# Recommended Books on Higher Mathematics

Epic textbooks, manuals 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 step-by-step guide on how the mathemagician Arthur Benjamin, a mathematician and a professional magician himself, manages to pull off gigantic arithmetic calculations off of top of his head through a combination of smart tricks and mnemonic techniques.

#### COVERED TOPICS

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

### A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

— By Paul Lockhart

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.

#### FEATURED THESES

• 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.

### Thinking Mathematically (2nd Edition)

— By John Mason, Leone Burton and Kaye Stacey

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.

#### NOTABLE FEATURES

• 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

### 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 a quasi-literary style by two prominent 20th-century mathematicians. It encourages the use of problem solving to develop new insight and understanding, and takes its readers gently to the different branches of higher mathematics.

#### COVERED TOPICS

• Elementary number theory
• Analytic geometry
• Topology
• Calculus

### The Mathematical Experience (Study Edition)

— By Philip Davis and Reuben Hersh

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.

###### FEATURED TOPICS
• 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

## AppliedCollege Mathematics

College algebra,  calculus, linear algebra, differential equations and the like.

### Introduction to Linear Algebra (5th Edition)

— By Gilbert Strang

A 500-page linear algebra classic by one of most sought-after educator in the field. It features conversational-style narratives, review sections and challenge problems, all of the while maintaining a focus on geometric intuition and real-life applications.

#### FEATURED TOPICS

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

### Calculus (8th Edition)

— By James Stewart

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.

#### FEATURED TOPICS

• Derivatives
• Derivative applications
• Integration techniques
• Partial derivatives
• Multiple integrals
• Lagrange multipliers
• Divergence / Curl
• Applications of integrals
• Differential equations

## Probability& Statistics

Discrete/continuous distributions,  hypothesis testing, analysis of variance, Bayesian inference and more.

### Statistics (13th Edition)

— By James McClave and Terry Sincich

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.

#### FEATURED TOPICS

• Descriptive statistics
• Basic probability
• Probability distributions
• Confidence interval
• Hypothesis testing
• Analysis of variance
• Linear regression
• Categorical data analysis
• Non-parametric statistics

### Mathematical Statistics with Applications (7th Edition)

— By Dennis Wackerly, William Mendenhall and Richard Scheaffer

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.

#### FEATURED 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 toProof

Basic logic,  proof techniques and their applications in higher mathematics.

### 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 accessible conversational style, it also features 15 video lectures and the solution manual on its student companion website.

#### FEATURED TOPICS

• Forward-backward method
• Construction method
• Choose method
• Generalisation
• 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 one proficiently. Notable features include conversational-style instructions, extensive proof illustrations and gently-crafted exercise sets. Accessible to high school students and beyond.

#### FEATURED TOPICS

• Basic logic
• Elementary set theory
• Proof techniques
• Mathematical induction
• Relation
• Function
• Infinite sets
• Schröder–Bernstein theorem

## Proof-Based Algebra

Abstract algebra,  proof-based linear algebra and other goodness.

### Elements of Modern Algebra (8th Edition)

— By Linda Gilbert

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.

#### FEATURED TOPICS

• Set theory
• Integer
• Group
• Ring
• Integral domain
• Field
• Complex number
• Polynomial

### Linear Algebra Done Right (3rd Edition)

— By Sheldon Axler

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.

#### FEATURED TOPICS

• Vector space
• Basis
• Linear mapping
• Polynomial
• Eigenspace
• Inner product space
• Duality
• Spectral theorem

## Geometry & Topology

Platonic solids,  fractals, manifolds and other highly-visual math.

### Introduction to Geometry (2nd Edition)

— By Richard Rusczyk

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.

#### FEATURED TOPICS

• Angle
• Similar / Congruent triangles
• Power of a point
• Polygon
• Circle
• 3D geometry
• Transformations

## Discrete Mathematics

Number theory,  combinatorics, graph theory... Very discrete.

### Discrete Mathematics with Applications (5th Edition)

— By Suzanna Epp

A modern, ultra-comprehensive textbook on much of what discrete mathematics has to offer, featuring clear, accessible explanations along with numerous solved examples and exercise sets.

#### FEATURED TOPICS

• Set Theory
• Logic
• Proof techniques
• Relation
• Sequence
• Recurrence relation
• Cardinality
• Counting techniques
• Graph
• Logic circuit
• Algorithmic efficiency
• Finite-state automata

### Discrete Mathematics and its Applications (8th Edition)

— By Kenneth Rosen

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

#### FEATURED TOPICS

• Set and 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

## ContinuousMathematics

Real analysis,  complex analysis, numerical analysis — among other "continuous math".

### Understanding Analysis (2nd Edition)

— By Stephen Abbott

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.

#### FEATURED TOPICS

• Limit
• Sequence
• Series
• Point-set topology
• Continuity
• Derivative
• Uniform convergence
• Integral

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

— By Robert Wrede and Murray Spiegel

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.

#### FEATURED TOPICS

• Sequence and series
• Function and limit
• Derivative
• Continuity
• Integral
• Partial derivative
• Multiple integral
• Line integral
• Surface integral
• Integral theorems
• Improper integral
• Fourier series
• Gamma / Beta functions
• Complex-variable function

### Complex Variables with Applications (3rd Edition)

— By David Wunsch

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.

#### FEATURED TOPICS

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

### Numerical Analysis (3rd Edition)

— By Timothy Sauer

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.

#### FEATURED TOPICS

• Numerical root-finding
• Interpolation
• System of linear equations
• Least-square methods
• Numerical differentiation
• Numerical integration
• Numerical ODE methods
• Numerical PDE methods

## Set Theory & Mathematical Logic

First-order logic,  proof systems, meta-theory and other foundation stuffs.

### A Concise Introduction to Logic (12th Edition)

— By Patrick Hurley

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

#### FEATURED TOPICS

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

### Language, Proof and Logic (2nd Edition)

— By David 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).

#### FEATURED TOPICS

• Formal / informal translations
• Truth table
• Fitch system
• Axiomatic set theory
• Mathematical induction
• Löwenheim-Skolem Theorem
• Compactness theorem
• Gödel's Incompleteness Theorems

## LaTeX

The awesome document-preparation system for typesetting math and other scientific goodness.

### LaTeX Beginner's Guide

— By Stefan Kottwitz

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.

#### FEATURED TOPICS

• Installing LaTeX distribution
• Formatting text
• Designing page layout
• Customizing lists
• Float environments
• Cross-reference
• Typesetting math

### More Math Into LaTeX (5th Edition)

— By George Grätzer

A most up-to-date (ours aside of course), comprehensive LaTeX manual that goes into the nuts and bolts of mathematical typesetting. Two decades in the making, with some notable features being the amount of examples (e.g., code vs. its output) and the dos and don'ts.

#### FEATURED TOPICS

• Symbols
• Preamble
• Font
• Spacing
• Page layout
• Math expressions
• Proclamation
• Proof
• Beamer
• Tikz
• Bibliography
• Index