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An Illustrated Theory of Numbers
An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history.
Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers.
Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition.
Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.
Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers.
Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition.
Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.
323 pages, Hardcover
Published August 8, 2017
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Displaying 1 - 5 of 5 reviews
May 16, 2022
This was a required textbook for the undergrad course in number theory that I just completed. Was I annoyed that I couldn’t simply access libgen and download a free copy like I had been doing throughout my entire degree? Absolutely. Was An Illustrated Theory of Numbers worth every penny? Absolutely.
I have some major gripes when it comes to mathematics education. Underfunded math departments and incompetent professors with no pedagogical training are only two nails in a very well-bolted coffin. In my experience, the problem stems from how concepts are introduced and how they are reinforced. More often than not, an undergrad class in higher-level mathematics will start by introducing a definition, completing its proof, solving a computational example, and moving on to the next definition. Lather, rinse, repeat. Definitions and proofs are all well and good but in order to hit home they need to be introduced properly, made relevant. We shouldn’t lose sight of the big picture. And yet we often do. Building an intuitive understanding of the concepts being taught is often eschewed for procedural know-how. It’s no surprise that many mathematics undergrads rely on algorithmically bullshitting their way through the program, quickly forgetting any material they were able to absorb as soon as they’ve finished writing finals.
An Illustrated Theory of Numbers goes up to bat and challenges this status quo. Weissman spends ample time introducing concepts and problems in plain language before delving into the nitty gritty, supplementing the text with data visualizations, visual explanations, and visual mnemonics. These illustrations aid logical reasoning and establish a metaphorical system that assists the reader in digesting abstract concepts. You'll be oft to find any decorative fluff here: each image is purposeful and has something to teach. Being that the topics covered in this book are delivered alongside visual explanations, proofs tend to be approached from geometric or dynamical perspectives. Weissman's insistence that we don't fall back on algebraic manipulation is a breath of fresh air.
Each chapter contains a section on the historical context of the concepts and ideas covered within. These sections bear no resemblance to the short blurbs students tend to see in the margins of ordinary math textbooks. Weissman approaches mathematics history from a scholarly perspective, taking us on a journey through time. The end result is that the reader gets to see how an idea was born and developed before it finally grew its wings.
Finally, we have the set of problems and exercises at the end of each chapter. Each set contains a mix of relatively simple drill exercises and hearty, proof-based mathematical explorations. These problems successively increase in difficulty. Unfortunately, Weissman has not provided a solution manual although there is speculation that one may be offered in the future. For that reason, this book may not be the best option for independent learners wishing to assess their work. This is a real shame considering that it is just about perfect in every other way.
An Illustrated Theory in Numbers struck pedagogical gold and revived my dying interest in math. Years later, number theory is one of the few mathematics courses I've taken that's stuck to me like glue. I'm genuinely convinced that the world would be a better place if every field in mathematics had a textbook written by Martin Weissman.
I have some major gripes when it comes to mathematics education. Underfunded math departments and incompetent professors with no pedagogical training are only two nails in a very well-bolted coffin. In my experience, the problem stems from how concepts are introduced and how they are reinforced. More often than not, an undergrad class in higher-level mathematics will start by introducing a definition, completing its proof, solving a computational example, and moving on to the next definition. Lather, rinse, repeat. Definitions and proofs are all well and good but in order to hit home they need to be introduced properly, made relevant. We shouldn’t lose sight of the big picture. And yet we often do. Building an intuitive understanding of the concepts being taught is often eschewed for procedural know-how. It’s no surprise that many mathematics undergrads rely on algorithmically bullshitting their way through the program, quickly forgetting any material they were able to absorb as soon as they’ve finished writing finals.
An Illustrated Theory of Numbers goes up to bat and challenges this status quo. Weissman spends ample time introducing concepts and problems in plain language before delving into the nitty gritty, supplementing the text with data visualizations, visual explanations, and visual mnemonics. These illustrations aid logical reasoning and establish a metaphorical system that assists the reader in digesting abstract concepts. You'll be oft to find any decorative fluff here: each image is purposeful and has something to teach. Being that the topics covered in this book are delivered alongside visual explanations, proofs tend to be approached from geometric or dynamical perspectives. Weissman's insistence that we don't fall back on algebraic manipulation is a breath of fresh air.
Each chapter contains a section on the historical context of the concepts and ideas covered within. These sections bear no resemblance to the short blurbs students tend to see in the margins of ordinary math textbooks. Weissman approaches mathematics history from a scholarly perspective, taking us on a journey through time. The end result is that the reader gets to see how an idea was born and developed before it finally grew its wings.
Finally, we have the set of problems and exercises at the end of each chapter. Each set contains a mix of relatively simple drill exercises and hearty, proof-based mathematical explorations. These problems successively increase in difficulty. Unfortunately, Weissman has not provided a solution manual although there is speculation that one may be offered in the future. For that reason, this book may not be the best option for independent learners wishing to assess their work. This is a real shame considering that it is just about perfect in every other way.
An Illustrated Theory in Numbers struck pedagogical gold and revived my dying interest in math. Years later, number theory is one of the few mathematics courses I've taken that's stuck to me like glue. I'm genuinely convinced that the world would be a better place if every field in mathematics had a textbook written by Martin Weissman.
November 28, 2022
An extremely visual way to learn the basics of number theory.
One highlight for me was his explanations of Gaussian primes and Eisenstein primes. And it was in this book that I learned of Conway's topographs, which are the doorway to intuitions about quadratic forms.
So, I strongly recommend. One quibble: it would be nice if the author spent more time explaining the motivation for his explanations before simply diving in.
One highlight for me was his explanations of Gaussian primes and Eisenstein primes. And it was in this book that I learned of Conway's topographs, which are the doorway to intuitions about quadratic forms.
So, I strongly recommend. One quibble: it would be nice if the author spent more time explaining the motivation for his explanations before simply diving in.
December 31, 2021
Excellent visual explanations that really help the reader gain an intuitive understanding of the field.
The only criticism is that I wish answers to the problem sets were included.
The only criticism is that I wish answers to the problem sets were included.
August 11, 2019
The modular worlds chapters were illuminating. Illustrations and visual examples really help
July 8, 2021
Approccio non convenzionale
Non è che sia tutta illustrata, la teoria dei numeri spiegata in questo libro. Io ci speravo anche un po', a dire il vero... Ma il compito sarebbe stato davvero improbo. Il testo raccoglie tre temi fondamentali della teoria dei numeri "algebrica", e in effetti l'affermazione che è sufficiente avere le basi della high school mi pare corretta, anche se devo ammettere che mi sono perso nella sezione sulle forme quadratiche. La cosa che mi è piaciuta di più nell'approccio, a parte i disegnini quando ci sono, è la scelta non convenzionale di come trattare la materia, che porta Weissman a cercare unificazioni che non mi era mai capitato di vedere. Dal lato negativo ho trovato alcuni refusi, che in un libro che non è certo regalato infastidiscono parecchio.
Non è che sia tutta illustrata, la teoria dei numeri spiegata in questo libro. Io ci speravo anche un po', a dire il vero... Ma il compito sarebbe stato davvero improbo. Il testo raccoglie tre temi fondamentali della teoria dei numeri "algebrica", e in effetti l'affermazione che è sufficiente avere le basi della high school mi pare corretta, anche se devo ammettere che mi sono perso nella sezione sulle forme quadratiche. La cosa che mi è piaciuta di più nell'approccio, a parte i disegnini quando ci sono, è la scelta non convenzionale di come trattare la materia, che porta Weissman a cercare unificazioni che non mi era mai capitato di vedere. Dal lato negativo ho trovato alcuni refusi, che in un libro che non è certo regalato infastidiscono parecchio.
Displaying 1 - 5 of 5 reviews