Personal Mathematics Books Recommendation
This is a list of books that I personally prefer in the study of tertiary-level mathematics, especially for self-study and further reading.
Discrete Mathematics
If you cannot handle the content of the books listed in this section, then university mathematics may not be for you (serious). I've learnt the hard way myself, so please do consider carefully.
| Book title and author | Brief review |
|---|---|
| Proofs and Fundamentals: A First Course in Abstract Mathematics by Ethan D. Bloch | Comprehensive and rigorous, yet still accessible and beginner-friendly; covers a wide range of essential topics in fundamental discrete mathematics. |
| Book of Proof, by Richard Hammack | Beginner-friendly; covers fundamental topics in discrete mathematics, plus an introductory exposition to advanced set theory. |
| Mathematical Reasoning: Writing and Proof, by Ted Sundstrom | Extensive and accessible; puts great focus on proper mathematical proof writing; relatively slower in pacing. |
Linear Algebra
| Book title and author | Brief review |
|---|---|
| Introduction to Linear Algebra, by Gilbert Strang | Comprehensive and rigorous; includes extra interesting topics that involve application of linear algebra, such as Fourier series. |
| Linear Algebra Done Right, by Sheldon Axler | A popular and enlightening linear algebra textbook that explains the topics in a more abstract and generalised manner, rather than a computation heavy approach. |
Probability and Measure Theory
| Book title and author | Brief review |
|---|---|
| Introduction to Probability, by Joseph K. Blitzstein and Jessica Hwang | Features a lot of examples that aid in understanding; adopts a more 'data science oriented' or 'applied' approach to the subject; contains tutorials to the programming language R. |
| An Introduction to Measure Theory, by Terence Tao | A graduate-level textbook that discusses measure theory, an axiomatic generalisation of the concept of probability. Although it contains solid material, it is written in a more traditional style, and thus may be a challenging read to some. |
| Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias M. Stein and Rami Shakarchi | The third instalment of the highly acclaimed mathematics text tetralogy Princeton Lectures in Analysis. Explanations are clear, rigorous and adequately linked to established fundamental concepts. |
Multivariable Calculus
| Book title and author | Brief review |
|---|---|
| Multivariable Calculus, by Don Shimamoto | Detailed explanation of standard topics in multivariable calculus with lots of clear illustrations; also contains an introductory segment of the more advanced concept of differential forms. |
info
Although this technically is not a textbook, the Calculus III notes on the website 'Paul's Online Math Notes' is also an excellent material for studying multivariable calculus.
Real/Mathematical Analysis and Topology
| Book title and author | Brief review |
|---|---|
| Introduction to Real Analysis, by Robert G. Bartle and Donald R. Sherbert | An accessible yet still rigorous treatment of the fundamentals of real analysis; also features a basic exposition to the subject of metric spaces and topology. |
| Principles of Mathematical Analysis, by Walter Rudin | An enduring classic in the study of mathematical analysis; adopts a more traditional approach in the presentation of topics; a more rigorous and challenging text. |
| Topology, by James Munkres | Comprehensive, extensive and rigorous; difficulty increases with the progression of chapters, but still manages to handle the fundamentals in a beginner-friendly manner. |
Complex Analysis
| Book title and author | Brief review |
|---|---|
| Complex Analysis, by Elias M. Stein and Rami Shakarchi | The second volume of the famous mathematics text series Princeton Lectures in Analysis; covers fundamental concepts in complex analysis which are explained in an illustrative manner. |
| Functions of One Complex Variable, by John B. Conway1 | An extensive and rigorous treatment of graduate-level complex analysis topics. |
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Not technically a book, but Complex Analysis: A Visual and Interactive Introduction created by Juan Carlos Ponce Campuzano is an excellent online resource featuring interactive visuals.
Functional Analysis
| Book title and author | Brief review |
|---|---|
| Introductory Functional Analysis with Applications, by Erwin Kreyszig | An accessible presentation of fundamental concepts in functional analysis, featuring related topics such as Fourier series and approximation theory. |
| An Invitation to Operator Theory, by Y. A. Abramovich and C. D. Aliprantis | A graduate-level textbook on the study of operators and related topics in functional analysis; can be regarded as an extension of linear algebra. |
| Functional Analysis: Introduction to Further Topics in Analysis, by Elias M. Stein and Rami Shakarchi | The fourth book of the mathematics text tetralogy Princeton Lectures in Analysis. A more rigorous and advanced treatment of topics in functional analysis. |
Graph/Hypergraph Theory
| Book title and author | Brief review |
|---|---|
| Graph Theory, by Reinhard Diestel | A rigorous treatment of the subject of graph theory, with connections to some topics in abstract algebra, particularly morphisms. |
| Hypergraph Theory: An Introduction, by Alain Bretto | A comprehensive text that discusses the topic of hypergraphs, which is a generalisation of graph theory. |
note
A more applied, 'computer science oriented' treatment of graph theory can be found in Chapter 4 of the book Competitive Programming 2, written by Steven Halim, Felix Halim and Suhendry Effendy.
Abstract Algebra
| Book title and author | Brief review |
|---|---|
| Abstract Algebra, by I. N. Herstein | Beginner-friendly yet still rigorous; topics are presented in a logical sequence. |
| Abstract Algebra, by David S. Dummit and Richard M. Foote | A classic and more traditional text; a more challenging read. |
| Algebra: A Graduate Course, by I. Martin Isaacs | A more comprehensive, rigorous and advanced treatment of topics in abstract algebra. |
Number Theory
| Book title and author | Brief review |
|---|---|
| An Introduction to the Theory of Numbers, by G. H. Hardy and E. M. Wright | A classic text in the study of number theory. |
| An Introduction to the Theory of Numbers, by Ivan Morton Niven, H. S. Zuckerman and Hugh L. Montgomery | A comprehensive and accessible treatment of core topics in number theory. |
| The Book of Numbers, by John H. Conway1 and Richard K. Guy | A fascinating book that introduces readers to important results in the study of various number types. |
Differential Equation
Ordinary Differential Equation
| Book title and author | Brief review |
|---|---|
| Theory of Ordinary Differential Equations, by Earl A. Coddington and Norman Levinson | A detailed and extensive presentation of standard undergraduate-level ordinary differential equation topics. |
Partial Differential Equation
| Book title and author | Brief review |
|---|---|
| Partial Differential Equation: An Introduction, by Walter A. Strauss | A more accessible treatment of fundamental topics in partial differential equations, featuring its applications in physics. |
info
Although not technically a book, I personally recommend more the Differential Equations notes on the website 'Paul's Online Math Notes', which is available for free and features explanations that are easy to understand.
Differential Geometry
| Book title and author | Brief review |
|---|---|
| Differential Geometry of Curves and Surfaces, by Manfredo P. do Carmo | A commonly used textbook in high-level undergraduate differential geometry courses; more accessible but less rigorous. |
| Differential Geometry: A First Course in Curves and Surfaces, by Theodore Shifrin | A gentle and accommodative treatment of core topics in differential geometry, featuring illustrative examples and exercises of varied difficulties. |
| Introduction to Differential Geometry, by Joel W. Robbin and Dietmar A. Salamon | A more advanced and rigorous treatment of differential geometry, discussing the more generalised concept of manifolds, making this a more challenging text to study. |
Set Theory and Mathematical Logic
| Book title and author | Brief review |
|---|---|
| An Introduction to Set Theory, by William A. R. Weiss | Features graduate-level set theory topics, presented in a concise manner. |
| A Mathematical Introduction to Logic, by Herbert Enderton | A commonly used textbook in advanced undergraduate mathematical logic courses; a relatively manageable read. |
Miscellaneous
- Paul's Online Math Notes: a collection of very accessible class notes with clearly written explanations and helpful graphics; topics covered include high-school level algebra, introductory single- and multi-variable calculus, plus introductory differential equation.
- 3Blue1Brown: a collection of explainer videos of interesting mathematical topics with clear and interactive animated visuals.
- Rational Points on Elliptic Curves, by Joseph H. Silverman and John Tate: an accessible treatment of the study of elliptic curves, and its application in cryptography.
- The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, by Colin Adams: an introductory treatment of knot theory, a fascinating topic that involves geometry and topology.
- Fourier Analysis: An Introduction, by Elias M. Stein and Rami Shakarchi: the first volume of the mathematics book series Princeton Lectures in Analysis; a rigorous and solid text on the subject of Fourier analysis.
- The Sensual (Quadratic) Form, by John H. Conway1: a fascinating treatment of quadratic forms, which is a common research topic in number theory, featuring the interesting concept of topographs.
Footnotes
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Do not confuse John B. Conway, an American mathematician, with John H. Conway, a British mathematician. ↩ ↩2 ↩3