Principles And Techniques In Combinatorics
Review: Math undergrads will find Principles and Techniques in Combinatorics to be a detailed but easy to read book. This is a much needed textbook that can truly be classified as introductory. The authors take careful consideration not to over-elaborate key concepts and thereby confuse those readers who are not as advanced in mathematics as others. Students will enjoy walking step by step through precisely detailed combinatorial proofs as well as reading the greatly in depth chapter on Recurrence Relations . An abundance of combinatorial problems that are perfect for math competition trainers and participants can be found at the end of each chapter, adding even more value to this already low-priced gem. More info.
Mathematicians: An Outer View Of The Inner World
Review: In this uniquely interesting book, photographer Mariana Cook offers readers high quality black and white photographs of 92 noteworthy mathematicians. Cooks unusual concept for this book has raised several eyebrows. However it provides a much needed break from the rather austere climate that the math world is usually comprised of. Her selected mathematicians come from diverse backgrounds and have all reached their authoritative status in equally different ways. Each photograph is accompanied by a quick, informative and often enlightening essay by the mathematician at hand, frequently revealing the passion and deep love for their discipline that each mathematician possesses. Cook does a wonderful job of capturing her subjects in an honest and purely human light. As such, this title is the ideal coffee table book for math geeks. More info.
Linear Algebra Done Right
Review: This is an exemplary book requiring a small level of mathematical maturity. Axler takes a thoughtful and theoretical approach to the work. He skillfully takes the focus off from matrices and shifts the readers attention more towards linear mappings. This makes his proofs elegant, simple, and pleasing. Conscious of the readers possible unfamiliarity as well as time frame, Axler does a fine job of preparing and developing readers understanding rather than fully detailing application methods and formulas. He leaves the reader with unsolved exercises which many will find to be thought-provoking and stimulating. An understanding of working with matrices is required. This book works great as a supplementary or second course introduction to linear algebra. More info.
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High School Math Subjects
The standard US textbooks used by schools are pretty widely derided, although Ido give links to some of them near the bottom of this document. I’ve focused onother alternatives here. I begin with Pre-Algebra because, although it’s not usually considered high school level, it isthe first course that has many adult-style textbooks dedicated to it.
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An Invitation To Modern Number Theory
Review: Advanced undergrads interested in information on modern number theory will find it hard to put this book down. The authors have created an exposition that is innovative and keeps the readers mind focused on its current occupation. The subject of modern number theory is complex and therefore this book is intended for the more experienced student. However, the authors tackle the subject in a well-paced yet rigorous style that is more than commendable. Each page exudes brilliance, birthing an underlying deeper awareness of the topic. As described in the title this book really is an invitation and curious readers would be wise to accept it. More info.
The Art Of Problem Solving
I gather this series began as two books by Richard Rusczyk,The Art of Problem Solving volumes 1 and 2, which cover the pre-algebracurriculum and are aimed at Mathcountscontestants. However, the series has been expanded to include textbooks frompre-algebra through all of high school mathematics. People rave about them, sothey’re probably pretty good. I’ve only looked at Introduction to Algebra,but I found it interesting and thorough in its explanations.
The main high school curriculum subjects covered here , aren’t usually given a challenging treatment in the otheravailable books, so these make very attractive alternatives.
I don’t know if there’s as much need for the calculus book, as students alreadyhave lots of different options for that ,but what I like about it judging from the samples available online is that itgives nice, detailed walkthroughs of more complex problems than you see in theworked examples of a typical textbook.
Since there is often a lot of overlap between courses on “Algebra II” and”Precalculus”, I compared the contents of Intermediate Algebra andPrecalculus. They are mostly different, with Intermediate Algebra focusingpolynomials, other functions, and series, whereas Precalculus is almostentirely devoted to trigonometry and an introduction to linear algebra.
AoPS also has a set of books called Beast Academy which is planned to covergrades 2-5, although currently only grades 3 and 4 are complete. I have no ideaif those are any good.
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An Introduction To Probability Theory And Its Applications
Review: In this first volume, William Feller paints a clear picture of probability theory and several of its interesting applications from the discrete viewpoint. The material is a bit advanced and is only recommended for students going into their third or fourth years. His writing brims with examples that help establish an accurate conception of discrete probability, and it includes sound insight into the history and development of probability theory. Readers will walk away with an intuitive understanding and sharper awareness of the subject. It is a must read item for any intermediate to advanced student who is working in the field of probability theory. More info.
Numerical Analysis With Cd
Review: Sauer has created a book that is more than suitable for first course studies in numerical analysis. He highlights the five critical areas of the subject which are: Convergence, Complexity, Conditioning, Compression, and Orthogonality, and makes well-planned connections to each throughout the book. The proofs are exacting but not too intricate and will firmly satisfy students. Each chapter is laden with insight, and not just analysis. Sauer attentively infuses his book with numerous problems, some to be completed by hand and others through the use of the Matlab numerical computing package. Complete with Matlab code in the back of the book and an accompanying CD-ROM, students will find Numerical Analysis a pleasure to own. More info.
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Precalculus Mathematics In A Nutshell: Geometry Algebra Trigonometry
Review: George Simmons takes newbies and out of practice scholars alike, through a refreshing crash course in three basic mathematical practices in their simple but often hated form. High school graduates and others on the way to their first college calculus course will be thoroughly prepared to take on the intimidating realm of college level mathematics. Simmons shows readers just how uncomplicated and enjoyable mathematics can be all in a transparent and fluid tone. He goes into adequate depth while still maintaining enough brevity to encourage the reader to think on their own. He cuts to the chase and afterwards leaves readers feeling capable
A First Course In Complex Analysis With Applications
Review: This book gives students an accessible introduction to the world of complex analysis and how its methods are used. A First Course in Complex Analysis is reader-friendly to the newcomer and therefore is ideal for use by both undergrads as well as graduates. For undergrads, the authors refrain from abstractness and maintain an appreciated level of transparency. While for graduates, they effortlessly fill in the gaps that many standard course texts tend to leave wide open. Each chapter is followed by a section detailing the applications of the previously discussed topic. Additionally a quick review quiz for further verification and cultivation of skills is also included with each chapter. More info.
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Introduction To Algorithms Third Edition
Review: Introduction to Algorithms is a distinctly theoretical but all-around comprehensive book. Its use is not only limited to those taking algorithms courses but can also be utilized by anyone as an extensive reference source. Readers will learn quintessential algorithms as well as concepts such as what makes an algorithm efficient and why. Students will need a bit of mathematical background to get from cover to cover, however those who are able to do so will be intrigued by the content depth and wide spectrum of topics covered. These topics run the gamut from classical algorithms to computational geometry. More info.
Ams Mathematical World Series
From the AMS website: “This accessible series brings the beauty and wonder of mathematics to the advanced high school student, the mathematics teacher, the scientist or engineer, and the lay reader with a strong interest in mathematics. Mathematical World features well-written, challenging expository works that illustrate the fascination and usefulness of mathematics.”
Here’s a link to the whole series on the AMS website:
Some titles that have caught my attention:
- Prasolov, Intuitive Topology
- Farmer, Groups and Symmetry: A Guide to Discovering Mathematics
- Farmer and Stanford, Knots and Surfaces: A Guide to Discovering Mathematics
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The Calculus Lifesaver: All The Tools You Need To Excel At Calculus
Review: Lengthy yet vital, this resource book is jam packed with straight-forward explanations and loads of solved problems for students to easily learn from. Excelling many of its contemporaries by leaps and bounds, The Calculus Lifesaver truly lives up to its title. Students who are tired of dreary calculus textbooks that provide no motivation behind the concepts will be gladly surprised by the detailed and informal approach Banner uses to capture their attention. He fills all gaps and leaves readers feeling satisfied and enlightened. This book dually holds the characteristics of both an instructive primary aid as well as that of a supplementary read. More info.
Probability Theory: The Logic Of Science
Review: Jaynes writes a fantastic prose that views probability theory beyond the usual context. The ideas found within this book are innovative and the author takes a welcomed path away from the conventional. Readers will enjoy the intellectual journey into the fundamentals of probability theory coupled with the books relaxed and informal tonality. It is strangely akin to receiving a one-on-one lesson from the author himself. Jaynes should be praised for taking a huge step away from mainstream probability theory and into this fresher approach. The only disappointment to this masterpiece is that, sadly, Jaynes died before completely finishing it, causing the editor to step in and thinly inject the missing pieces. More info.
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Principles Of Mathematical Analysis Third Edition
Review: Rudin has written an exquisite book on analysis. Before approaching, students should have a modest understanding of mapping, set theory, linear algebra and other basic topics. Rudins problem sets appear tough, but once figured out students will be more than grateful for his rigor. For readers, the key to really benefiting from this book is not only in reading it but in completing Rudins exercises and proofs on their own. The challenge will train them to think intuitively and effectively. This book is also known as baby rudin and its a real classic. More info.
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Classic Set Theory For Guided Independent Study
Review: This is a clearly written and expertly arranged independent study guide designed to make the topic of set theory comprehensible and easy to grasp for self-study students. Without a doubt, this books more than delivers. Readers can expect a smooth ride devoid of complexity and assumed pre-exposure to the subject. Goldreis book brings forth an abundance of both solved an unsolved exercises, illustrations and thorough explanations. Ideas, commentaries and recommendations that are resourcefully placed alongside the main text delightfully height the learning experience. This is one of those unfortunately rare but wonderfully rigorous independent study math books that many students stumble across and never seem to put down. More info.
The Princeton Companion To Mathematics
Review: This is an extraordinary book that every student and mathematician should absolutely have. The PCM carries the true signature of a math encyclopedia in that it is versatile and capable of being all things to all learners in every field of mathematics, and on all levels also. In light of its broad spectrum of topics, the editors have managed to keep this book cohesive and well knit together. The PCM includes specialized articles from contributors on a variety of math topics that even the most advanced pros can learn from. Non-mathematicians who are curious about the trade can also learn a great deal of information from the PCM due to its overall accessible nature. This is the kind of book that will still be read a hundred years from now, and it truly is the nicest book I own. More info.
A Mathematical Introduction To Logic Second Edition
Review: This is one of the finest introductory texts on logic that any student can read. Enderton is cohesive in his explanations and skillfully covers all the essential bases, from number theory to first and second order logic as well as several theorems, including Godels. While not a mandatory requirement, it is highly recommended that the reader has a slight understanding of math logic. This will make it easier to complete the many exercises found throughout. Complete with many well-chosen examples and a wider scope than most of its peers, Id suggest this book to anyone seeking to learn or better understand math logic. More info.
Old Books In The Public Domain
Leonhard Euler wrote an introduction to algebra called Elements of Algebra:
- Scott Hecht’s MS Word typesetting:
- Copies of 1800’s editions of Hewlett’s translation:
These are apparently still really popular in India. Since they are in the public domain, you can find them available online for free. PDFs are available on archive.org, probably elsewhere as well. They are still in print, though not always as cheap as you’d hope given that the copyrights have expired. Be warned that since they’re old, the writing style isn’t what you’d expect from a contemporary book.
- Hall and Knight, Elementary Algebra for Schools )
- Chrystal, Algebra: an Elementary Text-book , Vol II @Archive.org Vol I @Archive.org , Vol II @Archive.org )
Elementary Algebra for Schools is probably unnecessary these days. It covers roughly the equivalent of Algebra I and maybe Algebra II in US high schools, but modern textbooks do a better job with the material.
I don’t know if anyone still uses these other old books or not, but some people like them.
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Introduction To Probability Models Tenth Edition
Review: Introduction to Probability Models differs from many probability books in that it covers a variety of disciplines. It has been widely used by a number of professors as the main text for many first courses. This elementary introduction provides ample instruction on probability theory and stochastic processes, and insight into its application in a broad range of fields. Ross has filled each chapter with loads of exercises and clear examples. He also takes his time in explaining the thinking and intuition behind many of the theorems and proofs. Even those who dont specialize in the field will find it a joy to read this book. More info.
Computational Science And Engineering
Review: Gilbert Strang has a reputation for writing ample, pragmatic, and insightful books. During the course of reading this one, it will become blatantly clear to the reader that the author has created this work out of passion and a genuine love for the subject. Every engineer can benefit deeply from reading this. Strangs writing style and his instructive approach are unrivaled. He covers all aspects of computational science and engineering with experience and authority. The topics discussed include applied linear algebra and fast solvers, differential equations with finite differences and finite elements, and Fourier analysis and optimization. Strang has taught this material to thousands of students. With this book many more will be added to that number. More info.
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Mathematics: From The Birth Of Numbers
Review: Gullbergs takes an amazing dive into the deep grounded roots of mathematics and its origins. The astounding part about all of it is that Jan Gullberg is a doctor and not a mathematician. Despite this, he has managed to write a thorough book that captures the entire history of math along with answers to the why and how that students often ask. The enthusiasm he exhibits throughout will spread onto readers like wildfire. This work is clearly a labor of love, not self-exaltation. Readers will appreciate that Gullberg is simply a man who has fallen in love with and holds an immense adoration for one of the most important components of human civilization. More info.