Recommended Math Books

Epic textbooksmanuals and guides for the mathematically-inclined, along with other less mathy masterpieces here and there!

Recreational Mathematics

Arithmetics, math puzzles, mathematical initiation and all that goodness.

Secrets of Mental Math: The Mathematgician's Guide to Lightning Calculation and Amazing Math Tricks

— By Arthur Benjamin and Michael Shermer

A how-to book filled with step-by-step instructions as to how the mathemagician Arthur Benjamin — being both a mathematician and a professional magician himself — manages to pull off gigantic calculation off of top of the head. Highly stimulating and practical.

COVERED TOPICS
  • Addition
  • Substraction
  • Multiplication
  • Division
  • Memorizing large numbers
  • Calendar-related calculation
  • Guess-estimation
  • Vedic division

What is Mathematics? An Elementary Approach to Ideas and Methods (2nd Edition)

— By Richard Courant and Herbert Robbins

An all-time classic in mathematical initiation written in quasi-literary style by two prominent 20th-century mathematicians. It encourages problem solving as a means of developing new insight and genuine comprehension in higher mathematics.

COVERED TOPICS
  • Elementary number theory
  • Analtytic geometry
  • Topology
  • Calculus
What is Mathematics? by Richard Courant and Herbert Robbins

Applied College Mathematics

College Algebra calculus, linear algebra, differential equations and the like.
Introduction to Linear Algebra (5th Edition) by Gilbert Strang

Introduction to Linear Algebra (5th Edition)

— By Gilbert Strang

The culmination of decades of teaching by one of the world's most prolific educator in linear algebra. In ~500 pages. It features conversational-style narratives coupled with review sections and challenge problems, while maintaining a focus on geometric intuition and real-life applications.

COVERED TOPICS
  • Matrices
  • System of linear equations
  • Vector space
  • Orthogonality
  • Determinant
  • Eigenvector
  • Singular value decomposition
  • Linear transformation

Calculus (8th Edition)

— By James Stewart

Accessible, ultra-comprehensive textbook on almost everything calculus has to offer. 30 years in the making and focuses on conceptual understanding while balancing the need for readability and rigor. Each section comes with a motivating discussion, followed by several solved examples and tons of problem sets afterwards.

COVERED TOPICS
  • Derivatives
  • Derivative applications
  • Integration Techniques
  • Partial derivatives
  • Multiple integrals
  • Lagrange multipliers
  • Divergence / Curl
  • Applications of integrals
  • Differential equations
Calculus (8th Edition) by James Stewart

Probability & Statistics

Discrete/continuous distributions hypothesis testinganalysis of variance, Bayesian inference and more.
Mathematical Statistics With Applications by Wackerly et al.

Mathematical Statistics with Applications (7th Edition)

— By Dennis Wackerly, William Mendenhall and Richard Scheaffer

A solid textbook on probability and statistics from the mathematical standpoint. It requires calculus and strikes a fine balance between applications and rigor.

COVERED TOPICS
  • Probability distributions
  • Moment-generating functions
  • Law of large numbers
  • Central Limit Theorem
  • Methods of estimation
  • Analysis of variance
  • Non-parametric statistics
  • Bayesian inference

Introduction to Proof

Basic logic proof techniques and their applications in higher mathematics.
How to Read and Do Proofs (6th Edition) by Daniel Solow

How to Read and Do Proof: An Introduction to Mathematical Thought Processes (6th Edition)

— By Daniel Solow

A unique, 170-page book with a systematic approach in identifying, categorizing and explaining a mathematical proof and its making. Written in conversational style with clarity and accessibility in mind. Also featured are 15 video lectures and solution manual on the book's website.

COVERED PROOF METHODS
  • Forward-Backward Method
  • Construction Method
  • Choose Method
COVERED FRAMEWORKS
  • Generalization
  • Unification
  • Dual Representation
  • Abstraction

How to Prove It: A Structured Approach (2nd Edition)

— By Daniel Velleman

A 300-page classic which takes a reader from not knowing what a proof is, to being able to carry out a proof proficiently. Notable features include conversational-style instructions, extensive proof illustrations and gently-crafted exercise sets. Highly accessible and does not require more than high-school algebra. 

COVERED TOPICS
  • Basic logic
  • Elementary set theory
  • Proof techniques
  • Mathematical induction
  • Relation
  • Function
  • Infinite set
  • Schröder–Bernstein theorem
How to Prove It: A Structured Approach (2nd Edition) by Daniel Velleman

Proof-Based Algebra

Abstract algebra proof-based linear algebra and other goodness.

Element of Modern Algebra (8th Edition)

— By Linda Gilbert and Jimmie Gilbert

A solid, accessible, 400-page textbook on first-year abstract algebra. Clearly formatted with straight-to-the-point presentation and a surprising amount of examples both within and after a section. Ideal for both self-study and reference purpose.

COVERED TOPICS
  • Set
  • Integer
  • Group
  • Ring
  • Integral domain
  • Field
  • Complex number
  • Polynomial

Linear Algebra Done Right (3rd Edition)

— By Sheldon Axler

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

COVERED TOPICS
  • Vector space
  • Basis
  • Linear mapping
  • Polynomial
  • Eigenspace
  • Inner product space
  • Duality
  • Spectral Theorem
Linear Algebra Done Right (3rd Edition) by Sheldon Axler

Discrete Mathematics

Number theory combinatorics, graph theory... Very discrete.
Discrete Mathematics With Applications (4th Edition) by Susanna Epp

Discrete Mathematics with Applications (4th Edition)

— By Susanna Epp

A 800-page survey as to what discrete mathematics has to offer. It's written in clear, accessible prose and include tons of  solved examples and exercise sets.

COVERED TOPICS
  • Set Theory
  • Logic
  • Proof techniques
  • Relation
  • Sequence
  • Recurrence relations
  • Cardinality
  • Counting techniques
  • Graphs
  • Logic circuits
  • Algorithmic efficiency
  • Finite-state automata

Discrete Mathematics and its Applications (7th Edition)

— By Kenneth Rosen

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

COVERED TOPICS
  • Set / Logic
  • Function
  • Algorithmic complexity
  • Cryptography
  • Recursive structure
  • Spanning tree
  • Boolean algebra
  • Logic circuit
  • Turing machine
  • Structural induction
  • Cardinality
  • Modular arithmetic
  • Recurrent relation
  • Generating function
  • Counting techniques
Discrete Mathematics and Its Applications (7th Edition) by Kenneth Rosen

Continuous Mathematics

Real Analysis complex analysis, numerical analysis — among other "continuous math".
Understanding Analysis (2nd Edition) by Stephen Abbott

Understanding Analysis (2nd Edition)

— By Stephen Abbott

A solid-yet-accessible textbook which presents an unifying view of first-year real analysis, where each section starts with some historical motivation and contains many engaging examples and the thought process through which an analysis problem can be solved.

COVERED TOPICS
  • Limit
  • Sequences
  • Series
  • Point-set topology
  • Continuity
  • Derivatives
  • Uniform convergence
  • Integrals

Schaum's Outline of Advanced Calculus  (3rd Edition)

— By Robert Wrede and  Murray Spiegel

A 400-page review guide covering 1370+ solved and unsolved problems in anything advanced calculus. Particularly useful for people in need of boosting their advanced  calculus skill in a short amount of time.

COVERED TOPICS
  • Sequence
  • Series
  • Derivatives
  • Integrals
  • Partial derivatives
  • Multiple integrals
  • Fourier series
  • Complex variables
Schaum's Outlines to Advanced Calculus — 3rd Edition
Complex Variables With Applications (3rd Edition) by David Wunsch

Complex Variables with Applications (3rd Edition)

— By David Wunsch

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

COVERED TOPICS
  • Complex transcendental functions
  • Analyticity
  • Contour integration
  • Cauchy integral formula
  • Laurent series
  • Residue mapping
  • Conformal mapping

Numerical Analysis  (2nd Edition)

— By Timothy Sauer

A clearly-written, well-organized textbook on numerical analysis which strikes a fine balance between rigor and applicability. It features clear expositions, solved examples and relevant MATLAB codes all throughout the text.

COVERED TOPICS
  • Numerical root-finding
  • Interpolation
  • System of equations
  • Least-square methods
  • Numerical differentiation
  • Numerical integration
  • Numerical ODE methods
  • Numerical PDE methods
Numerical Analysis (2nd Edition) by Timothy Sauer

Set Theory & Mathematical Logic

First-order logic proof systems, meta-theory and other foundation stuffs.
A Concise Introduction to Logic (12 Edition) by Patrick Hurley

A Concise Introduction to Logic (12th Edition)

— By Patrick Hurley

A comprehensive, mostly-non-technical survey on how critical thinking and different forms of logic plays out in our daily routine, with Part II of the book dedicated to the development of formal logic and other math-based reasoning.

COVERED TOPICS
  • Propositional logic
  • Predicate logic
  • Natural deduction
  • Rules of inference
  • Inductive reasoning
  • Legal reasoning
  • Probabilistic reasoning
  • Statistical reasoning

Language, Proof, Logic  (2nd Edition)

— By Dave Barker-Plummer, Jon Barwise and John Etchemendy

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.

COVERED TOPICS
  • Translation
  • Turth tables
  • Fitch
  • Axiomatic set theory
  • Mathematical induction
  • Löwenheim-Skolem Theorem
  • Compactness theorem
  • Gödel's Incompleteness Theorems
Language, Proof and Logic (2nd Edition) By Barker-Plummer et al.

Geometry & Topology

Platonic solids fractals, manifolds along with other highly-visual math.
The Art of Problem Solving (AOPS) — Introduction to Geometry (2nd Edition) by Richard Rusczyk

Introduction to Geometry (2nd Edition)

— By Richard Rusczyk

A standalone, 500-page textbook in introductory geometry written by a former winnder of USA Mathematical Olympiad. It features concise explanations along with 900+ problems from trivial to very challenging. Useful for developing one's geometric intuition and spatial reasoning skill.

COVERED TOPICS
  • Angle
  • Similar/congruent triangles
  • Quadrilaterals
  • Power of a point
  • Polygons
  • Circles
  • 3D geometry
  • Transformation

LaTeX

The awesome document-preparation system for mathematical typesetting and scientific publishing.
LaTeX Beginner's Guide by Stefan Kottwitz

LaTeX Beginner's Guide

— By Stefan Kottwitz

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

COVERED TOPICS
  • Installing LaTeX distribution
  • Formatting text
  • Designing page layout
  • Customizing lists
  • Float environments
  • Cross-reference
  • Typesetting math
  • Adjusting font
  • Hyperlink

More Math Into LaTeX  (5th Edition)

— By George Grätzer

An up-to-date, standalone 500-page LaTeX manual that goes into the nuts and bolts of mathematical typesetting. Two decades in the making and contains a ton of examples (e.g., code vs. its output) and the do-and-don't.

COVERED TOPICS
  • Symbols
  • Preamble
  • Font
  • Spacing
  • Page layout
  • Math expressions
  • Proclamation
  • Proof
  • Beamer
  • Tikz
  • Bibliography
  • Index
More Math Into LaTeX (5th Edition) by George Gratzer