Epic textbooks, manuals and guides for the mathematically-inclined, generally not at the K-12 level, along with some other less mathy masterpieces here and there!
Arithmetics, Math Puzzles, Mathematical Initiation and all that goodness.
Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks
— by Arthur Benjamin and Michael Shermer
Despite the fancy title, Secrets of Mental Math is a how-to book filled with insights on how the mathemagician Arthur Benjamin manages to perform gigantic calculations off the top of the head in a way that seems impossible to most human beings. Being both a math professor and a professional magician, Dr. Benjamin reveals in his book the mental arithmetic tricks he personally uses since very early age to compensate for his fragile short-term memory.
Starting from addition / subtraction to multiplication / division, Dr. Benjamin goes on to introduce a mnemonic technique for memorizing large numbers that allows him to hijack his memory to perform even larger computations. Along with some tricks in calendar calculation and number estimation, this ~200-page book covers just enough practical tricks for pretty much everyone in an accessible and stimulating manner, justifiably making it a popular read among mental math enthusiasts around the world.
What is Mathematics? An Elementary Approach to Ideas and Methods (2nd Edition)
— by Richard Courant and Herbert Robbins
Touted as an all-time classic introducing higher mathematics to the general audience, What is Mathematics? — originally written by two 20th-century prominent mathematicians — covers a wide range of topics such as elementary number theory, analytic geometry, topology and calculus in the format of a quasi-literary masterpiece. Even though the first edition of the book was published in the 1940s, its content still remains as relevant and lucid as ever.
As advocates for "putting meaning back into mathematics", Courant and Robbins challenges the viewpoint of mathematics being merely a game of derivation with formal symbols. Indeed, it can be seen throughout the book that the authors encourage problem solving primarily as a means for developing insight and genuine comprehension of mathematics. With just a bit of background in high school mathematics, almost everyone can use this book to initiate their passion and pursuit towards higher mathematics
APPLIED COLLEGE MATHEMATICS
College Algebra, Calculus, Linear Algebra, Differential Equations and the like.
Introduction to Linear Algebra (5th Edition)
— by Gilbert Strang
Written by the one and the only MIT veteran Gilbert Strang, Introduction to Linear Algebra is the culmination of decades of teaching by one of the world's most prolific educator in linear algebra. It is also used as the textbook for the Linear Algebra course 18.06 at MIT, and features conversation-style narratives along with numerous review and challenge problems to get the juice flowing.
From matrices, system of linear equations, vector space, orthogonality to determinant, eigenvector, singular value decomposition and linear transformation, this ~500-page comprehensive textbook leaves no stone unturned. Along with relevant, real-life applications and a focus on geometric intuition, coupled with online solution manual and video lectures. it's tough to not acquire a solid foundation on linear algebra after finishing this book and its ancillary content.
Calculus (8th Edition)
— by James Stewart
If you're looking for an accessible, ultra-comprehensive treatment on almost everything calculus has to offer, then this book probably has your back. Written with conceptual understanding and calculus users in mind, Calculus is the 30-year-lifework of the renowned Canadian mathematician James Stewart, and covers everything from single-variable calculus (e.g., derivatives, curve sketching, integration techniques) to multi-variable calculus (i.e., partial derivatives, multiple integrals, Lagrange multipliers, divergence and curl), along with plenty of applications — such as Newton's method, volume calculation, curvature and differential equations — in between the text.
With over 1200 pages of materials and an intermediate approach in delivering readability and rigor, its sections usually begin with some motivating discussions before throwing in the key concepts/theorems along with 5 or 6 solved examples in between and a ton of problem sets thereafter. Indeed, the fact that the proofs of most theorems are given — but are secondary in driving the discussion — makes it a very suitable read for students in applied sciences and higher-math enthusiasts alike.
PROBABILITY & STATISTICS
Discrete/Continuous Distributions, Hypothesis Testing, Analysis of Variance and more.
Mathematical Statistics with Applications (7th Edition)
— by Wackerly et al.
If you want to know more how the standard (and the less standard) theorems in probability and statistics come about, and are willing to grind through the proofs and calculations, then Mathematical Statistics With Applications is probably for you. From probability distributions, moment-generating functions, law of large numbers, central limit theorem to methods of estimation, analysis of variance, non-parametric statistics and Bayesian inference, this book pretty much covers all the usual university-level statistics — save with a theoretical twist.
As its title suggests, this book primarily focuses on the mathematical treatment of statistics itself, and hence is probably not geared towards students of applied statistics with little exposure to multivariate calculus, However, it is nevertheless very well-written and strikes a fine balance between rigour and clarity, making it a valuable resource for both reference and self-study.
INTRODUCTION TO PROOF
Basic Logic, Proof Techniques, plus their Applications in higher mathematics.
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes (6th Edition)
— by Daniel Solow
A unique book with a systematic approach in identifying, categorizing and explaining a mathematical proof and its making, How to Read and Do Proofs does exactly what it claims to do. In around 170 pages, Solow introduces the different kinds of proof techniques under the sun (e.g., the forward-backward method, the construction method, the choose method), and provides a framework of thinking processes (e.g., generalization, unification, dual representation, abstraction) by which proofs can be understood and constructed.
With short paragraphs and clear formatting throughout the text, this conversational-style guide on proofs delivers on an accessibility scale that is unmatched by most proof-based math textbooks in the market. Along with 15 video lectures and a solution manual (to selected questions) on the book's website, there is definitely more than enough resources to master mathematical proofs even in one's own spare time. If you're new to the world of higher mathematics, this is probably one of those foundational books that you don't want to miss.
How to Prove It: A Structured Approach (2nd Edition)
— by Daniel Velleman
A standalone classic among higher mathematics enthusiasts, How to Prove It: A Structured Approach takes a reader from not knowing what a proof is, to becoming proficient in crafting it. It covers the basics of logic and set theory, and gradually takes the readers to the various techniques in proving different kinds of statements (e.g., mathematical induction). It also elaborates a bit on some applications of the proof techniques through the topics of relation, function and infinite set.— before finishing off with a proof of the infamous Bernstein-Schroder theorem.
Among the transition-to-higher-mathematics books, this one seems to rise above the cloud due to its accessibility, extensive proof illustrations and gently-crafted exercises. Indeed, we get the impression throughout the book that the author is eager to — in a conversational tone — take us through every step of a proof sequentially, while giving away key advice in areas where it's especially needed. With just a bit of background in high-school algebra, this 300-page classic will help you better understand — perhaps more than most books on the same topic — what mathematicians do and how they think. If anything, this is probably your stepping stone towards almost all other topics in higher mathematics!
Abstract Algebra and (Theory-Based) Linear Algebra.
Elements of Modern Algebra (8th Edition)
— by Linda Gilbert and Jimmie Gilbert
Being one of the rare books in abstract algebra to have gone through an eighth iteration, Elements of Modern Algebra is written with first-year math undergraduate students in mind. It takes them through the materials from the very basics, to just a bit more than what a typical semester of abstract algebra would cover.
Well-structured with contents presented in a clean format, the authors introduce the readers to fundamental concepts such as set and integers, all the way to abstract structures such as group, ring and field with all the standard theorems in between. Coupled with straight-to-the-point presentations and a surprising amount of examples both within and after each section, at around 400 pages of meaty content, this abstract algebra textbook is excellent for both self-study and reference purpose.
Linear Algebra Done Right (3rd Edition)
— by Sheldon Axler
Intended for students who finished their first course in applied linear algebra, Linear Algebra Done Right has the single aim of covering materials of a second linear algebra course via a non-standard route. That is, it begins by introducing its readers to the concepts such as vector space, basis and linear map, and proceeds to develop the theory in a clean fashion without resorting to the determinants — which is only introduced in the last chapter for the sake of completeness. Among the topics included are the Fundamental Theorem of Linear Map, isomorphic vector space, duality, polynomials, eigenspace, inner product space and the Spectral Theorem.
Published almost a decade after its 2nd edition, the 3rd edition represents a major improvement in terms of exercise-set expansion (~300 new exercises), formatting and visual appeal, At around 250 pages, the scope of this compact textbook nevertheless goes beyond what can be covered in one single semester, and while it’s not geared towards students of applied linear algebra, its alternative theoretical treatment of linear algebra makes it a very exciting read — and a great companion for self-study.
Number Theory, Combinatorics, Graph Theory... Very discrete.
Discrete Mathematics with Applications (4th Edition)
— by Susanna Epp
Primarily written for students in computer science, Discrete Mathematics With Applications is a 800-page-comprehensive, introductory-level survey on what discrete mathematics has to offer. It will guide you from the basics of proof-based mathematics (e.g., set, logic, proof techniques) to the higher mathematical concepts (e.g., relations, sequences, recurrence relation, cardinality, counting techniques, graphs) — before moving onto the more computer-science-oriented topics (e.g., logic circuit, algorithm efficiency, finite-state automata).
While admittedly a huge textbook, it is by no means intimidating. Instead, one can readily see that it`s formatted professionally according to modern standards — with tons of accessible proses, solved examples and exercise sets in between. In addition, the coverage is neither pedantic nor shallow, and has a level of lucidity that makes it an interesting read. Definitely an excellent resource for those who are interested to pick up a bit of discrete mathematics on their own.
Discrete Mathematics and Its Applications (7th Edition)
— by Kenneth Rosen
As the de-facto bible of on the subject, Discrete Mathematics and Its Applications introduces us from the very basics (e.g., logic, set, functions) to a plethora of "sci-fi" topics such as algorithmic complexity, cryptography, recursive structure. spanning tree, boolean algebra, logic circuit and the so-called Turing machine. While primarily written for students in computer science, the coverage of this book on topics such as structural induction, set cardinality, modular arithmetic, recurrence relation, generating function, inclusion-exclusion principle and Bayes' Theorem makes it an equally interesting read for higher-mathematics enthusiasts as well.
Unlike some bland, black-and-white textbooks students of pure mathematics get so used to, this application-driven, 1000-page textbook is actually relatively accessible, and is filled with great visuals and more problems an average reader would solve, making it an excellent resource for both reference and self-study.
Real Analysis, Complex Analysis, Numerical Analysis — Among other "Continuous Math".
Understanding Analysis (2nd Edition)
— by Stephen Abbott
As the title suggests, Understanding Analysis is written to help students better understand single-variable real analysis (as taught at the first-year undergrad level). It covers essential topics such as limit, sequence, series, point-set topology, continuity, differentiation, uniform convergence and integration, and includes special topics such as Cantor Set, Fourier Series and Weierstrass Approximation Theorem — for the immensely curious.
Despite being a textbook on a highly theoretical subject, this book actually reads more like a novel with a geeky twist. Packed with clear, accessible expositions, engaging examples and historical motivation throughout the text, Understanding Analysis will guide the readers through the actual thought process of solving analysis problems — instead of merely showcasing the proofs like many of the texts do. It also nurtures a unifying view of real analysis without sacrificing rigor, and as such prepares the reader with a solid foundation for the more advanced work in mathematical analysis For anyone looking to pick up real analysis on their own, this is an outstanding resource on the topic that is hard to beat.
Schaum's Outline of Advanced Calculus (3rd Edition)
— by Robert Wrede and Murray Spiegel
Looking for an abnormal amount of calculus and analysis questions both with and without answers? Then look no further. With 1370 solved problems (plus other supplemental problems), Schaum's Outline of Advanced Calculus covers pretty much every major topic you need to know from sequences, series, derivatives, integrals to partial derivatives, multiple integrals, fourier series and complex variables in around 400 pages.
While primarily designed for applied-math students looking for a serious boost in calculus skills, this book contains so many interesting (and sometimes proof-based) problems that it would be unfair to dub it as a resource exclusively reserved for the advanced calculus test-takers. True, the shallowness of the outlines means that it's by no means a replacement for a proper textbook, but for those who are aching find more advanced calculus / analysis problems to solve, this will keep their brain busy for a while. 🙂
Complex Variables With Applications (3rd Edition)
— by David Wunsch
A rare textbook on the subject that stands the test of time, Complex Variables With Applications demystifies complex analysis through a remarkably clear writing and tons of examples and problems. Primarily geared towards students of engineering and other applied sciences, it covers the usual topics of complex transcendental functions, analyticity, contour integration, Cauchy Integral Formula, Laurent series, residue and conformal mapping (among others) through a series of accessible discussions, solved examples and problem sets scattered around the text.
While most textbooks on complex analysis tend to be a bit dry and proof-driven, this one is not: it cuts through the noises by building the theory intuitively without getting too bogged down with rigour, and by adopting a modern formatting similar to that of a textbook on applied calculus. As a result, it caters to both the mathematicians and the engineers/physicists within us. Definitely one of the most well-written textbooks on the subject — for those who want to pick up complex analysis on their own.
Numerical Analysis (2nd Edition)
— by Timothy Sauer
A modernly-designed textbook which capitalizes on mathematical intuition and insights, Numerical Analysis exemplifies a clearly written, well-organized resource on the subject. It covers the usual topics of numerical root-finding, interpolation, system of equations, least-square methods, numerical differentiation, numerical integration and numerical ODE / PDE methods (among others), and has plenty of examples where the author illustrates how a problem can be solved either algebraically or through the computing software MATLAB.
Integrating clear expositions with examples and relevant MATLAB codes throughout the text, this book also strikes a good balance between theory and applications by including just enough derivations to ensure that the important theoretical gaps are filled, but not so much as to distract the students from understanding the big picture. It's an excellent resource for math students interested in exploring the theoretical underpinning of the various numerical techniques, and a great reference for programmers looking to code their computations into MATLAB as well.
SET THEORY & MATHEMATICAL LOGIC
First-Order Logic, Proof Systems, Meta-Theorems — among other foundation stuffs.
A Concise Introduction to Logic (12th Edition)
— by Patrick Hurley
More of a textbook on critical thinking and symbolic logic, A Concise Introduction to Logic is a mostly-non-technical, comprehensive survey into what different kinds of logic has to offer in our daily routines. While primarily written for philosophy students, the Part II portion of the book does provide an excellent coverage in entry-level topics surrounding the so-called formal logic. These includes the chapters on propositional logic, predicate logic, natural deduction and its associated rules of inference.
Though not exactly a math textbook per se, its elegant writing and elaborated formating/design really does seem to make the pursuit of higher mathematics more interesting and meaningful. Furthermore, the sections on inductive and legal reasoning are a rather refreshing read, and the chapters on probabilistic and statistical reasoning actually cover a bit of math as is taught in an introductory course in statistics!
Language, Proof and Logic (2nd Edition)
— by Barker-Plummer et al.
If you're looking for a comprehensive-yet-accessible textbook on First-Order Logic (FOL), then Language, Proof and Logic should have you covered. While primarily geared towards non-mathematics students, it's nevertheless a complete, application-driven textbook on FOL at both the introductory and the intermediate level, with the first part of the book focused on the basics of Propositional / First-Order Logic (e.g., translation, Fitch, truth table),and the second part on the important meta-theoretical results of FOL (e.g.,axiomatic set theory, mathematical induction, Löwenheim-Skolem Theorem, Compactness Theorem, Gödel's Incompleteness Theorem).
Unlike most proof-driven textbooks on mathematical logic, this one exposes you to the meta-theoretical intuition and have you fill in the theoretical details yourself, and while many of the exercises in the book rely on software developed by the authors themselves, the book remains all the same a valuable resource towards a more rigorous pursuit in First-Order Logic.
GEOMETRY & TOPOLOGY
Platonic Solids, Fractals, Manifolds along with other highly-visual math.
Introduction to Geometry (2nd Edition)
— by Richard Rusczyk
Intended for high-performing pre-college students, Introduction to Geometry offers a no-fuss, complete course in introductory geometry that is unrivaled by those offered in most public and private schools. Written by a former USA Mathematical Olympiad winner, it offers in-depth, rigorous coverage on topics such as angle, similar / congruent triangles, quadrilaterals, power of a point, polygons, circles, 3D geometry and transformation — thereby preparing the students towards the more serious pursuit in higher mathematics.
With short explanations and a heavy emphasis on problem solving (e.g., ~900 problems from trivial to challenging), this 500-page textbook will get your brain thinking. Indeed, as long as you are a math enthusiast who has not yet taken a proper course in geometry, this book will stretch your geometrical reasoning skill a bit and get those visual cortex neurons firing like they're crazy!
The awesome document-preparation system for Mathematical Typesetting and Scientific Publishing.
LaTeX Beginner's Guide
— by Stefan Kottwitz
Authored by the LaTeX celebrity Stefan Kottwitz (who blogs at TeXblog.net), LaTeX Beginner’s Guide does a phenomenal job in laying out what LaTeX has to offer — without getting bogged down by what many learners would perceive as unnecessary verbiage. At around 300 pages, it offers an optimal coverage on LaTeX for entry-level users both in terms breadth and depth, all of the while being one of the most up-to-date, teach-by-example LaTeX book on the market.
At the most fundamental level, this guide covers a wide range of topics that an average LaTeXer needs to know. These includes LaTeX distribution installation, text formatting, page layout design, list structure customization, float environments, cross-references, math typesetting, font adjustment, and hyperlink customization, so that by the end of book, a typical reader would be able to create a professionally typeset document on their own — be it a simple article, a 10-page report, a doctoral thesis or a flowchart diagram.
More Math Into LaTeX (5th Edition)
— by George Grätzer
If you're looking to explore further into the intricacy of LaTeX, this is it. As a manual whose history spans two decades, More Math Into LaTeX introduces and elaborates on the nuts and bolts of LaTeX for those who are in need of some serious mathematical typesetting. As a standalone 500-page manual, it covers the absolute fundamentals (e.g., symbols, formatting, preamble) in two chapters, along with the nitty-gritty of a document (e.g., font, spacing, layout), a math article (e.g., math expressions, proclamations, proofs), a presentation (i.e., beamer), a vector graphics (e.g., TikZ) and a book (e.g., bibliography, index) — and doing so with a level of details on customization that is unmatched by most of the LaTeX guides currently out there.