Real Analysis and Walter Rudin's Principles of Mathematical Analysis.

There's also Michael Spivak's Calculus on Manifolds, which focuses purely on multi-variable calculus on manifolds. Torus calculus!

How would you rather split beginner vs intermediate/advanced ?

My feeling was that Ben Lambert's book would be a good intro and that Bayesian Data Analysis would be a good next ?

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.

This isn't an online resource, but this book is awesome for learning Calc 1.

https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

I'm not much of a visual person so I didn't learn much from skimming it, but many people stand by it and it's a beautiful exposition (of a topic I also think is beautiful) indeed!

Thanks for sharing I'll look into that one! Thanks:)

​

Edit: They actually write in that course "The book Numerical Linear Algebra by Trefethen and Bau is recommended." so It might be some further applications!

Rudin covers Hilbert spaces and Banach spaces in his Real and Complex Analysis, which is why he jumps straight into topological vector spaces in his book on functional analysis. So perhaps you could read those chapters from Real and Complex Analysis. Alternatively, check out the classic Functional Analysis by Reed and Simon or Conway's book. The reviews published by the MAA might also be interesting to you. And of course, there are many lecture notes available on the web. :-)

Try the "for dummies" books (for real). I went to a top ten uni for maths and didnt really go to my calculus lectures (they were monday and tuesday mornings). I went through "differential equasions for dummies" and it got me a high 2:1. Plus, they are loads cheaper than most other text books.

Also, this book is good for general. "How to think like a mathematician" - http://www.amazon.co.uk/How-Think-Like-Mathematician-Undergraduate/dp/052171978X

You'll usually find the following recommended:

• Axler's Linear Algebra Done Right
  
• Friedberg's Linear Algebra
  
  I've personally used Friedberg's text, and I found it to be pretty well written.

It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.

You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.

Mac Lane for category theory, of course.

I think people would agree on Hartshorne as the algebraic geometry reference.

Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.

Spivak for differential geometry.

Rotman is the group theory book for people who like group theory.

As a physics person, I must have a copy of Fulton & Harris.

I personally think you should brush up on frequentist statistics as well as linear models before heading to Bayesian Statistics. A list of recommendations directed at your background:

• For Frequentist Statistics and Probability: Introduction to Probability, Mathematical Statistics With Applications or since you are more interested in applications of Stats to Machine Learning, All of Statistics.
  
  
• For Bayesian Statistics, I would start with Statistical Rethinking and Introduction to Bayesian Statistics for an intro.
  
  

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).

It mentions Rosenlicht at the bottom. Lucky you, that book's only 8 bucks! It's a good book, too.

Since you are already going to take Machine Learning and want to build a good statistical foundation, I highly recommend Mathematical Statistics with Applications by Wackerly et al.

I read Mathematics: From the Birth of Numbers in high school / early college. It's a long book, but it's definitely worth checking out.