## 6 Best Linear Algebra Textbooks ( Review)

Linear algebra is one of the most important aspects of mathematics out there if you have an interest in machine learning and artificial intelligence, but as you might expect, it’s hardly a piece of cake to learn, so most students find themselves seeking the best linear algebra textbooks to guide them through their studies.

**What are the Best Linear Algebra Textbooks to read?**

For those who are beginners or have not been in a linear algebra class for years, a good guide that clearly explains the field is even more essential, but there are so many books on linear algebra on the market that it can be really difficult to pick which one teaches the skills you need best, especially if you are a novice with no idea of what you need to know.

### Best Linear Algebra Textbooks: Our Top 6 Picks

Fortunately, we’ve done all the necessary research to make the decision as easy for you as possible. In this post, we bring you a list of the 6 best linear algebra textbooks.

**1. Linear Algebra: Step by Step**

Linear Algebra by Kuldeep Singh is arguably one of the best texts on linear algebra out there, with the author offering a broad outlook covering its applications in many fields, including business, computer science, and engineering, with some interesting exposition on how Google makes use of linear algebra to rank search results.

The major strength of this book is in the number of examples it presents step-by-step to make the operations as easy to follow as possible, and the fact that the solutions to problems are all available online, making it the perfect companion for self-guided study or distance learners.

The exercises themselves are designed to test your understanding of each section, drawn from past questions in college examinations, and geared towards boosting your confidence with the wording and style used ahead of the real deal.

*Linear Algebra* is one of the most dynamic mathematics textbooks ever. Each chapter opens with a short biography of a leading player in the field of linear algebra who relates to the context being discussed within, and these—coupled with interviews with leading experts—ensure your understanding expands from the math to the state of the field as a whole.

Readers don’t have a bad word to say about the book and love the interesting facts they learn as they read.

**Author**: Kuldeep Singh**Publisher**: OUP Oxford; 1 edition (October 31, )**Pages**: pages**Edition**: First Edition

**2. No Bullshit Guide to Linear Algebra**

The author of the No Bullshit Guide to Linear Algebra, Ivan Savov, utilizes his year experience as a lecturer to paint linear algebra as a cornerstone of engineering and science that is key to the development of computer graphics, machine learning, and quantum mechanics.

The book attempts to create a geometric interpretation of linear algebra and its theoretical foundations, making it the perfect tool to supplement the studies of university students in particular.

One of the strengths of this book is the way it presents the subject in a concise and precise manner. Definitions, diagrams, formulas, explanations, and real-world examples are used to educate the reader in an accessible and clear way we’re sure readers will love.

To cement the reader’s understanding, there are math problems to solve after each module that encourage students to boost not only their linear algebra skills but their appreciation of the potential applications to real-life situations. Readers report that using this book exposes you to several concepts surrounding linear algebra which allow you to understand it better.

Aside from the quality of knowledge relayed to the reader, buyers have commented that the book is also light-hearted and fun to read, so it’s no surprise that the *No bullshit guide to linear algebra* has been rated as one of the best guides when it comes to teaching yourself the intricacies of the subject.

**Author**: Ivan Savov**Publisher**: Minireference Co.; 2^{nd}edition (April 2, )**Pages**: pages**Edition**: Second Edition

**3. Linear Algebra Done Right**

Linear Algebra Done Right is a book designed for undergraduates in mathematics as well as those in graduate school. It’s a bestseller and is highly recommended by both students and faculty alike.

The author makes use of a novel approach in writing this book which tackles the determinants in the latter half to ensure readers fully understand the central goal of the subjects covered first, as well as how they can be applied in the real world.

Furthermore, *Linear Algebra Done Right* takes extra steps to offer well broken-down explanations and make proofs simpler, rendering the book a valuable tool even for novices.

Each chapter ends with linear algebra problems that the student is required to solve in order to self-evaluate how well they understand the material covered.

This is the third edition of this book, ensuring the material covered is as current as possible and offering an impressive new exercises compared to previous versions. A few new topics have also been added, including dual spaces, product spaces, and quotient spaces, and both print and e-book copies offer a clean, modern design.

If you can only afford one textbook to help with your studies, this covers a wide range of topics and exercises, and just might be the pick for you.

**Author**: Sheldon Axler**Publisher**: Springer; 3^{rd}Edition (November 6, )**Pages**: pages**Edition**: Third Edition

**4. Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares**

We consider the Introduction to Applied Linear Algebra to be a very special book. Yet another bestseller, we love it for its to-the-point explanations of linear algebra, and effective explanations that take the sting out of study hall.

To test your understanding, the book offers a huge range of easy-to-understand practical examples designed to teach you the skills you need to ace your class, whether you’re a seasoned math pro expanding your skills or a total newbie to the field.

The book covers several essential aspects of linear algebra, including matrices, least squares, and vectors. These are then related to their applications in engineering, machine learning, tomography, and artificial intelligence, to name a few.

The range of problems the book offers provide the ideal opportunity to test your knowledge after each module, making use of real-world problems that allow the reader to translate what they have learned into such situations.

As a unique extra, you are able to access additional material online; including lecture slides, notes, more exercises, and data sets, making *Introduction to Applied Linear Algebra* a frontrunner for both self-study and extra-curricular study. If you are a beginner, this book provides you with a strong foundation to help you understand linear algebra better, but even those who are more experienced can use the extra materials to deepen their understanding and test the limits of their knowledge.

Readers regard this book as a fantastic introduction to linear algebra that boasts better print, more examples, and all-round superior content to much of the competition.

**Authors**: Steven Boyd and Lieven Vandenberghe**Publisher**: Cambridge University Press; 1^{st}edition (August 23, )**Pages**: pages**Edition**: First Edition

**5. Linear Algebra (Dover Books on Mathematics)**

Linear Algebra is a presentation of Georgi Shilov’s original text that has been reworked by Richard Silverman. Taking Shilov’s comprehensive teachings, Silverman breaks them down into bite-size elements, bringing the text’s already impressive accessibility to a whole new level that we’re sure students of all abilities will love.

The text covers topics like vector arguments, coordinate transformations, unitary spaces, Euclidian spaces, quadratic forms, and much more, guiding you from the simpler aspects of linear algebra to the more challenging topics.

There are a host of problems provided for users to solve, with answers and hints provided at the end of the book for either self-review or peer marking in a classroom environment. The problems and examples themselves are accessible and easily understandable, meaning you don’t need a teacher to translate and the book is versatile enough to be used both in the classroom and alone.

**Author**: Georgi E. Shilov (Original Author), Richard Silverman**Publisher**: Dover Publications; Dover Books on Mathematics edition (June 1, )**Pages**: pages**Edition**: Dover Books on Mathematics Edition

**6. Linear Algebra and its Applications (5th Edition)**

The main selling point of Linear Algebra and its Applications is the way it gets its ideas across. We already know that understanding linear algebra can be difficult, but this book simplifies it in a unique way that will ensure even the most inexperienced reader can tackle the material with confidence.

Even advanced concepts can feel like a breeze thanks to the way this book is structured. It builds students’ knowledge of fundamental aspects first to give them a solid grounding, gradually expanding this to more complex areas of linear algebra to ensure they have a solid foundation for learning and comprehensive working knowledge.

Thanks to the quality of the writing, any novice can pick up this book and expand their knowledge in a linear, sensical manner, which we think makes for a refreshing change to how textbooks are usually written.

We’d recommend confirming the ISBN you need with your instructor to ensure you get the right edition. Some readers have reported the newest is quite costly, so if you’re looking to save a few cents, ask your teacher if an older one will work – we promise these are just as useful.

**Authors**: David C. Lay and Steven R. Lay**Publisher**: Pearson; 5 edition (January 3, )**Pages**: pages**Edition**: Fifth Edition

**Conclusion **

Any of the above books are equipped to provide you with a plethora of knowledge about linear algebra in the simplest terms possible, so we’re sure that whichever one you go for, it will prove a useful tool to change your understanding of linear algebra for the better. Good luck, and happy learning!

**Related**

Filed Under: Sciences

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A short summary of this paper. It's highly recommended by everyone but the lectures seem to be a bit all over the place. Report Save. Gerry Strange. Does anyone have links to good lectures of "upper-level" courses in algebra, such as group theory and such? He received the S.B. Download Full PDF Package. Wellesley-Cambridge Press Book Order from Wellesley-Cambridge Press Book Order for SIAM members Book Order from American Mathematical Society Can somebody tell me, is there much difference between the two editions, 4th and 5th editions ? was it worth it?I am 4 videos in and finding it a bit scattered. I kind of agree. It is possible to learn calculus by watching Professor Leonard's videos without prior knowledge, but i dont think its feasible to do so with Strang's lectures. There's also a condensed version of it on iTunes U. Super excited! Strang, Gilbert. Download for print-disabled Nonlinear partial differential equations in applied science by Peter D. Lax, Gilbert Strang. The students gave the lecturer a standing ovation at the end. He came to MIT upon completing the Ph.D. from UCLA in under the direction of Peter Henrici. [Book] Differential Equations and Linear Algebra () - GILBERT STRANG. Linear Algebra for Everyone Gilbert Strang. share. save hide report. Initially I was hoping to get a low priced copy of the 4th edition. MIT Room , Department of Mathematics Cambridge MA First Review of the Book; Table of Contents; Preface ; For orders and requests, email [email protected] Gilbert Strang’s most popular book is Linear Algebra and Its Applications. “He is a favorite; there is no way around it,” says OCW Director Curt Newton. I love his spontaneous aha moments that just take you along for the ride. Report Save. Matrix multiplication always looked very artificial to me, this guy helped me develop an intuition about matrices. My first exposure to linear algebra was from the Linear Algebra and its Application textbook. Gilbert Strang's mit OCW course is the best Linear algebra course I have taken. I'm no Strang (who is? It is this geometric approach that I find best about his material. 4. Im trying to buy the "Introduction to Linear Algebra" book by Prof Gilbert Strang to study for Machine Learning. I particularly enjoy his contributions to Finite Element Analysis. This year it is his teaching that has received unusual attention. 55 short videos have been created to present the main ideas for differential equations in an active way. 10 years ago. Definitely one of the lecturers I've ever heard. The MIT website for the lectures state multi-variable calculus as a prerequisite. Khan academy's Linear algebra is great you should try that if you are just getting started with linear algebra. from MIT in , and the B.A. I'm about 5 lectures in and I'm finding it pretty hard to follow. EMBED EMBED (for Gilbert Strang. Share. I was reviewing Linear algebra from his course but it completely changed my outlook on Linear algebra. Gilbert Strang (S.B. I don't know why it's not on iTunes U. I just started a group theory course at University of Reddit. His Ph.D. was from UCLA and since then he has taught at MIT. share. You jelly. During his year career at MIT, Professor Gilbert Strang (S.B. Have already taken Linear Algebra but your post intrigued me! If you run into Gilbert Strang in a dark alley, you will know linear algebra, bitch. Posted this (even earlier) on Reddit ML it didn't get as much interest. Trying to learn again. Linear Algebra OCW or something like that. Press question mark to learn the rest of the keyboard shortcuts. level 1. This article is in list format, but may read better as prose. If it’s easy to understand it means it’s probably too easy. Gilbert Strang. 2 years ago. level 1. I recommend it to anyone who has no background in linear algebra and is going to be taking a theory-heavy course. I guess working on a few problems from the text after each chapter/lecture would give you a better grounding of the concepts. Accurate intuition and proper explanation of algorithms were discussed. I love his enthusiasm when he teaches. No notes. is much more better, IMO. Continue this thread level 1. Wellesley-Cambridge Press Book Order from Wellesley-Cambridge Press Book Order for … But not to those at OCW. ) of the Department of Mathematics has become one of the most recognized mathematicians in the world, thanks to his expertise in both teaching and writing. The series was severe overkill for the course material (it stops at eigenvalues and eigenvectors), but I'm still glad I watched, for I ended up being part math major. New comments cannot be posted and votes cannot be cast, Press J to jump to the feed. (January ) William Gilbert Strang (born November 27, ), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. I have only done SC but I will look into looks interesting, New comments cannot be posted and votes cannot be cast, Press J to jump to the feed. Download [Strang G.] Linear algebra and its … 5th ed. Thanks for sharing! 44 to rent $ to buy. Gilbert Strang: Linear Algebra lectures [MIT] (a wonder during my student years) Close. Get it as soon as Mon, May best. ISBN September Wellesley-Cambridge Press [email protected] I much prefer READ PAPER [Strang G.] Linear algebra and its applications(4)[].PDF . Yeah, just noticed. Sort by. I have not seen Professor Leonard’s videos. You may find the lectures more exciting when you watch them at x or 2x the normal speed (keeping the pitch of your voice constant). Dr. Strang is such a wonderful man. Share. However, this book is still the best reference for more information on the topics covered in each lecture. Learn Differential Equations: Up Close with Gilbert Strang … Gilbert Strang and Cleve Moler provide an overview to their in-depth video series about differential equations and the MATLAB® ODE suite. Totally gonna watch it. He has made many contributions to mathematics education, including publishing seven mathematics textbooks and one monograph. Gilbert Strang is the MathWorks Professor of Mathematics as of , the first holder of this faculty chair. William Gilbert Strang (born November 27, ), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. 2 years ago. Share . [read more bio] Curriculum Vitae: Curriculum Vitae. You can help us understand how dblp is used and perceived by answering our user survey (taking 10 … I was supposed to take linear algebra this semester, but coronavirus made it so that I didn't really learn much. Press question mark to learn the rest of the keyboard shortcuts. Share to Reddit. I'm about halfway through the course and really have been enjoying it. Report Save. Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. Calc II is the bane of every students' existence at my school, partly because it has a linear algebra component. Matrix Methods in Data Analysis, Signal Processing, and Machine Learning “That was a surprise to me,” says Strang. That’s the kind of math that makes Professor Gilbert Strang one of the most recognized mathematicians in the world. I saw the lectures alongside as well. how far did you go? I'm about to start it! Share via email. Hardcover. Share. I just saw some of the MIT courses online(by Gilbert Strang). These video lectures of Professor Gilbert Strang teaching were recorded in Fall and do not correspond precisely to the current edition of the textbook. Share to Pinterest. I really love that all of his lectures are basically improve. $ $ Strang provides the intuition and geometric viewpoint to make a book like Axler or Hoffman and Kunze exceedingly clear. Other books by Gilbert Strang Dummy. Video lectures for teaching mathematics by Gilbert Strang at MIT. From Professor Gilbert Strang, acclaimed author of Introduction to Linear Algebra, comes Linear Algebra and Learning from Data, the first textbook that teaches linear algebra together with deep learning and neural nets. Unpopular opinion but i dont like Gilbert Strang's lecture videos too much. Gilbert Strang Wellesley-Cambridge Press (USA) Cambridge University Press (UK) Book Order Form. Can someone explain why Strang is so highly recommended here? Differential Equations and Linear Algebra (Gilbert Strang) by Gilbert Strang | Feb 12, It's a great transitional set of lectures to heavily theoretical linear algebra. Email: [email protected] Glibert Strang’s Linear Algebra course at MIT OpenCourseWare is exquisite! Matrix multiplication always looked very artificial to me, this guy helped me develop an intuition about matrices. The moment you go beyond 3 dimensions (or are even asked to think of mappings from one 3-d space to another), your intuition starts to fail you. I felt that in the pace was kind of slow and the lectures were unnecessarily verbose. level 2. Wellesley-Cambridge Press, This paper. His Ph.D. was from UCLA and since then he has taught at MIT. Only 1 left in stock - order soon. Gilbert Strang can explain more in a single paragraph than other authors can in an entire chapter. Download PDF. I am a hobbyist you can say, really sucked at math during my grads. I would like to continue on my math knowledge. % Upvoted . His lectures are fun he teaches based on examples and then shows the need for the abstract stuff. and M.A. Woke up the next day, no money, no pants, 4 cracked ribs, and an A in linear algebra. Share to Tumblr. [Strang G.] Linear algebra and its applications(4)[].PDF. Took this class in the late 70's. Some of the course highlights are: - Matrix focus (that is, more concrete and less abstract) - An interesting take on “the fundamental theorem of linear algebra” Great instruction. Gerry Strange. These lectures follow the same structure as his textbook ‘Linear Algebra and its Applications’. Lecture videos from Gilbert Strang's course on Linear Algebra at MIT. GILBERT STRANG Professor of Mathematics - Massachusetts Institute of Technology. Biography: Gilbert Strang was an undergraduate at MIT and a Rhodes Scholar at Balliol College, Oxford. But in general, a first course in calculus tends to be much lower in abstraction that a first full-fledged course in linear algebra. Needs Digitizing. Loved the course and would love to attend one of his clases in real life. The only better MIT lecture series (videos) are those by Walter Lewin. More Buying Choices $ (34 used & new offers) Introduction To Linear Algebra, 5Th Edn . ), but it should be helpful. You should have to actively work to read a math book! 15 Full PDFs related to this paper. Himanshu Ahuja. And an excellent textbook, too. Which one or ? Posted by 8 years ago. out of 5 stars We use cookies on our websites for a number of purposes, including analytics and performance, functionality and advertising. I've had profs who have tried doing that and it went terribly. I had a tiny bit of linear algebra knowledge beforehand (the 3b1b series) and I'd imagine being completely lost coming in with no prior knowledge at all. This thread is archived. Archived. dsiegel on May 17, by Gilbert Strang | Jan 1, I ran into the guy down in the 9th ward in New Orleans once. It's a true sign of his insane intelligence. by Gilbert Strang. I spent the summer prior to entering watching this series, and passed the placement exam with a 97% (-3% for a misplaced minus sign). It already has most of its answers at the back if you need any further help you can watch lecture videos online on youtube or mit ocw where gilbert strang himself teaches in the c. Linear algebra and its applications 4th edition solutions manual strang pdf change your habit to hang or waste the time to only chat with your friends. I hate when his examples have errors that don't get corrected until the next lecture. I have a signed copy of his Linear Algebra textbook made out to "Ben." Gilbert Strang: Linear Algebra lectures [MIT] (a wonder during my student years) ocw.mit.edu/course 18 comments. Publication date Usage Attribution-Noncommercial-No Derivative Works Topics Calculus Publisher Wellesley-Cambridge Press Collection opensource Language English. ) Credits: Photo / Donna Coveney Previous image Next image. I feel like he's not on par with Professor Leonard. Strang is an excellent lecturer - his videos for (Linear Algebra) were instrumental for me in relearning the subject matter more than twenty years after I last studied it. I was reviewing Linear algebra from his course but it completely changed my outlook on Linear algebra. He joined the MIT mathematics faculty in , professor in Yes: google abstract algebra Harvard extension course. FREE Shipping by Amazon. save hide report. The only way to train your intuition for this N-dimensional thinking is by doing a lot of problems, which Strang’s aforementioned textbook offers and even emphasises. Introduction to Linear Algebra, Fifth Edition () by Gilbert Strang ([email protected]) ISBN : Table of Contents and Preface; New ideas in Linear Algebra for Everyone; Section Section ; Image compression by the SVD, Demonstration created by Tim Baumann ; A Vision of Linear Algebra (videos) Solution … 1. Many people watch the lecture videos on YouTube: Lectures by Gil Strang: MIT (Spring ) on YouTube - scroll to bottom of this page for overview of videos by topic. But I see that there is a new edition available (5th Edition). as a Rhodes Scholar from Oxford University in New comments cannot be posted and votes cannot be cast. In his material Strang tries to show you how things work in 2 and 3 dimensions geometrically, and then how those principles can be extrapolated to higher dimensions. Definitely one of the lecturers I've ever heard. ISBN (13): ISBN (10): Publisher: Wellesley-Cambridge Press (12 February ) 1 comment. ISBN: Dr. Gilbert Strang's course is super fun. Introduction to Linear Algebra. Himanshu Ahuja. But at least some introductory familiarity with calculus and matrices is required I suppose. Thank you! Linear Algebra and Learning from Data () by Gilbert Strang ([email protected]) ISBN : First published in 1 edition — 1 previewable Not in Library. 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## How to select the right books? [closed]

There is no one answer to your question. It depends where you are in your mathematical journey. Take linear algebra for example. Perhaps one should begin with: (this may be too silly for some of us, not me)

The Manga Guide to Linear Algebra

Easy reading, fun, maybe good for highschool. A bit later, say after you've had some calculus at the university, you might look at a book like that by Anton, Strang, Larson or Lay. The list of these standard university texts for the Sophomore/Junior level linear course is endless. Almost certainly you should buy an edition from circa as those have a bit more content and a bit less pandering to the slothful student. Two nice examples, recently published by Dover,

A Course in Linear Algebra by David B. Damiano, John B. Little

or, at a bit higher level,

Linear Algebra by Sterling K. Berberian

But, at some point, you realize you need more, then you start reading things like Halmos, Curtis, Insel-Spence-Friedberg. Yet, still, something is missing, so, you move on to the relevant chapters in Dummit and Foote or Roman's Advanced Linear Algebra. But, even then, you will still have questions. This is the great part of it all, it's endless.

Back to your original question: I don't think there is an efficient answer. In some sense, knowing the answer to "what is the right book on X" amounts to knowing what you know and what you don't know about X. But, how can you know that unless you know X? That said, you can get approximate knowledge by reading the reviews on various websites and seeing the recommendations made in answers such as

what are the best books on linear algebra

## Books reddit linear algebra best

## Neil Sainsbury

For over six years now, I've been studying mathematics on my own in my spare time - working my way through books, exercises, and online courses. In this post I'll share what books and resources I've worked through and recommend and also tips for anyone who wants to go on a similar adventure.

Self-studying mathematics is hard - it's an emotional journey as much as an intellectual one and it's the kind of journey I imagine many people start but then drop off after a few months. So I also share (at the end) the practices and mindset that have for me allowed this hobby to continue through the inevitable ups and downs of life (raising two young boys, working at a startup, and moving states!)

### How it all began for me

I used to love mathematics. Though I ended up getting an engineering degree and my career is in software development, I had initially wanted to study maths at university. But the reality is, that's a very tough road to take in life - the academic world is, generally speaking, a quite tortuous path with low pay, long hours, and rife with burnout. So I took the more pragmatic path and as the years went by never really found the time to reconnect with math. That was until about six years ago when I came across Robert Ghrist's online course Calculus: Single Variable (at the time I took it, it was just a Coursera course but now it's freely available on YouTube). Roughly 12 weeks and many filled notebooks later, I had reignited my interest in math and felt energized and excited.

Robert, if you read this: thanks for being such an inspiring teacher.

### Why learn mathematics?

Growing up I always loved puzzles and problem solving. I would spend hours working my way through puzzle books, solving riddles, and generally latching on to anything that gives you that little dopamine hit.

If you're similar, mathematics might just be for you. Mathematics is hard. Seriously hard. And then suddenly, what was hard is easy, *trivial*, and you continue your ascent on to the next hard problem. It deeply rewards patience, persistence, and creativity and is a highly engaging activity - it's just you quietly working away, breaking down seemingly impossible problems and making them possible. I can't say enough how deeply satisfying and personally enriching it is to make the impossible, possible through your own hard work and ingenuity.

One thing many people don't know as well is that the mathematics you learn at most high schools is actually quite different from what you're exposed to at the university level. The focus turns from being about rote computation to logic, deduction, and reasoning. A great quote I read once is that for most of us, when we learn mathematics at school, we learn how to play a couple of notes on a piano. But at university, we learn how to write and play music.

### Picking the right books and courses for self-learning

As a self-learner, it's critical to pick books with exercises and solutions. At some point later on you can swap to books without exercises and/or solutions, but in the beginning you need that feedback to be able to learn from your mistakes and move forward when you're stuck.

The books you pick as a self-learner are also sometimes different from what you would work use if you were engaged in full-time study at a university. Personally, I lean more towards books with better exposition, motivation, and examples. In a university setting, lecturers can provide that exposition and complement missing parts of books they assign for courses, but when you're on your own those missing bits can be critical to understanding.

I recommend avoiding the Kindle copies of most books and always opting for print. Very few math books have converted to digital formats well and so typically contain many formatting and display errors. Incidentally, this is often the main source for bad reviews of some excellent books on Amazon.

I'd be remiss as well if I didn't mention the publisher **Dover**. Dover is a well known publisher in the math community, often publishing older books at fantastically low prices. Some of the Dover books are absolutely brilliant classics - I own many and have made sure to make note of them in my recommendations below. If you don't have a big budget for learning, go for the Dover books first.

In several places I also recommend courses from MIT OpenCourseware. These courses are completely free and often have full recorded video lectures, exam papers with solutions, etc. If you like learning by video instruction and find at various points that you're getting a bit lost in a book, try looking up an appropriate course on MIT OpenCourseware and seeing if that helps get you unstuck.

Pretty much all my books I recommend below focus on undergraduate level math, with an emphasis on pure vs applied. That's just because that's the level that I'm at and also the kind of maths I like the most!

And also, just a final note that the order of books I recommend below is not exactly the order I worked through them - rather, it's the order I think they should be worked through. Sometimes I picked up a book that was too hard and had to double back and wait until I was ready. And some books have only just come out recently as well (eg. Ivan Savov's "No BS" books) so weren't available to me when I was at that stage of learning. In short, you get to benefit from my hindsight and missteps along the way.

Alright, let's jump in to the recommendations!

### Foundations

I'm going to assume a high-school level of maths is where you last left-off and that it's been some time since then that you've last done any maths. To get going, there's a couple of books I recommend:

**The Art of Problem Solving, Vol 1 & 2 (with solutions manuals)** - Lehoczky and Rusczyk

*The Art of Problem Solving* books are wonderful starter books. They're oriented heavily towards exercises and problem solving and are fantastic books to get you off to a start actually *doing* maths and also doing it in a way that's not just repetitive and boring. Depending on your level of mathematical maturity, you may only want to work through volume 1 and come back to volume 2 after you've worked through a proofs book first though (the second volume has many more questions involving writing proofs which you may not yet be comfortable enough to do at this stage). Volume 2 has many excellent exercises though, so don't skip it!

**No bullshit guide to math and physics** - Savov

If your calculus is a little rusty or you never really understood it in high-school, I recommend working through this book. It's compact, free of long-winded explanations, and contains lots of exercises (with solutions). This book teaches calculus in a contextually motivated way by teaching it alongside mechanics, which is how I think calculus should always be taught initially (I almost recommended Kline's Calculus: An Intuitive and Physical Approach here instead, as a book I also very much like, but Kline's book is just so thick and verbose. If you do like that additional exposition, you may want to consider this book as an alternative).

Also, of course I must mention the course that started it all for me, Calculus: Single Variable. It appears Coursera has now broken the course up into several parts and as I mentioned you can also find the full lessons on YouTube. Work through either this or Savov's book - depending on whether you prefer learning from books or online courses.

### A historical (and motivated) perspective

I think it's useful early on in the learning journey to have a broad map of where math has been, what has motivated its development to date, and also where it's going.

**Mathematics for the Nonmathematician (Dover)** - Kline

For a historical view, I highly recommend reading through Kline's *Mathematics for the Nonmathematician*. It contains a small handful of exercises, but they're not the main focus - this is one of the few math books I recommend that you can just leisurely read.

**Concepts of Modern Mathematics (Dover)** - Stewart

While Kline provides the historical perspective, Stewart will provide you with the modern perspective. This is one of the first math books I read that genuinely made me excited and deeply want to understand topology - up until then, I was only somewhat dimly aware of the subject and thought it was a bit silly. Like the Kline book, this book also has no exercises - but for me it was a springboard and motivator to open other related books and dig in and do some hands-on math.

**Mathematics and Its History** - Stillwell

I consider *Mathematics and Its History* to be somewhat optional at this point, but I want to mention it because it's so darn good. If you read through Kline's and Stewart's books and thought "You know what, these ideas are really nice but I'd love to go more hands-on with them with some exercises" then this book is for you. Want to try to do some gentle introductory exercises from fields like noneuclidean geometry, group theory, and topology, not just idly read about them? This book might be for you.

**BBC's A Brief History of Mathematics (Podcast)** - du Sautoy

If you prefer listening over reading, I recommend listening to the part podcast *A Brief History of Mathematics* that focuses on the interesting lives and personalities of some of the driving historical forces in mathematics (Galois, Gauss, Cantor, Ramanujan, etc.).

### Proofs and mathematical logic

For many, your first proof book is where everything clicks and you begin to understand that there is more to math than just calculation. For this reason, many people have very strong feelings about their favourite proofs book and there are indeed several that are quite good. But my favourite of all of them is:

**An Introduction to Mathematical Reasoning** - Eccles

I think what I love most about *An Introduction to Mathematical Reasoning* is how it successfully pairs explanation with exercises, which is a recurring theme in books that I tend to gravitate to. Good exercises are an extension of the teaching journey - they tell their own story and have progression and meaning. And at the time I worked through this book, the difficulty was *just* right. A good chunk of the book is occupied with applying the proof techniques you learn to different domains like set theory, combinatorics, and number theory, which is also something that personally resonated with me.

**Book of Proof** - Hammack

*Book of Proof* is a nice little proofs book. It's not too long and has a good number of exercises. If you're looking for a gentler introduction to proofs this is the one to go for. For the edition I used, it contained solutions for every second problem with full solutions available on the author's personal website, which I believe is still the case today.

### Calculus/Real Analysis

**Calculus** - Spivak

Spivak's *Calculus* is the among the best maths book I have ever worked through but don't be fooled by the name - this is an introductory book to real analysis and is very different from the Calculus books mentioned earlier which emphasize computation. The emphasis for this book is on building up the foundations step by step for single variable calculus (starting from the construction of real numbers). It is a wonderfully coherent and realized book and what's also great about it is once again the exercises complement and expand on the content so well. Speaking of the exercises, some are **seriously hard**. This book took me about 6 months to work through because at the time I was still committed to solving every single exercise on my own. I almost burned out and I discuss what I learned from that experience coming up.

Everybody should own this book.

**Calculus** - Apostol

Spivak's book can be genuinely too hard for some people at this stage. For that reason, there are two other books I'm happy to recommend as alternatives.

The first is Apostol's *Calculus*. Apostol proceeds at a more leisurely pace compared to Spivak, and is happy to spend time building up your intuition with examples and geometric arguments before diving in to more rigorous proofs. Interestingly enough, the book also goes a fairly long way in introducing linear algebra in the final few chapters. I do like as well that this book introduces integration before differentiation, which you'll know if you read through Kline's historical book, is more historically accurate.

One thing to note is this book can be quite hard to find and unfortunately the edition I have has some weird print quality issues. Your mileage may vary.

**Introduction to Analysis** - Mattuck

The second introductory analysis book I'm happy to recommend is Mattuck's *Introduction to Analysis*. As with Spivak, this book mostly focuses on real-valued functions of a single variable and only in the last few chapters goes beyond. Mattuck makes clear in the introduction that this book was written for those struggling with analysis and less confident with proof techniques and I think it succeeds really well in what it set out to do. If you're struggling with Spivak, pick this book up.

### Linear Algebra

For Linear Algebra, I can suggest two different learning paths. If you prefer to work through books only, I recommend working through Savov and Axler for both the applied and pure views of linear algebra. However, if you're comfortable with video instruction, I really like Gilbert Strang's Linear Algebra material (both the book and the online course).

Starting with Savov and Axler:

**No bullshit guide to linear algebra** - Savov

*No bullshit guide to linear algebra* is a book that has only come out quite recently, and it's a book I read only after already making my way through more advanced texts. That necessarily changes your perspective, but nonetheless I still think this is an excellent book for a first introduction to linear algebra. One quality of the book I really appreciated was the healthy coverage of applications in the final few chapters - looking at the application of linear algebra to problems in cryptography, fourier analysis, probability, etc.

**Linear Algebra Done Right** - Axler

*Linear Algebra Done Right* is a quite well known book, famous for its non-standard treatment of determinants, only really introducing them towards the end of the book. Incidentally, this is what the "Done Right" in the title refers to: Axler has a bone to pick with determinants and doesn't hide it. It's a pure-math proofs focused book with nothing in the way of applications, which is also why I suggest pairing this book with Savov's which is more applied/computational. The proofs in the book are excellent and are a model for clarity and simplicity - working through this book really helped me build up a strong foundational intuition for linear algebra. The book unfortunately does not come with solutions to the exercises, but many can be found online.

**Introduction to Linear Algebra** and **MIT OCW Linear Algebra** - Strang

Here I treat both the book and the course as one, because they complement each other so well. What that does mean however is that you'll often find yourself jumping back and forwards between the book and the videos - for some people that's going to be a plus and enhance the learning experience (in many ways, it's much like it would be if you attended MIT in person) while for others that's a big detracting point. I'll leave it for you to decide which is best for you. In terms of the material, the focus is mostly on the applied side of things but still managing to touch on the theory in places. The exercises are quite good, plus you also get access to the MIT exams (with solutions).

**Coding the Matrix: Linear Algebra through Applications to Computer Science** - Klein

If you're a software developer like me, I wholeheartedly recommend that you pick up *Coding the Matrix*. I originally learned about this book through a course on Coursera but never took the online course and instead went straight for the book. The emphasis of the book is on programming (with Python), applying techniques from linear algebra to applied problems such as compression, image manipulation, machine learning, etc. It's the kind of book that I think works best if you've already had some introduction to linear algebra by working through any of the above above recommended books, giving your theoretical foundations a really strong applied grounding.

### A brief interlude to talk about study, exercises, and taming the completionist inside you

Ok, let me take a moment to share some hard-won advice. When it comes to solving exercises as you work through these books, the completionist mindset will destroy you. When I first set out on my adventure, I would refuse to move forward in a book until I had solved **every problem**. This worked up to a pointand that point was when I met Spivak and had to finally concede defeat.

It's hard to put a formal rule on how long you should spend on any given exercise before "giving up" and looking at the solution. I would say a lower bound should be maybe 20 minutes or so. But the upper bound? I don't know. For some problems, it can be fruitful and productive to spend days (weeks even?) thinking about them provided you're making some kind of incremental progress or still have tricks in your bag you want to apply.

For me personally, I spend around 10 hours most weeks doing math and I've had brief bursts where I've done as much as 30hrs/week when I'm between contracts, etc. What that does mean though is that one hard exercise can completely block your progress for weeks if you let it, and I'll say right now, when that does happen, it can be quite demoralizing.

Ultimately, my advice is to let your intuition and energy levels guide you - do you have the energy to chase this really hard problem down or are you in the mood for just learning the answer now? If you just want the answer now so you can move forward, look it up! There's no penalties here. You didn't lose.

I know it feels unnatural to not set hard rules around how long you should spend on a given problem, but remember that ultimately you're optimizing for your enjoyment (that's why you're doing this, right?) and long term consistency.

You will do yourself a disservice (in maths, and in life) if you burn out hot and fast.

### Multivariable Calculus

**MIT OCW Multivariable Calculus**

The *MIT OCW Multivariable Calculus* course is the best resource I've found for getting comfortable with multivariable calculus. I don't really know of any good books covering this subject (no, please don't say Stewart) and I found the MIT course to be really enjoyable. Good collection of problems and worked solutions, clear material and lectures. And if I'm honest, as a self-learner who went to a good but not great university, there is also something quite satisfying about "sitting" an exam from MIT in the same conditions as an MIT student would experience and completely acing the exam.

### Differential Equations

**Ordinary Differential Equations (Dover)** - Tenenbaum and Pollard

ODEs get a bit of a bad rap for being a quite boring subject, and I think many people view the process of learning about differential equations as being about internalizing an almost random collection of "tricks" to solve certain ODE forms. But it doesn't have to be that way, and I learned that from working through this book. While this book leans heavily into the applied/computational side of mathematics, it also does a great job complementing the examples and exercises with theory. It does feel a bit dated at times (it was written in ) and could probably benefit in certain places with some visual aids, but if you're comfortable with Python and plotting/graphical libraries you can make it up yourself as you go along. Overall, the book moves along at a fairly gentle pace - you're unlikely to get stuck as you work through the book as long as you're diligent, all the exercises have solutions, and being a Dover book, it's really cheap!

### Analysis

**Principles of Mathematical Analysis** - Rudin

Rudin's *Principles of Mathematical Analysis*, also affectionately known as *Baby Rudin*, is a difficult, serious book but if you made your way through Spivak and all the books I've shared so far, you've got all the tools you need to succeed to make it through this. Occasionally you'll see this recommended as the first book people should learn analysis from, but I think that's a big mistake for us self-learners: it's dry, contains little in the way of motivation/exposition and has quite a few "fill in the blanks" moments.

Note that although this book does not have any official solutions, there is a reddit community that has been working through the book, crowd-sourcing and documenting solutions. The community is at https://www.reddit.com/r/babyrudin/ and the crowd-sourced solutions document can be found here.

### Algebra

**A Book of Abstract Algebra (Dover)** - Pinter

I'm really excited to recommend Pinter's *A Book of Abstract Algebra* and the truth is if the subject of Abstract Algebra is of real interest to you, this is the kind of book you can pick up and work through very early on - as early as after finishing your first book of proofs (and if you're clever, even before that). The chapters are quite short, with each chapter accompanied by a pretty hefty set of exercises which expand on the content. The book covers all the usual suspects: groups, rings, etc. and ultimately builds its way up to Galois Theory. One important thing to note is the book only contains solutions to selected exercises but the exercises are not too hard, so this shouldn't be too much of a problem.

One funny sidenote: I've heard it said on many occasions that mathematicians typically fall into one of two camps: those who like algebra (algebraists), and those who like analysis (analysts). It's even been observed that analysts tend to eat corn in spirals, while algebraists in rows. After now getting your first taste of algebra, which camp do you think you're in and does the corn theory hold true?

**Abstract Algebra** - Dummit and Foote

While Pinter's *A Book of Abstract Algebra* is excellent, it only covers a small sliver of topics from the field of abstract algebra. By contrast, *Abstract Algebra* is massive, and covers a lot of ground. You could almost affectionately call it a reference, but at the same time it also has a lot of those same qualities that make it excellent for self-study - lots of examples and exercises, good motivation and descriptions, etc. Just a note though that it does not come with solutions to the exercises, but given its popularity, it's quite easy to find them online.

To reiterate, this is a big book and I haven't myself worked my way through it entirely, but rather just picked it up now and then and absorbed small bits and pieces at a timein many ways treating it like a reference instead of something to work through from start to finish.

If you can, buy the hardcover. Books this big don't fare as well as paperback.

### Topology

**Topology** - Munkres

Focusing on point set Topology, Munkres' *Topology* is widely considered to be one of the best introductory books in the field. Topology is a quite difficult subject to grasp and it took me several attempts to get going - I admit to feeling much like Hitler in Hitler learns topology for a very long time until finally things started to click. Part 2 of the book contains a nice introduction to algebraic topology but I'll admit now at the time I got to this material I was already struggling a bit and have never revisited the material.

**Introduction to Topology (Dover)** - Mendelson

This was the book that really helped me get Topology. If you're finding Munkres a bit difficult to get going, this book is both cheaper and has a bit more exposition - there's a lot more hand-holding and the book takes its time to help you build up the intuition for why things work. In my opinion, this is the better book of the two for self-learners.

### Number Theory

**Elementary Number Theory** - Jones

*Elementary Number Theory* is a great book for anyone who wants to jump in to number theory first and may have skipped some of the previously mentioned books on algebra, as it assumes very little prior knowledge. This was certainly the case for me, where I worked through this book shortly after my first proofs book. In terms of difficulty, it was perfect for me at the time, but on reflection now it might be too easy for some. If that's the case, take a look at An Introduction to the Theory of Numbers which I think would be the next best step up in difficulty and maturity.

**Number Theory (Dover)** - Andrews

George Andrews is one of the leading experts in the field of partitions and this book provides an excellent first introduction to the field along with a good selection of other topics (with a combinatorial focus). It's quite a short book, but manages to pack a lot in and has a great selection of exercises (with solutions to selected exercises). If you've ever heard the "Rogers-Ramanujan identities" and wanted to learn more, this book is the perfect study companion.

### Probability

**Introduction to Probability, Statistics, and Random Processes** - Pishro-Nik

While focusing more on the applied side of probability and statistics (this book doesn't go anywhere near measure theory), this book ended up being a really pleasant surprise for me in a field that I wasn't particularly interested in and have always found a bit dry. It also doesn't really assume much in the way of prior knowledge except for a little multivariable calculus and linear algebra, making it very approachable. I think engineers working in the fields of machine learning and data science in particular would really benefit from working through this book.

### Other

What follows is a collection of books and resources I also highly recommend, but somewhat off the beaten path and chosen by me more for personal interest.

**What Is Mathematics?** - Courant and Robbins

*What Is Mathematics* is a classic and it's one that I thoroughly enjoyed readingeventually. Why eventually? While it's claimed to be an "elementary" book for the "beginner", in truth, if you gave this book to any but the most gifted high-school graduate it would absolutely crush them within the first ~20 pages. And such was the case with me, where I picked this book up and tried to work through it very early on in my learning journey. Unless you're brilliant (and I surely am not), I really would recommend only tackling this book after working through a proofs book and some of the foundation books I recommended earlier. The book itself is quite broad in the subject matter it covers - number theory, geometric constructions, projective geometry, topology, and finally on to calculus. It's actually a really playful and fun book and manages really well to avoid getting bogged down too much in any one area. I highly recommend itjust make sure you don't start it before you're ready.

**Naive Set Theory (Dover)** - Halmos

Naive Set Theory gets right to the heart of the set-theoretic underpinnings of mathematics and does it extremely well. It is an extremely readable book, and in my opinion Halmos was one of the best mathematical expositors ever (interesting aside: Halmos invented the "iff" notation for that wonderful phrase "if and only if" and also the ∎ notation to signify the end of a proof!). It contains only a small handful of exercises. This book was my first exposure to proper set theory and I really enjoyed it.

**The Cauchy-Schwarz Master Class** - Steele

You can almost think of *The Cauchy-Schwarz Master Class* as an exercises/problem solving book for inequalitiesas I worked through it I couldn't help but think this book would basically be the perfect bootcamp for dealing with tricky inequalities in a competitive math setting. It's got that quality of books like Polya's *How to Solve It* and others in that its always prodding you to think creatively and *work the problems*. And working through it, you also come to appreciate just how damn versatile the Cauchy-Schwarz inequality is!

**Prime Numbers and the Riemann Hypothesis** - Mazur and Stein

I'm currently reading this book right now!

### Connecting with the broader math community

At least for me, studying mathematics has been a fairly isolated experience and I've never really found a good community to participate in. There are a couple of communities I'm aware of though, and of these, /r/math and the AoPS Community seem to be the most active. There's also the Math StackExchange, which is very active, but I find it hard to classify that as a community.

I also really like Grant Sanderson's 3Blue1Brown YouTube Channel - he has a lot of excellent general interest math videos with just the right amount of rigor.

Incidentally, if you're interested in talking math with someone, I'd love to hear from you and probably the best place to connect with me is on Twitter: @neilwithdata

### What's next

There's a couple of books I've heard great things about that I'm really interested in reading next. They are (in no specific order):

Mathematics: Its Content, Methods and Meaning - I actually own the Kindle edition of this but as with many math books, it has not been converted to digital format well. I'm really looking forward to picking up the print copy and having a go at this.

Visual Complex Analysis - Some chapters in the TOC look quite familiar to me, and others I have no idea aboutwhich is quite exciting!

Introductory Functional Analysis with Applications - I've heard this is one of the best introductory books for functional analysis.

### Some parting advice

I'm currently 35 years old, and while studying mathematics have had two wonderful kids and co-founded a startup (and I've also just started a new small business building Slack and Microsoft Teams apps and integrations). Suffice to say, life has been busy. So I wanted to offer some more general advice for anyone who also wants to study math while also staying sane. Here is what has helped me:

Exercise. It turns out that you're not just a brain in a tank and the mind-body connection has an immense impact on your daily happiness and well-being. Exercise every day.

Take regular breaks and go for walks. Seemingly intractable problems have a surprising way becoming tractable after a long walk. Sunshine also has a wonderful effect on improving your overall mood and helping you sleep better.

Alternate between easier and harder content. One way you're sure to burn out is if you are always pushing, without periods of rest. Make sure you follow up hard books with easier, almost routine, books. Sometimes it can even be beneficial to work on two books at once - one easy, one hard. Make sure to pepper in little easy wins everywhere in between the big meaty hard problems. Always follow up "failure" (ie. this problem is too hard, I give up) with "success".

Spend time with friends and family. Self-studying maths can be a bit isolating, but ultimately we're wired to be social creatures. Don't neglect your friends and family. Invest in good relationships with people you love to be around.

## Reddit reviews: 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.

- 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.
- 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.
- 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.
- 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.
- 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.
- Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
- 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. - Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
- 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.
- 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.
- Khan Academy: Has a nice beginning LinAlg course.
- 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.
- 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. - 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.
- Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
- 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.
- 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. - Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
- Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
- 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. - 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.
- Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
- 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.

I hope I am not too late.

You can post this to /r/suicidewatch.

Here is my half-baked attempt at providing you with some answers.

First of all let's see, what is the problem? Money and women. This sounds rather stereotypical but it became a stereotype because a lot of people had this kind of problems. So if you are bad at money and at women, join the club, everybody sucks at this.

Now, there are a few strategies of coping with this. I can tell you what worked for me and perhaps that will help you too.

I guess if there is only one thing that I would change in your attitude that would improve anything is learning the fact that "there is more where that came from". This is really important in girl problems and in money problems.

When you are speaking with a girl, I noticed that early on, men tend to start being very submissive and immature in a way. They start to offer her all the decision power because they are afraid not to lose her. This is a somehow normal response but it affects the relationship negatively. She sees you as lacking power and confidence and she shall grow cold. So here lies the strange balance between good and bad: you have to be powerful but also warm and magnanimous. You can only do this by experimenting without fearing the results of your actions. Even if the worst comes to happen, and she breaks up with you you'll always get a better option. There are billion ladies on the planet. The statistics are skewed in your favor.

Now for the money issue. Again, there is more where that came from. The money, are a relatively recent invention. Our society is built upon them but we survived for 3 million years without them. The thing you need to learn is that your survival isn't directly related to money. You can always get food, shelter and a lot of other stuff for free. You won't live the good life, but you won't die. So why the anxiety then?

Question: It seems to me you are talking out of your ass. How do I put into practice this in order to get a girlfriend?

Answer: Talk to people. Male and female. Make the following your goals:

Talk to 1 girl each day for one month.

Meet a few friends each 3 days.

Make a new friend each two weeks.

Post your romantic encounters in /r/seduction.

This activities will add up after some time and you will have enough social skill to attract a female. You will understand what your female friend is thinking. Don't feel too bad if it doesn't work out.

Question: The above doesn't give a lot of practical advice on getting money. I want more of that. How do I get it?

Answer: To give you money people need to care about you. People only care about you when you care about them. This is why you need to do the following:

Start solving hard problems.

Start helping people.

Problems aren't only school problems. They refer to anything: start learning a new difficult subject (for example start learning physics or start playing an instrument or start writing a novel). Take up a really difficult project that is just above the verge of what you think you are able to do. Helping people is something more difficult and personal. You can work for charity, help your family members around the house and other similar.

Question: I don't understand. I have problems and you are asking me to work for charity, donate money? How can giving money solve anything?

Answer: If you don't give, how can you receive? Helping others is instilling a sense of purpose in a very strange way. You become superior to others by helping them in a dispassionate way.

Question: I feel like I am going to cry, you are making fun of me!

Answer: Not entirely untrue. But this is not the problem. The problem is that you are taking yourself too serious. We all are, and I have similar problems. The true mark of a person of genius is to laugh at himself. Cultivate your sense of humor in any manner you can.

Question: What does it matter then if I choose to kill myself?

Answer: There is this really good anecdote about Thales of Miletus (search wiki). He was preaching that there is no difference between life and death. His friends asked him: If there is no difference, why don't you kill yourself. At this, he instantly answered: I don't kill myself because there is no difference.

Question: Even if I would like to change and do the things you want me to do, human nature is faulty. It is certain that I would have relapses. How do I snap out of it?

Answer: There are five habits that you should instill that will keep bad emotions away. Either of this habits has its own benefits and drawbacks:

- Mental contemplation. This has various forms, but two are the best well know: prayer and meditation. At the beginning stage they are quite different, but later they begin to be the same. You will become aware that there are things greater than you are. This will take some of the pressure off of your shoulders.
- Physical exercise. Build up your physical strength and you will build up your mental strength.
- Meet with friends. If you don't have friends, find them.
- Work. This wil give you a sense of purpose. Help somebody else. This is what I am doing here. We are all together on this journey. Even though we can't be nice with everyone, we need to at least do our best in this direction.
- Entertainment. Read a book. Play a game. Watch a movie. Sometimes our brain needs a break. If not, it will take a break anyway and it will not be a pretty one. Without regular breaks, procrastination will occur.

Question: Your post seems somewhat interesting but more in an intriguing kind of way. I would like to know more.

Answer: There are a few good books on these subjects. I don't expect you to read all of them, but consider them at least.

For general mental change over I recommend this:

http://www.amazon.com/Learned-Optimism-Change-Your-Mind/dp//ref=sr_1_1?ie=UTF8&amp;qid=&amp;sr=

http://www.amazon.com/Generous-Man-Helping-Others-Sexiest/dp/

For girl issues I recommend the following book. This will open up a whole bag of worms and you will have an entire literature to pick from. This is not going to be easy. Remember though, difficult is good for you.

http://www.amazon.com/GAME-UNDERCOVER-SOCIETY-PICK-UP-ARTISTS/dp//ref=sr_1_1?ie=UTF8&amp;qid=&amp;sr= (lately it is popular to dish this book for a number of reasons. Read it and decide for yourself. There is a lot of truth in it)

Regarding money problem, the first thing is to learn to solve problems. The following is the best in my opinion

http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/X

The second thing about money is to understand why our culture seems wrong and you don't seem to have enough. This will make you a bit more comfortable when you don't have money.

http://www.amazon.com/Story-B-Daniel-Quinn/dp//ref=sr_1_3?ie=UTF8&amp;qid=&amp;sr= (this one has a prequel called Ishmael. which people usually like better. This one is more to my liking.)

For mental contemplation there are two recommendations:

http://www.urbandharma.org/udharma4/mpe.html . This one is for meditation purposes.

http://www.amazon.com/Way-Pilgrim-Continues-His/dp/ . This one is if you want to learn how to pray. I am an orthodox Christian and this is what worked for me. I cannot recommend things I didn't try.

For exercising I found bodyweight exercising to be one of the best for me. I will recommend only from this area. Of course, you can take up weights or whatever.

http://www.amazon.com/Convict-Conditioning-Weakness-Survival-Strength/dp//ref=sr_1_1?ie=UTF8&amp;qid=&amp;sr= (this is what I use and I am rather happy with it. A lot of people recommend this one instead: http://www.rosstraining.com/nevergymless.html )

Regarding friends, the following is the best bang for your bucks:

http://www.amazon.com/How-Win-Friends-Influence-People/dp//ref=sr_1_1?ie=UTF8&amp;qid=&amp;sr= (again, lots of criticism, but lots of praise too)

The rest of the points are addressed in the above books. I haven't given any book on financial advices. Once you know how to solve problems and use google and try to help people money will start coming, don't worry.

I hope this post helps you, even though it is a bit long and cynical.

Merry Christmas!

> I assume I ought to check it out after my discrete math class? Or does CLRS teach the proofs as if the reader has no background knowledge about proofs?

Sadly it does not teach proofs. You will need to substitute this on your own. You don't need deep proof knowledge, but just the ability to *follow* a proof, even if it means you have to sit there for minutes on one sentence just to understand it (which becomes much easier as you do more of this).

> We didn't do proof by induction, though I have learned a small (very small) amount of it through reading a book called Essentials of Computer Programs by Haynes, Wand, and Friedman. But I don't really count that as "learning it," more so being exposed to the idea of it.

This is better than nothing, however I recommend you get very comfortable with it because it's a cornerstone of proofs. For example, can you prove that there are less than 2 ^ (h+1) nodes in any perfect binary tree of height h? Things like that.

> We did go over Delta Epsilon, but nothing in great detail (unless you count things like finding the delta or epsilon in a certain equation). If it helps give you a better understanding, the curriculum consisted of things like derivatives, integrals, optimization, related rates, rotating a graph around the x/y-axis or a line, linearization, Newton's Method, and a few others I'm forgetting right now. Though we never proved why any of it could work, we were just taught the material. Which I don't disagree with since, given the fact that it's a general Calc 1 course, so some if not most students aren't going to be using the proofs for such topics later in life.

That's okay, you will need to be able to do calculations too. There are people who spend all their time doing proofs and then for some odd reason can't even do basic integration. Being able to do both is important. Plus this knowledge will make dealing with other math concepts easier. It's good.

> I can completely understand that. I myself want to be as prepared as possible, even if it means going out and learning about proofs of Calc 1 topics if it helps me become a better computer scientist. I just hope that's a last resort, and my uni can at least provide foundation for such areas.

In my honest opinion, a lot of people put too much weight on calculus. Computer science is very much in line with discrete math. The areas where it gets more 'real numbery' is when you get into numerical methods, machine learning, graphics, etc. Anything related to theory of computation will probably be discrete math. If your goal is to get good at data structures and algorithms, most of your time will be spent on discrete topics. You don't *need* to be a discrete math genius to do this stuff, all you need is some discrete math, some calc (which you already have), induction, and the rest you can pick up as you go.

If you want to be the best you can be, I recommend trying that book I linked first to get your feet wet. After that, try CLRS. Then try TAOCP.

Do not however throw away the practical side of CS if you want to get into industry. Reading TAOCP would make you really good but it doesn't mean shit if you can't program. Even the author of TAOCP, Knuth, says being polarized completely one way (all theory, or all programming, and none of the other) is not good.

> From reading ahead in your post, is Skiena's Manual something worth investing to hone my skills in topics like proof skills? I'll probably pick it up eventually since I've heard nothing but good things about it, but still. Does Skiena's Manual teach proofing skills to those without them/are not good at them? Or is there a separate book for that?

You could, at worst you will get a deeper understanding of the data structure and how to implement them if the proof goes over your head which is okay, no one on this planet starts off good at this stuff. After you do this for a year you will be able to probably sit down and casually read the proofs in these books (or that is how long it took me).

Overall his book is the best because it's the most fun to read (CLRS is sadly dry), and TAOCP may be overkill right now. There are probably other good books too.

> I guess going off of that, does one need a certain background to be able to do proofs correctly/successfully, such as having completed a certain level of math or having a certain mindset?

This is developed over time. You will struggle trust me. There will be days where you feel like you're useless but it continues growing over a month. Try to do a proof a day and give yourself minutes to think about things. Don't try insane stuff cause you'll only demoralize yourself. If you want a good start, this is a book a lot of myself and my classmates started on. If you've never done formal proofs before, you will experience exactly what I said about choking on these problems. Don't give up. I don't know anyone who had never done proofs before and didn't struggle like mad for the first and second chapter.

> I mean, I like the material I'm learning and doing programming, and I think I'd like to do at least be above average (as evident by the fact that I'm going out of my way to study ahead and read in my free time). But I have no clue if I'll like discrete math/proving things, or if TAOCP will be right for me.

Most people end up having to do proofs and are forced to because of their curriculum. They would struggle and quit otherwise, but because they have to know it they go ahead with it anyways. After their hard work they realize how important it is, but this is not something you can experience until you get there.

I would say if you have classes coming up that deal with proofs, let them teach you it and enjoy the vacation. If you really want to get a head start, learning proofs will put you on par with top university courses. For example at mine, you were doing proofs from the very beginning, and pretty much all the core courses are proofs. I realized you can tell the quality of a a university by how much proofs are in their curriculum. Any that is about programming or just doing number crunching is *literally* missing the whole point of Computer *Science*.

Because of all the proofs I have done, eventually you learn forever how a data structure works and why, and can use it to solve other problems. This is something that my non-CS programmers do not understand and I will always absolutely crush them on (novel thinking) because its what a proper CS degree teaches you how to do.

There is a lot I could talk about here, but maybe such discussions are better left for PM.

My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.**Neuroscience**: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.**General chemistry, organic chemistry/synthesis**: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.**Biochemistry**: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.**Mathematics**: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.**Computer Science**: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.**Physics**: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.**Other**: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.

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.

Your mileage with certifications may vary depending on your geographical area and type of IT work you want to get into. No idea about Phoenix specifically.

For programming work, generally certifications aren't looked at highly, and so you should think about how much actual programming you want to do vs. something else, before investing in training that employers may not give a shit about at all.

The more your goals align with programming, the more you'll want to acquire practical skills and be able to demonstrate them.

I'd suggest reading the FAQ first, and then doing some digging to figure out what's out there that interests you. Then, consider trying to get in touch with professionals in the specific domain you're interested in, and/or ask more specific questions on here or elsewhere that pertain to what you're interested in. Then figure out a plan of attack and get to it.

A lot of programming work boils down to:

**Using appropriate data structures**, and algorithms (often hidden behind standard libraries/frameworks as black boxes), that help you solve whatever problems you run into, or tasks you need to complete. Knowing when to use a Map vs. a List/Array, for example, is fundamental.- Integrating 3rd party APIs. (e.g. a company might Stripe APIs for abstracting away payment processing or Salesforce for interacting with business CRM countless 3rd party APIs out there).
- Working with some development framework. (e.g. a web app might use React for an easier time producing rich HTML/JS-driven sites or a cross-platform mobile app developer might use React-Native, or Xamarin to leverage C# skills, etc.).
- Working with some sort of platform SDKs/APIs. (e.g. native iOS apps must use 1st party frameworks like UIKit, and Foundation, etc.)
- Turning high-level descriptions of business goals ("requirements") into code.
**Basic logic**, as well as**systems design**and**OOD**(and a sprinkle of FP for perspective on how to write code with reliable data flows and cohesion), is essential. - Testing and debugging. It's a good idea to write code with testing in mind, even if you don't go whole hog on something like TDD - the idea being that you want it to be easy to ask your code questions in a nimble, precise way. Professional devs often set up test suites that examine inputs and expected outputs for particular pieces of code. As you gain confidence learning a language, take a look at simple assertion statements, and eventually try dabbling with a tdd/bdd testing library (e.g. Jest for JS, or JUnit for Java, ). With debugging, you want to know how to do it, but you also want to minimize having to do it whenever possible. As you get further into projects and get into situations where you have acquired "technical debt" and have had to sacrifice clarity and simplicity for complexity and possibly bugs, then debugging skills can be useful.

As a basic primer, you might want to look at Code for a big picture view of what's going with computers.

For basic logic skills, the first two chapters of How to Prove It are great. Being able to think about conditional expressions symbolically (and not get confused by your own code) is a useful skill. Sometimes business requirements change and require you to modify conditional statements. With an understanding of Boolean Algebra, you will make fewer mistakes and get past this common hurdle sooner. Lots of beginners struggle with logic early on while also learning a language, framework, and whatever else. Luckily, Boolean Algebra is a tiny topic. Those first two chapters pretty much cover the core concepts of logic that I saw over and over again in various courses in college (programming courses, algorithms, digital circuits, etc.)

Once you figure out a domain/industry you're interested in, I highly recommend focusing on one general purpose programming language that is popular in that domain. Learn about data structures and learn how to use the language to solve problems using data structures. Try not to spread yourself too thin with learning languages. It's more important to focus on learning how to get the computer to do your bidding via one set of tools - later on, once you have that context, you can experiment with other things. It's not a bad idea to learn multiple languages, since in some cases they push drastically different philosophies and practices, but give it time and stay focused early on.

As you gain confidence there, identify a simple project you can take on that uses that general purpose language, and perhaps a development framework that is popular in your target industry. Read up on best practices, and stick to a small set of features that helps you complete your mini project.

When learning, try to avoid haplessly jumping from tutorial to tutorial if it means that it's an opportunity to better understand something you really should understand from the ground up. Don't try to understand everything under the sun from the ground up, but don't shy away from 1st party sources of information when you need them. E.g. for iOS development, Apple has a lot of development guides that aren't too terrible. Sometimes these guides will clue you into patterns, best practices, pitfalls.

Imperfect solutions are fine while learning via small projects. Focus on completing tiny projects that are just barely outside your skill level. It can be hard to gauge this yourself, but if you ever went to college then you probably have an idea of what this means.

The feedback cycle in software development is long, so you want to be unafraid to make mistakes, and prioritize finishing stuff so that you can reflect on what to improve.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

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To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

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Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):

Prelude to Mathematics

A Book of Set Theory - More relevant to my current course & have heard good things about it

Linear Algebra

Number Theory

A Book of Abstract Algebra

Basic Algebra I

Calculus: An Intuitive and Physical Approach

Probability Theory: A Concise Course

A Course on Group Theory

Elementary Functional Analysis

Imagine you have a dataset without labels, but you want to solve a supervised problem with it, so you're going to try to collect labels. Let's say they are pictures of dogs and cats and you want to create labels to classify them.

One thing you could do is the following process:

- Get a picture from your dataset.
- Show it to a human and ask if it's a cat or a dog.
- If the person says it's a cat or dog, mark it as a cat or dog.
- Repeat.

(I'm ignoring problems like pictures that are difficult to classify or lazy or adversarial humans giving you noisy labels)

That's one way to do it, but is it the most efficient way? Imagine all your pictures are from only 10 cats and 10 dogs. Suppose they are sorted by individual. When you label the first picture, you get some information about the problem of classifying cats and dogs. When you label another picture of the same cat, you gain less information. When you label the th picture from the same cat you probably get almost no information at all. So, to optimize your time, you should probably label pictures from other individuals before you get to the th picture.

How do you learn to do that in a principled way?

Active Learning is a task where instead of first labeling the data and then learning a model, you do both simultaneously, and at each step you have a way to ask the model which next example should you manually classify for it to learn the most. You can than stop when you're already satisfied with the results.

You could think of it as a reinforcement learning task where the reward is how much you'll learn for each label you acquire.

The reason why, as a Bayesian, I like active learning, is the fact that there's a very old literature in Bayesian inference about what they call Experiment Design.

Experiment Design is the following problem: suppose I have a physical model about some physical system, and I want to do some measurements to obtain information about the models parameters. Those measurements typically have control variables that I must set, right? What are the settings for those controls that, if I take measurements on that settings, will give the most information about the parameters?

As an example: suppose I have an electric motor, and I know that its angular speed depends only on the electric tension applied on the terminals. And I happen to have a good model for it: it grows linearly up to a given value, and then it becomes constant. This model has two parameters: the slope of the linear growth and the point where it becomes constant. The first looks easy to determine, the second is a lot more difficult. I'm going to measure the angular speed at a bunch of different voltages to determine those two parameters. The set of voltages I'm going to measure at is my control variable. So, Experiment Design is a set of techniques to tell me what voltages I should measure at to learn the most about the value of the parameters.

I could do Bayesian Iterated Experiment Design. I have an initial prior distribution over the parameters, and use it to find the best voltage to measure at. I then use the measured angular velocity to update my distribution over the parameters, and use this new distribution to determine the next voltage to measure at, and so on.

How do I determine the next voltage to measure at? I have to have a loss function somehow. One possible loss function is the expected value of how much the accuracy of my physical model will increase if I measure the angular velocity at a voltage V, and use it as a new point to adjust the model. Another possible loss function is how much I expect the entropy of my distribution over parameters to decrease after measuring at V (the conditional mutual information between the parameters and the measurement at V).

Active Learning is just iterated experiment design for building datasets. The control variable is which example to label next and the loss function is the negative expected increase in the performance of the model.

So, now your procedure could be: - Start with:
- a model to predict if the picture is a cat or a dog. It's probably a shit model.
- a dataset of unlabeled pictures
- a function that takes your model and a new unlabeled example and spits an expected reward if you label this example

- a model to predict if the picture is a cat or a dog. It's probably a shit model.
- Do:
- For each example in your current unlabeled set, calculate the reward
- Choose the example that have the biggest reward and label it.
- Continue until you're happy with the performance.

- For each example in your current unlabeled set, calculate the reward
- ????
- Profit

Or you could be a lot more clever than that and use proper reinforcement learning algorithms. Or you could be even more clever and use "model-independent" (not really) rewards like the mutual information, so that you don't over-optimize the resulting data set for a single choice of model.

I bet you have a lot of concerns about how to do this properly, how to avoid overfitting, how to have a proper train-validation-holdout sets for cross validation, etc, etc, and those are all valid concerns for which there are answers. But this is the gist of the procedure.

You could do Active Learning and iterated experiment design without ever hearing about bayesian inference. It's just that those problems are natural to frame if you use bayesian inference and information theory.

About the jargon, there's no way to understand it without studying bayesian inference and machine learning in this bayesian perspective. I suggest a few books:

- Information Theory, Inference, and Learning Algorithms, David Mackay - for which you can get a pdf or epub for free at this link.

Is a pretty good introduction to Information Theory and bayesian inference, and how it relates to machine learning. The Machine Learning part might be too introductory if already know and use ML. - Bayesian Reasoning and Machine Learning by David Barber - for which you can also get a free pdf here

Some people don't like this book, and I can see why, but if you want to learn how bayesians think about ML, it is the most comprehensive book I think. - Probability Theory, the Logic of Science by E. T. Jaynes. Free pdf of the first few chapters here.

More of a philosophical book. This is a good book to understand what bayesians find so awesome about bayesian inference, and how they think about problems. It's not a book to take too seriously though. Jaynes was a very idiosyncratic thinker and the tone of some of the later chapters is very argumentative and defensive. Some would even say borderline crackpot. Read the chapter about plausible reasoning, and if that doesn't make you say "Oh, that's kind of interesting", than nevermind. You'll never be convinced of this bayesian crap.

Sounds like you are about 4 years behind me (Future physics PhD candidate). Glad to know you have discovered Dover books, they really are great and so cheap. It also sounds like you know what you're doing so good job, keep at it and you might make a good case for graduate school (if that's your destination). But I will warn you that upper division mathematics courses are different. I have seen so many people who think they are really great at mathematics up to vector calculus and then get completely shit on by more abstract courses like real analysis, abstract algebra and topology. The reason for this is that it requires more formalism and is very rigorous as far as proofs go. You'll eventually learn that math is all about making sure you have checked every possible condition in order to move on. I think something you will need is mathematical logic before you tackle abstract courses. If you do collect textbooks (like I do) then I would also recommend this textbook. It teaches you how to think like a mathematician and the logic behind proofs. I think a mathematics logic course is essential to students and it's a shame many mathematics students don't go through a formal logic course before they tackle advanced courses. Of course, some don't need it but unless you are brilliant, I would recommend it (Even if you are brilliant it would be a easy read). Just dig deep and focus and good luck with your future work. Mathematics and Physics are two beautiful subjects and it's always great to talk to future mathematicians or physicists(or any aspiring scientist in that case!) and help them get inspired or motivated!

P.S. Funny story, I had a friend who thought it would be funny to make people believe that Euler is pronounce "you-ler" with the argument that Euclid is pronounced "you-clid". It was pretty funny seeing people believe him.

I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from If you're in the United States, I recommend BookFinder to save money buying these things used.

On the CS side:

- Basic automata/formal languages/Turing machines; Sipser is recommended here.
- Basic programming language theory; I used University of Washington CSE P online video lectures and materials and can recommend it.
- Formal semantics; Semantics with Applications is good.
- Compilers. You'll need several resources for this; my personal favorites for an introductory text are Appel's ML book or Programming Language Pragmatics, and Muchnick is mandatory for an advanced understanding. All of the graph theory that you need for this type of work should be covered in books such as these.
- Algorithms. I used several books; for a beginner's treatment I recommend Dasgupta, Papadimitriou, and Vazirani; for an intermediate treatment I recommend MIT's J on Open CourseWare; for an advanced treatment, I liked Algorithmics for Hard Problems.

On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis: - You run into abstract algebra a lot in program analysis as well as in cryptography, so it's best to begin with a solid foundation along those lines. There's a lot of debate as to what the best text is. If you're never touched the subject before, Gallian is very approachable, if not as deep and rigorous as something like Dummit and Foote.
- Order theory is everywhere in program analysis. Introduction to Lattices and Order is the standard (read at least the first two chapters; the more you read, the better), but I recently picked up Lattices and Ordered Algebraic Structures and am enjoying it.
- Complexity theory. Arora and Barak is recommended.
- Formal logic is also everywhere. For this, I recommend the first few chapters in The Calculus of Computation (this is an excellent book; read the whole thing).
- Computability, undecidability, etc. Not entirely separate from previous entries, but read something that treats e.g. Goedel's theorems, for instance The Undecidable.
- Decision procedures. Read Decision Procedures.
- Program analysis, the "accessible" variety. Read the BitBlaze publications starting from the beginning, followed by the BAP publications. Start with these two: TaintCheck and All You Ever Wanted to Know About Dynamic Taint Analysis and Forward Symbolic Execution. (BitBlaze and BAP are available in source code form, too -- in OCaml though, so you'll want to learn that as well.) David Brumley's Ph.D. thesis is an excellent read, as is David Molnar's and Sean Heelan's. This paper is a nice introduction to software model checking. After that, look through the archives of the RE reddit for papers on the "more applied" side of things.
- Program analysis, the "serious" variety. Principles of Program Analysis is an excellent book, but you'll find it very difficult even if you understand all of the above. Similarly, Cousot's MIT lecture course is great but largely unapproachable to the beginner. I highly recommend Value-Range Analysis of C Programs, which is a rare and thorough glimpse into the development of an extremely sophisticated static analyzer. Although this book is heavily mathematical, it's substantially less insane than Principles of Program Analysis. I also found Gogul Balakrishnan's Ph.D. thesis, Johannes Kinder's Ph.D. thesis, Mila Dalla Preda's Ph.D. thesis, Antoine Mine's Ph.D. thesis, and Davidson Rodrigo Boccardo's Ph.D. thesis useful.
- If you've gotten to this point, you'll probably begin to develop a very selective taste for program analysis literature: in particular, if it does not have a lot of mathematics (actual math, not just simple concepts formalized), you might decide that it is unlikely to contain a lasting and valuable contribution. At this point, read papers from CAV, SAS, and VMCAI. Some of my favorite researchers are the Z3 team, Mila Dalla Preda, Joerg Brauer, Andy King, Axel Simon, Roberto Giacobazzi, and Patrick Cousot. Although I've tried to lay out a reasonable course of study hereinbefore regarding the mathematics you need to understand this kind of material, around this point in the course you'll find that the creature we're dealing with here is an octopus whose tentacles spread in every direction. In particular, you can expect to encounter topology, category theory, tropical geometry, numerical mathematics, and many other disciplines. Program analysis is multi-disciplinary and has a hard time keeping itself shoehorned in one or two corners of mathematics.
- After several years of wading through program analysis, you start to understand that there must be some connection between theorem-prover based methods and abstract interpretation, since after all, they both can be applied statically and can potentially produce similar information. But what is the connection? Recent publications by Vijay D'Silva et al (1, 2, 3, 4, 5) and a few others () have begun to plough this territory.
- I'm not an expert at cryptography, so my advice is basically worthless on the subject. However, I've been enjoying the Stanford online cryptography class, and I liked Understanding Cryptography too. Handbook of Applied Cryptography is often recommended by people who are smarter than I am, and I recently picked up Introduction to Modern Cryptography but haven't yet read it.

Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.

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

- What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)

The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.

As a general rule, I'd say that working on proof-based contest questions that are*just*beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online. - What's your current mathematical background?

This may be especially true for things like logic and*very*elementary set theory. - What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?

Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.) - What resources are available to you for vetting your work?

Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's*orders of magnitude*more effective when you have someone who can guide you. - Are you trying to learn the basics of mathematical proofs, or genuinely
*rigorous*mathematical proofs?

Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone*else*? - What experience have you already had with proofs in particular?

Have you had at least, for example, a geometry class that's proof-based? - How would you characterize your general writing ability?

Proofs are all about*communicating ideas*. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.

With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.

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

Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.

How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!

There would have been a time that I would have suggested getting a curriculum

text book and going through that, but if you're doing this for independent work

I wouldn't really suggest that as the odds are you're not going to be using a

very good source.

Going on the typical

Arithmetic > Algebra > Calculus********

### Arithmetic

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

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

*********

### Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

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

I think that you could get the first (arithmetic) text as well, personally I

prefer having actual books for working. They're also valuable for future

reference. This search should be remembered and used liberally

for finding things such as worksheets etc (eg for a search).

Algebra by Gelfland

No where near as comprehensive as chrystals algebra, but interesting and well

written questions (search for 'correspondence series' by Gelfand).

### Calculus

Calculus made easy - Thompson

This text is really good imo, there's little rigor in it but for getting a

handle on things and bashing through a few practical problems it's pretty

decent. It's all single variable. If you've done the algebra and stuff before

this then this book would be easy.

Pauls Online Notes (Calculus)

These are just a solid set of Calculus notes, there're lots of examples to work

through which is good. These go through calc I, II, III So a bit further than

you've asked (I'm not sure why you state up to calc II but ok).

Spivak - Calculus

If you've gone through Chrystals algebra then you'll be used to a formal

approach. This text is only single variable calculus (so that might be calc I

and II in most places I think, ? ) but it's extremely well written and often

touted as one of the best Calculus books written. It's very pure, where as

something like Stewart has a more applied emphasis.******

### Geometry

I've got given any geometry sources, I'm not too sure of the best source for

this or (to be honest) if you *really* need it for the above. If someone has

good geometry then they're certainly better off, many proofs are given

gemetrically as well and having an intuition for these things is only going to

be good. But I think you can get through without a formal course on it I'm

not confident suggesting things on it though, so I'll leave it to others. Just

thought I'd mention it.********

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.

Good question OP! I drafted a blog article on this topic a while back but haven't published it yet. An excerpt is below.

With equations, I sometimes just visualize what I'd usually do on paper. For arithmetic, there are actually a lot of computational methods that are better suited to mental computation than the standard pencil-and-paper algorithms.

In fact, mathematician Arthur Benjamin has written a book about this called Secrets of Mental Math.

There are tons of different options, often for the same problem. The main thing is to understand some general principles, such as breaking a problem down into easier sub-problems, and exploiting special features of a particular problem.

Below are some basic methods to give you an idea. (These may not all be entirely different from the pencil-and-paper methods, but at the very least, the format is modified to make them easier to do mentally.)

ADDITION

(1) Separate into place values: 27+39= (20+30)+(7+9)=50+16=66

We've reduced the problem into two easier sub-problems, and combining the sub-problems in the last step is easy, because there is no need to carry as in the standard written algorithm.

(2) Exploit special features: + = + -2 =

We could have used the place value method, but since is close to , which is easy to work with, we can take advantage of that by thinking of as - 2.

SUBTRACTION

(1) Number-line method: To find , you move forward 6 units on the number line to get to 30, then 41 more units to get to 71, for a total of 47 units along the number line.

(2) There are other methods, but I'll omit these, since the number-line method is a good starting point.

MULTIPLICATION

(1) Separate into place values: 18*22 = 18*(20+2)=+36=

(2) Special features: 18*22=()*(20+2)==

Here, instead of using place values, we use the feature that 18*22 can be written in the form (a-b)*(a+b) to obtain a difference of squares.

(3) Factoring method: 14*28=14*7*4=98*4=()*4==

Here, we've turned a product of two 2-digit numbers into simpler sub-problems, each involving multiplication by a single-digit number (first we multiply by 7, then by 4).

(4) Multiplying by 11*52= (add the two digits of 52 to get 5+2=7, then stick 7 in between 5 and 2 to get ).

This can be done almost instantaneously; try using the place-value method to see why this method works. Also, it can be modified slightly to work when the sum of the digits is a two digit number.

DIVISION

(1) Educated guess plus error correction: /7 = ? Note that 7*20=, and we're over by We need to take away two sevens to get back under, which takes us to , so the answer is 18 with a remainder of 3.

(2) Reduce first, using divisibility rules. Some neat rules include the rules for 3, 9, and

The rules for 3 and 9 are probably more well known: a number is divisible by 3 if and only if the sum of its digits is divisible by 3 (replace 3 with 9 and the same rule holds).

For example, is not divisible by 9, since 5+6+5+4=20, which is not divisible by 9.

The rule for 11 is the same, but it's the alternating sum of the digits that we care about.

Using the same number as before, we get that is divisible by 11, since +=0, and 0 is divisible by

PRACTICE

I think it's kind of fun to get good at finding novel methods that are more efficient than the usual methods, and even if it's not that fun, it's at least useful to learn the basics.

If you want to practice these skills outside of the computations that you normally do, there's a nice online arithmetic game I found that's simple and flexible enough for you to practice any of the four operations above, and you can set the parameters to work on numbers of varying sizes.

Happy calculating!

Greg at Higher Math Help

Edit: formatting

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

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

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

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

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

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

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

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

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

Computer scientist here I'm not a "real" mathematician but I do have a good bit of education and practical experience with some specific fields of like probability, information theory, statistics, logic, combinatorics, and set theory. The vast majority of mathematics, though, I'm only interested in as a hobby. I've never gone much beyond calculus in the standard track of math education, so I to enjoy reading "layman's terms" material about math. Here's some stuff I've enjoyed.

Fermat's Enigma This book covers the history of a famous problem that looks very simple, yet it took several hundred years to resolve. In so doing it gives layman's terms overviews of many mathematical concepts in a manner very similar to jfredett here. It's very readable, and for me at least, it also made the study of mathematics feel even more like an exciting search for beautiful, profound truth.

Logicomix: An Epic Search for Truth I've been told this book contains some inaccuracies, but I'm including it because I think it's such a cool idea. It's a graphic novelization (seriously, a graphic novel about a logician) of the life of Bertrand Russell, who was deeply involved in some of the last great ideas before Godel's Incompleteness Theorem came along and changed everything. This isn't as much about the math as it is about the people, but I still found it enjoyable when I read it a few years ago, and it helped spark my own interest in mathematics.

Lots of people also love Godel Escher Bach. I haven't read it yet so I can't really comment on it, but it seems to be a common element of everybody's favorite books about math.

I double majored in math and CS as an undergrad and I enjoyed math more than CS. I'm a graduate student right now planning on doing research in a mathy area of CS. Everything I write below comes from that perspective.

- In my experience Wikipedia has some pretty good math articles. Many articles do a decent job of explaining the intuition behind of various concepts, not just the formalism.
- Math.StackExchange.com is similar to stackoverflow and I've found it to be quite helpful on occasion. Example of a question with some great answers
- /r/math is pretty active and has a very knowledgeable user base.
- One of the best known living mathematicians is Terrence Tao. He has a math blog but you might not have the background necessary to understand much of the material; I would guess that you need knowledge covering at least the standard undergraduate math major coursework to understand many of the posts.

But if you're interested in really digging in and understanding some math at an advanced undergraduate level (analysis, abstract algebra, topology, etc.) then I don't think there is any substitute for books. - A personal favorite is The Princeton Companion to Math. It has expository articles that provide high level overviews of different branches of math, important theorems, biographies of mathematicians, articles about the historical development of math, and more. It has some top notch contributors and was designed to be approachable by anyone with a good knowledge of calculus. This would be a great place to get a sense of the areas of study in math. I bought this book right after it came out after graduating high school and have loved it ever since. Everyone with a love of math should own this book.
- How to Prove It does a great job of introducing proofs and set theory which are both fundamental to higher math.
- Dover is a well loved publisher among math folks because they offer extremely cheap books on math that are of fairly high quality if a little old. You can find textbooks on any topic in the undergraduate math curriculum for less than $20 from Dover.

> 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.

It's less math intensive in the sense that you won't be solving calculus problems very often (or at all), but there are classes where a (basic) understanding of calculus will be helpful. For instance, I just completed algorithms and was pretty glad that I had taken Calculus. Knowing a lot about limits and knowing L'Hopital's rule made parts of asymptotic analysis a lot more intuitive than it otherwise would have been.

With that said, discrete math (which you'll cover in CS ) is a pretty big part of the program and computer science as a whole. You'll serve yourself well by getting a solid understanding of discrete math--even in classes where it's not an explicit requirement.

To give an example, in CS (operating systems), there was an assignment where we had to build a pretty simple dungeon-crawler game where a player moved through a series of rooms. Each time the player played the game, there needed to be a new random dungeon, and the connections between rooms needed to be two-way. Calculus isn't really going to help you solve this problem, but if you're good with discrete math, you'll quickly realize that this sort of problem can easily be solved with a graph. Further, you can represent the graph as a 2D array, and at that point the implementation becomes pretty easy.

So, there is math in the program, but not the type that you've probably been doing throughout your academic career. Discrete math comes naturally to some, and it's really difficult for others. I'd recommend picking up this book (which is used in the program) whenever you get a chance:

https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/

I'm almost done with the program, but I've been returning to that a lot to review concepts we covered in class and to learn new stuff that we didn't have time for in the term. It's a great book.

Good luck!

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp//ref=sr_1_1?ie=UTF8&amp;qid=&amp;sr=&amp;keywords=spivak%27s+calculus

This book starts with basic properties of numbers (associativity, commutativity, etc), then moves onto some proof concepts followed by a very good foundation (functions, vectors, polar coordinate). Be forewarned that the content is VERY challenging in this book, and will definitely require a determined effort, but it will certainly be good if you can get through it.

A more gentle introduction to Calculus is http://www.amazon.com/Thomas-Calculusth-George-B/dp//ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=&amp;sr=&amp;keywords=thomas%27+calculus and it is a much easier book, but you don't prove much in this one. Both of these can likely be found online for free. Also, if you want to get a decent understanding I recommend, http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp//ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=&amp;sr=&amp;keywords=how+to+prove+it or http://www.people.vcu.edu/~rhammack/BookOfProof/index.html the latter is definitely free.

You may also need a more introductory text for trig and functions. I can't find the book my school used for precalc, hopefully someone else can offer a good recommendation.

Also, getting a dummies book to read alongside was pretty helpful for me, and Paul's online notes(website) is very nice.

Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.

You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.

Two other books you may be interested in instead, that teach the same kinds of things:

Introduction to Mathematical Thinking which he wrote to use in his Coursera course.

How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.

Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.

What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.

Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics *without any freaky looking symbols*, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/philgmh/text/c01_pdf

Honestly if I were starting out I would love that last link, it looks fantastic actually.

There are a lot of good classics on /u/thebenson's list. I want to highlight the books that are good for what you'll be learning, and give you a sense of how the sequence works. And I'll add a few.* Calculus books: *Thomas' Calculus

*,*Calculus

*by James Stewart (not multivariable), and this cheap easy read by Morris Kline.*

Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.

Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.

Intro physics books:

*Fundamentals of Physics*(Halliday & Resnick),

*Physics for Scientists and Engineers*(Serway & Jewett),

*Physics for Scientists and Engineers*(Tipler & Mosca),

*University Physics*(Young), and

*Physics for Scientists and Engineers*(Knight) are all good. Gee, they get really unoriginal with the names, huh?

Each of these books assumes no background in physics, but you do need to use calculus. If you're going to take a class in basic mechanics that doesn't involve any calculus, you may find it more useful to get a book at that level. The only such book that I'm familiar with is

*Physics: Principles with Applications*by Giancoli. I know there are many others, but I can't speak for them.

*Engineering Mathematics*

Mathematical methods: Greenberg is way more than you need here. I think you would findMathematical methods: Greenberg is way more than you need here. I think you would find

*by Stroud & Booth more useful as a reference, since it covers a lot of the less advanced stuff that you may need a refresher on.*

Sequence: it's typical to start learning physics by learning about Newtonian mechanics, with or without calculus. After that, one often goes on to thermodynamics or to electricity and magnetism. It sounds like this is roughly how your program is going to work.

If you are learning mechanics with calculus, you can expect E&M to be even heavier on the calculus and thermodynamics to be less so. More calculus is not a bad thing. People often get scared of it, but it actually makes things easier to understand.

It is very typical that you will use only one book (from the intro books above) for all of these topics. You shouldn't need to get any books on specific topics.

**

*Spacetime Physics* by Taylor and Wheeler, since I don't want to imply that this is a background-heavy book. On the contrary, this is one of the most beginner-friendly physics books ever written, and it is my favorite introduction to special relativity. Special relativity is probably not something you need to learn about right now, but if you have any interest, I seriously recommend finding an old used copy of this book—it's a fun read aside from any other uses!*

The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.

But I do want to give some mention to

The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.

But I do want to give some mention to

Hey! This comment ended up being a lot longer than I anticipated, oops.

My all-time favs of these kinds of books definitely has to be ** Prime Obsession** and

**by John Derbyshire -**

*Unknown Quantity**Prime Obsession*covers the history behind one of the most famous unsolved problems in all of math - the Riemann hypothesis, and does it while actually diving into some of the actual theory behind it.

*Unknown Quantity*is quite similar to

*Prime Obsession*, except it's a more general overview of the history of algebra. They're also filled with lots of interesting footnotes. (Ignore his other, more questionable political books.)

In a similar vein,

**by Simon Singh also does this really well with Fermat's last theorem, an infamously hard problem that remained unsolved until The rest of his books are also excellent.**

*Fermat's Enigma*All of Ian Stewart's books are great too - my favs from him are

**,**

*Cabinet***, and**

*Hoard***which are each filled with lots of fun mathematical vignettes, stories, and problems, which you can pick or choose at your leisure.**

*Casebook*When it comes to fiction, Edwin Abbott's

**is a classic parody of Victorian England and a visualization of what a 4th dimension would look like. (This one's in the public domain, too.) Strictly speaking, this doesn't have any equations in it, but you should definitely still read it for a good mental workout!**

*Flatland*Lastly, the

**series is a Japanese YA series all about interesting topics like Taylor series, recursive relations, Fermat's last theorem, and Godel's incompleteness theorems. (Yes, really!) Although the 3rd book actually has a pretty decent plot, they're not really that story or character driven. As an interesting and unique mathematical resource though, they're unmatched!**

*Math Girls*I'm sure there are lots of other great books I've missed, but as a high school student myself, I can say that these were the books that really introduced me to how crazy and interesting upper-level math could be, without getting too over my head. They're all highly recommended.

Good luck in your mathematical adventures, and have fun!

Copying my answer from another post:

I was literally in the bottom 14th percentile in math ability when i was

One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College

Precalculus

It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is **extremely well organized.** Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.

Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.

Also, there is a metric *shitload* of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)

I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.**Edit: other resources**

Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through *application* of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.

Paul's Online Math Notes

This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions)

Stuff that's more important than you think (if you're interested in higher math after your GED)**Trigonometric functions:** very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.**Matrix algebra:**Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.**Edit 2: Electric Boogaloo****Other good, cheap math textbooks**

/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).

Search up **Dover Mathematics** on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math Exchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1, pages of Linear Algebra, Graph Theory, and Discrete Math text for $ If you prefer paper to .pdf, this is probably a good route to go.

Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.

> Are the deep mathematical answers to things usually very complex or insanely elegant and simple when you get down to it?

I would say that the deep mathematical answers to questions tend to be very complex and insanely elegant at the same time. The best questions that mathematicians ask tend to be the ones that are very hard but still within reach (in terms of solving them). The solutions to these types of questions often have beautiful answers, but they will generally require lots of theory, technical detail, and/or very clever solutions all of which can be very complex. If they didn't require something tricky, technical, or the development of new theory, they wouldn't be difficult to solve and would be uninteresting.

For any experts that happen to stumble by, my favorite example of this is the classification of semi-stable vector bundles on the complex projective plane by LePotier and Drezet. At the top of page 7 of this paper you'll see a picture representing the fractal structure that arises in this classification. Of course, this required a lot of hard math and complex technical detail to come up with this, but the answer is beautiful and elegant.

> How hard would it be for a non mathematician to go to a pro? Is there just some brain bending that cannot be handled by some? How hard are the concepts to grasp?

I would say that it's difficult to become a professional mathematician. I don't think it has anything to do with an individual's ability to think about it. The concepts are difficult, certainly, but given time and resources (someone to talk to, good books, etc) you can certainly overcome that issue. The majority of the difficulty is that there is *so much math!* If you're an average person, you've probably taken at most Calculus. The average mathematics PhD (i.e., someone who is just getting their mathematical career going) has probably taken two years of undergraduate mathematics courses, another two years of graduate mathematics courses, and two to three years of research level study beyond calculus to begin to be able tackle the current theory and solve the problems we are interested in today. That's a lot of knowledge to acquire, and it takes a very long time. That doesn't mean you can't start solving problems earlier, however. If you're interested in this type of thing, you might want to consider picking up this book and see if you like it.

I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.

EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).

I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.

Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.

This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:

- theoretical computer science (essentially a math text)
- set theory
- linear algebra
- algebra
- predicate calculus

Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!

(edit: can't count and thought five was four)

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

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

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

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

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

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

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

let me give you a shortcut.

You want to know how partial derivatives work? Consider a function with two variables: f(x,y) = x^2 y^3, for a simple example.

here's what you do. Let's take the partial derivative with respect to x. What you do, is you consider all the other variables to be constant, and just take the standard derivative with respect to x. In this case, the partial derivative with respect to x is: 2xy^3. That's it, it's really that easy.

What about taking with respect to y? Same thing, now x is constant, and your answer is 3x^2 y^2.

This is an incredibly deep topic, but getting enough of an understanding to tackle gradient descent is really pretty simple. If you want to full on jump in though and get some exposure to way more than you need, check out div curl and grad and all that. It covers a lot, including a fair amount that you won't need for any ML algorithm I've ever seen (curl, divergence theorem, etc) but the intro section on the gradient at the beginning might be helpful maybe see if you can find a pdf or something. There's probably other good intros too, but seriously the mechanics of actually performing a partial derivative really are that easy. If you can do a derivative in one dimension, you can handle partial derivatives.

edit: I misread, didn't see you were a junior in highschool. Disregard div curl grad and all that, I highly recommend it, but you should be up through calc 3 and linear algebra first.

To change my advice to be slightly more relevant, learn how normal derivatives work. Go through the Kahn Academy calc stuff if the format appeals to you. Doesn't matter what course you go through though, you just need to go through a few dozen exercises (or a few hundred, depending on your patience and interest) and you'll get there. Derivatives aren't too complicated really, if you understand the limit definition of the derivative (taking the slope over a vanishingly small interval) then the rest is just learning special cases. How do you take the derivative of f(x)g(x)? f(g(x))? There's really not too many rules, so just spend a while practicing and you'll be right where you need to be. Once you're there, going up to understanding partial derivatives is as simple as I described above if you can take a standard derivative, you can take a partial derivative.

Also: props for wading into the deep end yourself! I know some of this stuff might seem intimidating, but if you do what you're doing (make sure you understand as much as you can instead of blowing ahead) you'll be able to follow this trail as far as you want to go. Good luck, and feel free to hit me up if you have any specific questions, I'd be happy to share.

Thanks for the answer!

Glad to hear about Spivak! I've heard good things about that textbook and am looking forward to going through it soon :). Are the course notes for advanced algebra available online? If so, could you link them?

Is SICP used only in the advanced CS course or the general stream one, too? (last year I actually worked my way through the first two chapters before getting distracted by something else - loved it though!) Also, am I correct in thinking that the two first year CS courses cover functional programming/abstraction/recursion in the first term and then data structures/algorithms in the second?

That's awesome to know about 3rd year math courses! I was under the impression that prerequisites were enforced very strongly at Waterloo, guess I was wrong :).

As for graduate studies in pure math, that's the plan, but I in no way have my heart set on anything. I've had a little exposure to graph theory and I loved it, I'm sure that with even more exposure I'd find it even more interesting. Right now I think the reason I'm leaning towards pure math is 'cause the book I'm going through deals with mathematical logic / set theory and I think it's really fascinating, but I realize that I've got 4/5 years before I will even start grad school so I'm not worrying about it too much!

Anyways, thanks a lot for your answer! I feel like I'm leaning a lot towards Waterloo now :)

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:

- 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.

This is just my perspective, but . . .

I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.

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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.

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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:

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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/X ) covers basic strategies for problem solving in mathematics

Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/ ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.

Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/ ) does a good job of teaching how to prove mathematical conjectures.

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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/ ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.

&#xB;

Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:

&#xB;

A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/ )

&#xB;

Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/ ).

&#xB;

If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.

**What this expressions says**

First of all let's specify that the domain over which these statements operate is the set of all *people* say.

Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation *x loves y*.

This allows us to translate the statement:

∀x∀yP(x,y) -> ∀xP(x,x)*What does ∀x∀yP(x,y) mean?*

This is saying that *For all x, it is the case that For all y, x loves y*.

So you can interpret it as saying something like *everyone loves everyone*. *What does ∀xP(x,x) mean?*

This is saying that *For all x it is the case that x loves x*. So you can interpret this as saying something like *everyone loves themselves*.

So the statement is basically saying:*Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves*.

This translation gives us the impression that the statement is true. But how to prove it?**Proof by contradiction**

We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.

So let's assume:

∀x∀yP(x,y) -> *not* ∀xP(x,x)*not* ∀xP(x,x) is equivalent to ∃x not P(x,x).

In words this means *It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false*.

So let's *instantiate* this expression with something from the domain, let's call it *a*. Basically let's pick a person for whom we are saying a loves a is false.*not P(a,a)*

Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.

Let's instantiate the expression with the object *a* we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)

This happens in two stages:

First we instantiate y

∀xP(x,a)

Then we instantiate x*P(a,a)*

The statements *P(a,a)* and *not P(a,a)* are contradictory, therefore we have shown that the statement:

∀x∀yP(x,y) -> *not* ∀xP(x,x) leads to a contradiction, which implies that

∀x∀yP(x,y) -> ∀xP(x,x) is true.

Hopefully that makes sense.**Recommended Resources**

Wilfred Hodges - Logic

Peter Smith - An Introduction to Formal Logic

Chiswell and Hodges - Mathematical Logic

Velleman - How to Prove It

Solow - How to Read and Do Proofs

Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics

You should absolutely not give up.

- Axler is fairly advanced for a freshman course in linear algebra. The fact that it's making more sense the second time you go over it is much more important than failing to understand it the first time.
- Nobody can learn sophisticated math from a lecture if they haven't seen it before. Well, maybe geniuses can, but my guess is that the majority of successful mathematicians reach a point where the lecture medium becomes much less important. You have to read the textbook with a pencil in hand, proving lemmas yourself. Digest proofs at your own pace, there's nothing wrong or unusual with not understanding it the way your Professor presented it.
- About talking math with people - this just takes time. Hold off on judging yourself. You can also get practice by getting involved with math subreddits or math.stackexchange.
- It's pretty unlikely that you are "too stupid" to study math. I've seen people with a variety of natural ability learn a tremendous amount about math and related disciplines, just by working hard.

None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."

As for specific recommendations,*make the most of this summer*. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!

You do realize that there is guesswork but the extremes of the confidence interval are strictly positive right? In other words, no one is certain but what we are certain about is that optimum homework amount is **positive**. Maybe it's 4 hours, maybe it's 50 hours. But it's definitely not 0.

I don't like homework either when I was young. I dreaded it, and I skipped so many assignments, and I regularly skipped school. I hated school. In my senior year I had such severe senioritis that after I got accepted my grades basically crashed to D-ish levels. (By the way this isn't a good thing. It makes you lazy and trying to jumpstart again in your undergrad freshman year will feel like a huge, huge chore)

Now that I'm older I clearly see the benefits of homework. My advice to you is not to agree with me that homework is useful. My advice is to pursue your dreams, but when doing so be keenly aware of the pragmatical considerations. Theoretical physics demands a high level of understanding of theoretical mathematics: Lie groups, manifolds and differential algebraic topology, grad-level analysis, and so on. So get your arse and start studying math; you don't have to like your math homework, but you'd better be reading Spivak if you're truly serious about becoming a theoretical physicist. It's not easy. Life isn't easy. You want to be a theoretical physicist? Guess what, top PhD graduate programs often have acceptance rates **lower** than Harvard, Yale, Stanford etc. You want to stand out? Well everyone wants to stand out. But for every wannabe year-old theoretical physicists out there, only 1 has actually started on that route, started studying first year theoretical mathematics (analysis, vector space), started reading research papers, started really **knowing** what it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doing **work** that allows you to reach your dreams.

Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.

- There seems to be tons. At least I keep being told there are tons! My school has a lot of recruiters come by who are interested in math people!
- I can definitely recommend HMC, but I would also consider MIT, Caltech, Carnegie Melon, etc. I've heard UW is good, too!
- Most all of linear algebra is important later on. I will say that many texts treat linear algebra the same as "matrix algebra", which it is not. Linear algebra is much more general, and deals with things called vector spaces. Matrix algebra is a specific case of linear algebra. If you want a good linear algebra text (though it might be a bit difficult), check out http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/

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## Mathematics for Machine Learning

Linear Algebra

Is it just the linear algebra you're struggling with? Because that's really only % of CSE , basicaly the numpy sections. The rest is just python and learning how to use the language and its relevant libraries (pandas and numpy).

I took CSE before ISYE and even though I didn't have the background for some of the things we were doing (e.g., implementing PCA in python without understanding what PCA actually was), I still had very few problems with the course and found it relatively easy because I'm generally comfortable with python. Admittedly, I have familiarity with linear algebra from my undergrad years, so while I also found the numpy sections harder, I could still understand what's going on once I saw the solution and it easily 'clicked'.

If python/programming is your problem in general, then that's not necessarily good news because there's a lot of it in the program. But if you found ISYE easy, that also has a good amount of coding so I don't know

If not understanding the concepts behind the programming is an issue, remember that CSE 's focus is more on building your programming proficiency than with teaching you the mathematical concepts (like I said, I implemented PCA with ease without understand what it actually meant).

If the linear algebra alone is the problem, thenmixed news for you. There are some courses, like Deterministic Optimisation, High Dimensional Data Analytics, Computational Data Analytics, which all require linear algebra knowledge as a prerequisite. There are probably a few others I haven't mentioned. You'd struggle with these. On the other hand, none of these are really compulsory (ISYE CDA is, but only if you're taking the computational analytics track) so you can avoid them and take courses without significant knowledge of linear algebra if you want, though linear algebra really does form the foundation of a lot of data science because of its ability to quickly process information without running full-blown for loops so you'll likely encounter it again in different places.

It's just knowledge like any other, though. I've found that linear algebra is often taught in an obtuse manner, where the focus is more on the mechanical operations than the actual intuition and meaning behind those operations (e.g., how to find the inverse of a matrix instead of what the inverse means and why you'd want to use it in solving real problems). I've found that once you understand that intuition, linear algebra is generally kinda easy, but if it's not taught properly and all you learn is those mechanical operations and how to do them, you can easily start getting headaches.

But there are good courses that can help you get up to speed. Some of these are:

I'd recommend taking them in that order. The 3blue1brown videos will give you the intuition for what different LA operations are all about and the Coursera specialisation will reinforce this and teach how to compute (and program) them. Then, go back to the 3blue1brown videos to reinforce that intuition of what all those mechanical operations you learned to perform are actually useful for. If you do just these two, you should have at least a good foundation for OMSA. You can refer to the textbook if you want as well to build out your knowledge more formally.

If you want to learn even more/with more academic rigour, the LAFF courses have an excellent reputation and there are three follow-on courses for even more advanced linear algebra if you want to do that. Alternatively, you can also follow the MIT lectures. Gilbert Strang is the author of one of the most popular LA textbooks out there, so you know you're in good hands.

**1906**1907 1908