Recommended Books on Higher Mathematics

In this 140-page essay, Paul Lockhart decries the deplorable state of the traditional U.S. K-12 math educational system, and passionately calls for the need to teach mathematics as a creative endeavour and an artform.

• Mathematics is primarily an artform, not a tool.
• U.S. K-12 math education has lost touch with what mathematics is, and what mathematicians do. 
• By teaching math without the whys and the contexts, mathematics is in effect reduced to an empty shell.
• The essence of mathematics is abstraction — not the memorization or applications of facts and figures.

A book dedicated to taking one from the mere act of exercise solving to the core of mathematics — mathematical thinking. It showcases the different mental processes for dissecting a problem, and discusses how one can go from a master of tricks into an actual, standalone mathematical thinker.

• A thorough treatment on mathematical investigation
• A mental framework for dealing with intellectual plateaus
• Illustrative, beginner-level problems (with guidance or hints)
• An integrated approach to problem solving and the associated mental processes

A collection of humanistic, semi-technical essays about some of most fundamental questions and philosophical issues in mathematics, along with the historical development of the various subtopics and peripheral topics in mathematics. Some experience with undergrad math is recommended.

• Mathematical creation vs. discovery
• Mathematical beauty and elegance
• Math as an intermediate reality between the physical and the subjective
• Intro into Fourier analysis, non-Cantorian set theory, finite group theory, non-standard calculus and other topics

An accessible, ultra-comprehensive textbook on almost everything calculus has to offer. 30 years in the making, and strikes a balance between applicability and rigor. Each section comes with a motivating discussion, followed by several solved examples and many problem sets afterwards.

A comprehensive, time-tested, non-calculus-based textbook on much of what applied statistics has to offer. It's accessible to almost anyone with a background in high-school math, and features a wide range of explanations, examples, sample problems and exercises to choose from.

A solid, calculus-based textbook on probability and statistics from the mathematical point of view. It strikes a good balance between theory and applications, while featuring a wide range of problems to further solidify the mechanics and the understanding of concepts and theorems.

A 300-page classic which takes a reader from not knowing what a proof is, to being able to carry out one proficiently. Notable features include conversational-style instructions, extensive proof illustrations and gently-crafted exercise sets. Accessible to high school students and beyond.

An accessible, time-tested textbook on first-year abstract algebra, featuring neat formatting, clear presentation, and a surprising amount of examples both within and after a section. Ideal for both self-study and reference purpose.

A unique 250-page textbook covering second-year linear algebra through a non-standard route (i.e., without resorting to determinant — which is only covered in the last chapter of the book). The 3rd edition represents a major improvement in terms of both formatting and exercise set expansion.

A standalone, 500-page textbook on introductory geometry — written by a former winner of USA Mathematical Olympiad. Aside from the concise explanations, it also features 900+ problems which are Useful for developing one's geometric intuition and spatial reasoning skill.

The de-facto bible in discrete mathematics. Accessible, application-driven and rich in visuals and problem sets.

A solid and accessible textbook that provides a unifying view of first-year real analysis. Each section starts with some historical motivation behind the topic, followed by many engaging examples and thought processes through which a problem can be solved.

A 400-page review guide with 1370+ solved and unsolved problems in anything advanced calculus. While not suitable as a first text, it's nonetheless very useful for people who want to hone in their advanced calculus skill in a very efficient and cost-effective manner.

A unique textbook which demystifies complex analysis through a series of clear presentations, accessible discussions,  solved examples and problem sets. Ideal for students in engineering and other applied sciences.

A neatly-formatted, well-organized textbook on the various aspects of numerical analysis. It strikes a fine balance between theory and applicability, and features clear expositions, solved examples and relevant MATLAB codes throughout the text.

A comprehensive, accessible textbook on first-order logic and its associated meta-theory. It's based on the proof system Fitch and covers logic at both introductory and intermediate levels (for students majoring in philosophy).

A standalone, 300-page guide that teaches introductory LaTeX through short explanations and examples. It strikes a good balance between breadth and depth, and ensures that by the end of the book, the reader will be able to produce a professionally-typeset document on their own.