Fundamental Math

Full disclosure. The author of the following book is a friend of mine. However, I cannot overstate how much I appreciated reading his books

True Beauty of Math Volume I: (Set Theory)

This book begins in a place that is mainly untouched until entrance into an undergraduate math program. However, this book introduces the subject as fundamental concept of mathematics. The book builds upon only itself and some basic arithmetic to make some truly exciting claims. You will learn the basics of sets and functions, gain a new appreciation for infinity, get your mind blown by a proof of infinite primes, and understand the Cartesian coordinate system in a way that truly makes sense. If you're beginning to think, this isn't mathematics, this isn't what I clicked your link bait for, then that's perfect. Michael's book introduces you to the world of advanced mathematics in a truly easy and invigorating way.

True Beauty of Math Volume II: (Group Theory)

Try not to spend too much time in between Volumes I and II, because this one picks up right where the last one left off. This one gets into Group Theory and starts to introduce some more rigorous curriculum. If you are anything like me, your K-12 education tried to convince you many things just work and we should remember that, but this brings the secrets right out. While you still may not be able to solve some seriously tough math problems, you will begin to think of math as a series of building blocks, and you will have an understanding of how the discipline holds together as a series of fundamental truths. And you will start to understand and THINK in some very abstract mathematical terms. Group theory can really start to unleash the building blocks of our crazy mathematical world.

Calculus

Calculus Made Easy:

This book was written in 1908! And it is at the top of my list for a reason. This is a straight-forward, proven introduction into the whole Calculus discipline. It takes one step at a time and gives plenty of practice problems, so that by the time you get through this one you will feel very dangerous. It is a long book, but to me was fun in the way it writes the subject like a story. Do not feel the need to read it from start to finish; make some progress, move on to something else, and come back to finish it. It is a great book to truly get the fundamentals of Calculus ingrained into your system. The most remarkable thing about this book is simply that this teaches the entire discipline without reliance on formulas. It builds up from simple reasoning, one step at a time (the way math should be!).

Calculus Better Explained:

This book should probably be first on Calculus list since its the most remedial and fun to read. Its a very short book that just sort of wets the appetite for the subject, but I went to revisit the book and found that it really helps reinforce the subject. It takes the proven/regurgitated formulas from your intro class and gives them fresh life as they pertain to geometry and other subjects. If you're already familiar with Calculus concepts, feel free to skip this book, but it (and is) a great experience for me.w if you are a visual person, and love to see how to integrate a circle with pictures, this is the book for you.

Kahn Academy: (Single variable, Multi-variable and literally everything else)

If you have not stumbled upon Kahn Academy before, please do so now. It is one of the most amazing resources ever. It is helpful for both quick refreshers on subjects (I did this for logarithms and thoroughly enjoyed the lectures). Now, I have no doubt that all of the courses on this site are top notch; however, I want to speak specifically to one that I have taken recently. The Multi-variable Calculus course was one of the most rewarding MOOC's I have ever made it through. As is common theme with my recommendations, this course preaches the intuitive nature behind the mathematics. The focus on graphing and visualizations and understanding of the core components is such a fresh take on mathematics. It also gives some awesome references (or teasers!) into the relationship with Linear Algebra. This is not an easy subject, but it is presented in a very captivating way. It is very very visual and built up so much intuition in a very visible way.

Linear Algebra

Linear Algebra - Foundations to Frontiers (LAFF):

If there is one resource on this list that proves how awesome the 21st century is, it's the existence of this course. The content is absolutely phenomenal, and engaging, and tough, and empowering. The videos are very high quality, and the partnership with Matlab provides for awesome online resources. Professor Van de Geijn is very much a Linear Algebra programmer, and provides a lot of programming exercises to get you very familiar with the course content. It takes a little bit of struggle to get up and running in Matlab, but it was extremely helpful in understanding the content. There is also some very advanced content in this course (Optional), about the cutting edge of matrix math in computer programming. There are also some very fun applied projects around image compression and Markov Chains. I cannot say enough about this course.

Linear Algebra:

This is a textbook through and through. Again, this is not one that I would not recommend reading from front to back. There is a lot of content, a lot of practice problems and exercises. It will get dull, and it's ok to move on and come back to this. However, this has a lot of content and remains a great reference for me. It is a good one to have in the bookshelf, but maybe not the best learning material.

Applied Linear Algebra:

I am a sucker for anything applied. There are some pretty fun concepts/examples in this book. But again, it reads very similar to a textbook. If you want to prove your worth in Linear Alebra, going after these problems might be a fun time. But for the most part, this does not make the best learning material either. I'll be sure to update this section if I ever do make it through this book.

Statistics

Elements of Statistical Learning:

Pattern Recognition and Machine Learning:

To be honest, neither of these books are going to be "Statistics" books. They are very much fundamental Machine Learning books, but they're the only statistics books I have given a shot. And continuing on that honesty theme, I am pretty confident that getting through these books will have you feeling quite comfortable in your Statistics knowledge. I have only made it through a couple of chapters of each of these (they are very similar), and they both are extremely impressive in their depth and immediate focus on the value of the knowledge. They introduce subjects and the theory of each subject in an applied sense, giving you the why and how this can/is used. They are much more advanced books, but if you get through these you will have to feel like a champ.

Functional Programming (FP)

We are going to deviate from our purely mathematical pursuit of knowledge here. These books are not "mathematical" in the classic sense, but instead are just a programmer's pleasant way of using our mathematical concepts to make life easier. The True Beauty of Math series makes for great requisite knowledge to thoroughly enjoy this book. Some programming experience would also be required.

Professor Frisby's Mostly Adequate Guide to Functional Programming:

It reads in a very fun and informative manner, stepping step by step through some critical mathematical concepts such as (pure) functions, composition, and Hindley-Milner. It then goes on to show these ideas applied with Javascript code, and makes a serious case for why we've been doing it wrong for so long. These are programming concepts that have fanatic followings, and I personally believe will continue to shape the future. For any programmer, it may be the most entertaining read on this list.

Theorems for free!:

Ok. This is an academic paper. Maybe I am getting a little carried away with my enthusiasm here. However, this paper is a truly amazing argument for the power of FP (and it's not even a FP paper). It goes through some typical FP functions (i.e zip, concat, filter, head, tail) and explains how their Hindley-Milner form can lead us to proofs about their correctness. And as a programmer, I can say that there is (likely) nothing more that I care about in my working life.