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Very Short Introductions #447
Algebra: A Very Short Introduction
Explains algebra clearly through theory and example
Renews the reader's aquaintance with school mathematics before taking them progressively further
Elucidates the laws governing algebra in order to reveal the elegance and power of equations and inequalities
Provides an accessible account of algebra for the general reader
Part of the bestselling Very Short Introductions series - over seven million sold worldwide
Algebra marked the beginning of modern mathematics, moving it beyond arithmetic, which involves calculations featuring given numbers, to problems where some quantities are unknown. Now, it stands as a pillar of mathematics, underpinning the quantitative sciences, both social and physical.
This Very Short Introduction explains algebra from scratch. Over the course of ten logical chapters, Higgins offers a step by step approach for readers keen on developing their understanding of algebra. Using theory and example, he renews the reader's aquaintance with school mathematics, before taking them progressively further and deeper into the subject.
ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
Renews the reader's aquaintance with school mathematics before taking them progressively further
Elucidates the laws governing algebra in order to reveal the elegance and power of equations and inequalities
Provides an accessible account of algebra for the general reader
Part of the bestselling Very Short Introductions series - over seven million sold worldwide
Algebra marked the beginning of modern mathematics, moving it beyond arithmetic, which involves calculations featuring given numbers, to problems where some quantities are unknown. Now, it stands as a pillar of mathematics, underpinning the quantitative sciences, both social and physical.
This Very Short Introduction explains algebra from scratch. Over the course of ten logical chapters, Higgins offers a step by step approach for readers keen on developing their understanding of algebra. Using theory and example, he renews the reader's aquaintance with school mathematics, before taking them progressively further and deeper into the subject.
ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.
144 pages, Paperback
First published October 22, 2015
45 people are currently reading
500 people want to read
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Displaying 1 - 12 of 12 reviews
July 1, 2020
A very good book, but I got lost once I entered the matrix ... I was hoping to review a lot of the precalculus material that I studied in high school and the first five or six chapters did cover that fairly well.
The problem with many of these technical very short introductions is that they necessarily speed through material, so if you don't understand a paragraph the following text is well nigh impassable. I read through and garnered what I could, but it reminded me of one of my high school math teachers who taught to the *only* kid in class who'd already had the material while leaving the rest of us eager, but baffled.
The author has great enthusiasm, but this book is more of a summary that should follow on the heels of an introductory textbook on algebra. The proofs looked interesting and I'll probably revisit the book after I backtrack and do more foundational math work. I just don't know how far back to go ... algebra I, algebra II, or precalculus? Hey, go for the gusto, right?
The problem with many of these technical very short introductions is that they necessarily speed through material, so if you don't understand a paragraph the following text is well nigh impassable. I read through and garnered what I could, but it reminded me of one of my high school math teachers who taught to the *only* kid in class who'd already had the material while leaving the rest of us eager, but baffled.
The author has great enthusiasm, but this book is more of a summary that should follow on the heels of an introductory textbook on algebra. The proofs looked interesting and I'll probably revisit the book after I backtrack and do more foundational math work. I just don't know how far back to go ... algebra I, algebra II, or precalculus? Hey, go for the gusto, right?
January 26, 2023
Algebra : A Very Short Introduction (2015) by Peter Higgins is a very quick introduction to a lot of content about Algebra. Higgins is an Australian born professor of mathematics and the author of numerous popular books on math.
The book provides an overview of Algebra from variable substitution through Group Theory, Matrices and finishes with Vector Spaces. It covers what about one and a half years or more of University level math courses cover. There are lots of equations. This is an issue. If you understand all the equations and cover the material it's likely that you've studied much of the material previously. It would be surprising if someone could gain understanding from the book itself. But perhaps a good overview would be obtained.
Algebra : A Very Short Introduction is a book that sets itself a very difficult task but doesn't really get there. Perhaps a book three times the length with exercises would work better. Or alternatively, as suggested in further reading, an introductory Algebra textbook.
The book provides an overview of Algebra from variable substitution through Group Theory, Matrices and finishes with Vector Spaces. It covers what about one and a half years or more of University level math courses cover. There are lots of equations. This is an issue. If you understand all the equations and cover the material it's likely that you've studied much of the material previously. It would be surprising if someone could gain understanding from the book itself. But perhaps a good overview would be obtained.
Algebra : A Very Short Introduction is a book that sets itself a very difficult task but doesn't really get there. Perhaps a book three times the length with exercises would work better. Or alternatively, as suggested in further reading, an introductory Algebra textbook.
August 5, 2019
The book, as the series would suggest, is a very short introduction to the vastly comprehensive field of algebra. The book introduces to the reader the fundamental concepts such as addition, subtraction, multiplication and division and intuitively defines the natural numbers, real number, etc. around these four operations. The book gives a brief glimpses on the laws that govern algebra, such as identities, quadratic equations and their solutions, binomial theorem etc.
No algebra book would be complete without a mention of matrices and this book provides plenty from a brief information about its history to the principles of determinants, ranks and other properties of linear algebra.
I would recommend this book to anyone who wants to get back at mathematics but I must mention that the book does contain quite a lot of equations, which are not bad and I believe are necessary for a book about algebra. It is just a simple footnote I would like to mention in this review.
No algebra book would be complete without a mention of matrices and this book provides plenty from a brief information about its history to the principles of determinants, ranks and other properties of linear algebra.
I would recommend this book to anyone who wants to get back at mathematics but I must mention that the book does contain quite a lot of equations, which are not bad and I believe are necessary for a book about algebra. It is just a simple footnote I would like to mention in this review.
January 8, 2018
Probably 4 stars for a reader with a bigger head start or a fresher memory of algebra from school. For me, found it far too condensed to be a useful review — too much knowledge (or possibly quickness of grasp) taken for granted and only one or two examples of each procedure, with occasional bits of information skipped at that. But the earlier chapters did by accident show me how an algebraic equation can be represented by a graph; that was an unexpected plus.
June 15, 2025
This book will please nobody. There’s nothing new for experts and it’s too dense for novices, especially towards the end. The last chapter is ostensibly on vector spaces but it’s basically an index of terms. I was hoping that this VSI could offer a new perspective on algebra, or its applications, or prime numbers, or finite fields, or something. But the author insisted on instructing… with textbook mathematical rigour and few concrete examples… nobody.
May 26, 2024
This is probably a very good book, if you understand it. I felt I would have needed to go back to maths classes before reading this book. I appreciate that this is not necessarily a subject matter which can be easily explained without using a large amount of equations and formulae, but for me it felt it jumped into the deep end quite quickly.
March 25, 2021
Deserves a much higher average rating than it currently has (3.36/5.0 at the time of writing). There's no royal road to geometry, and for that matter there's no royal road to algebra either. One person's terse is another's concise.
January 1, 2018
Great book, starts very simply with high school stuff, equations, linear systems etc. and ends with groups. In between there is linear algebra, determinants, rotation symmetries etc.
Want to read
October 6, 2024DMPL EXAMINED 24_10_04. NOT USEFUL GOES SOUTH IMMEDIATELY.
July 7, 2025
clearly set out, logical introduction to algebra. Do not recommend reading as part of an all-day readathon once brains are already frazzled.
January 13, 2018
There's tons of information here, but it's all just crammed up without any sense of logic, so it's hard to actually learn anything...
May 31, 2024
I highly recommend Peter Higgins’ Algebra: A Very Short Introduction (a title of the VSI-series published by OUP) as an excellent and accessible introduction (as the title says) to some of the major fields of advanced algebra. The brief but comprehensive 137-page volume eases you into the subject in its first few chapters by explaining in detail the fundamental laws and rules of arithmetic and demonstrating how more complex algebraic laws and theorems can be inferred from these axioms. Higgins thus provides a deeper insight into basic algebraic knowledge, showing the algebraic reasoning and proofs behind well-known theorems (Sätze) learnt in maths lessons at school.
In the following chapters, the author delves into linear and quadratic equations, illuminating, for example, how the famous quadratic formula to solve such equations was derived. After a challenging excursion into the subject area of polynomials and cubic equations, Peter Higgins next presents a succinct overview over the defining features of groups, rings and fields before moving on to the topic of modular or clock arithmetic which involves calculations based on equivalences between elements of least residue classes.
The last and most extensive section of the book is dedicated to linear algebra, a major field concerned with networks, matrices and linear (i.e. matrix) transformations of points in n-dimensional space. You will learn about matrix inverses, determinants and eigenvectors and -values, among other things.
In the last chapter Higgins briefly delves into the topics of vector spaces, which are fundamental to linear algebra, and of finite fields - these latter being central to cryptography and other areas of applied mathematics and also playing an important role in the famous proof of Fermat’s Last Theorem by Sir Andrew Wiles in the 1990s.
In summary, Algebra: A Very Short Introduction by Peter M. Higgins is a concise, rigorous and intellectually rewarding excursion into the (for non-academics) often unknown territories of advanced algebra. A must-read primer and excellent reference work for anyone interested in higher mathematics.
In the following chapters, the author delves into linear and quadratic equations, illuminating, for example, how the famous quadratic formula to solve such equations was derived. After a challenging excursion into the subject area of polynomials and cubic equations, Peter Higgins next presents a succinct overview over the defining features of groups, rings and fields before moving on to the topic of modular or clock arithmetic which involves calculations based on equivalences between elements of least residue classes.
The last and most extensive section of the book is dedicated to linear algebra, a major field concerned with networks, matrices and linear (i.e. matrix) transformations of points in n-dimensional space. You will learn about matrix inverses, determinants and eigenvectors and -values, among other things.
In the last chapter Higgins briefly delves into the topics of vector spaces, which are fundamental to linear algebra, and of finite fields - these latter being central to cryptography and other areas of applied mathematics and also playing an important role in the famous proof of Fermat’s Last Theorem by Sir Andrew Wiles in the 1990s.
In summary, Algebra: A Very Short Introduction by Peter M. Higgins is a concise, rigorous and intellectually rewarding excursion into the (for non-academics) often unknown territories of advanced algebra. A must-read primer and excellent reference work for anyone interested in higher mathematics.
Displaying 1 - 12 of 12 reviews