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Very Short Introductions #636
Number Theory: A Very Short Introduction
Number theory is the branch of mathematics that is primarily concerned with the counting numbers. Of particular importance are the prime numbers, the 'building blocks' of our number system. The subject is an old one, dating back over two millennia to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no-one has ever been able to resolve them.
But number theory has also recently become of great practical importance - in the area of cryptography, where the security of your credit card, and indeed of the nation's defence, depends on a result concerning prime numbers that dates back to the 18th century. Recent years have witnessed other spectacular developments, such as Andrew Wiles's proof of 'Fermat's last theorem' (unproved for over 250 years) and some exciting work on prime numbers. In this Very Short Introduction Robin Wilson introduces the main areas of classical number theory, both ancient and modern. Drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler, and Gauss, he situates some of the most interesting and creative problems in the area in their historical context.
ABOUT THE
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.
But number theory has also recently become of great practical importance - in the area of cryptography, where the security of your credit card, and indeed of the nation's defence, depends on a result concerning prime numbers that dates back to the 18th century. Recent years have witnessed other spectacular developments, such as Andrew Wiles's proof of 'Fermat's last theorem' (unproved for over 250 years) and some exciting work on prime numbers. In this Very Short Introduction Robin Wilson introduces the main areas of classical number theory, both ancient and modern. Drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler, and Gauss, he situates some of the most interesting and creative problems in the area in their historical context.
ABOUT THE
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
Published August 1, 2020
44 people are currently reading
269 people want to read
About the author
Robin Wilson
89 books9 followersRobin Wilson (born September 26, 1969) is an expert in the Clean Design, wellness and sustainability advocate. She is the founder of Robin Wilson Home, an interior design firm based in New York City – and she is chief creative officer of the licensing division of her eponymous brand which has generated over $80 million in branded revenue from textiles and cabinetry since 2010.
Wilson’s design work emphasizes the integration of eco-friendly and sustainable design with a focus on social good. Her clients have included Panasonic USA, the White House Fellows office, and the Lake Nona Laureate Park development. In addition, she has worked on showhouses, and for both residential, developer and commercial clients.
She is author of 'Clean Design: Wellness for Your Lifestyle' (Greenleaf, 2015). Her first book was 'Kennedy Green House: Designing an Eco-Friendly Home from the Foundation to the Furniture' (Greenleaf, 2010), with the foreword written by Robert F. Kennedy, Jr. She is an ambassador to the Asthma and Allergy Foundation of America and she has served on the Board of the Sustainable Furnishings Council.
In May 2013, her furniture line, Nest Home by Robin Wilson, premiered at the International Contemporary Furniture Fair (ICFF) in New York.
She regularly appears on the speakers circuit, on television and offers commentary in print on design, wellness, sustainability and allergy & asthma issues.
Wilson’s design work emphasizes the integration of eco-friendly and sustainable design with a focus on social good. Her clients have included Panasonic USA, the White House Fellows office, and the Lake Nona Laureate Park development. In addition, she has worked on showhouses, and for both residential, developer and commercial clients.
She is author of 'Clean Design: Wellness for Your Lifestyle' (Greenleaf, 2015). Her first book was 'Kennedy Green House: Designing an Eco-Friendly Home from the Foundation to the Furniture' (Greenleaf, 2010), with the foreword written by Robert F. Kennedy, Jr. She is an ambassador to the Asthma and Allergy Foundation of America and she has served on the Board of the Sustainable Furnishings Council.
In May 2013, her furniture line, Nest Home by Robin Wilson, premiered at the International Contemporary Furniture Fair (ICFF) in New York.
She regularly appears on the speakers circuit, on television and offers commentary in print on design, wellness, sustainability and allergy & asthma issues.
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Displaying 1 - 14 of 14 reviews
September 4, 2022
لعالم الرياضيات الألماني الشهير جاوس مقولة شهيرة: "الرياضيات ملكة العلوم، ونظرية الأعداد ملكة الرياضيات".
.نظرية الأعداد مجال مثير للاهتمام وفيه مسائل مفتوحة عديدة للآن، بعضها قد يمنح صاحب حلها مليون دولار(^ن^)
الكتاب متنوع وذكر عدة موضوعات مختلفة في نظرية الأعداد منها إنجازات حديثة في المجال. وطريقة تناوله بسيطة وأظن مناسبة لأي شخص درس بعض الرياضيات في المدرسة، ومع هذا كانت فيه بعض النظريات والمعلومات التي لم أكن على علم بها وودت لو كنت قرأته قبل دراستي لمقرر نظرية الأعداد بالجامعة، بالإضافة إن به فصل عن مشروع تخرجي، ولكن على أي حال هو بالتأكيد زاد من اهتمامي وإعجابي بالمجال.
.نظرية الأعداد مجال مثير للاهتمام وفيه مسائل مفتوحة عديدة للآن، بعضها قد يمنح صاحب حلها مليون دولار(^ن^)
الكتاب متنوع وذكر عدة موضوعات مختلفة في نظرية الأعداد منها إنجازات حديثة في المجال. وطريقة تناوله بسيطة وأظن مناسبة لأي شخص درس بعض الرياضيات في المدرسة، ومع هذا كانت فيه بعض النظريات والمعلومات التي لم أكن على علم بها وودت لو كنت قرأته قبل دراستي لمقرر نظرية الأعداد بالجامعة، بالإضافة إن به فصل عن مشروع تخرجي، ولكن على أي حال هو بالتأكيد زاد من اهتمامي وإعجابي بالمجال.
November 9, 2020
Modular arithmetic, factorisation, prime numbers, and many more areas are covered. For a lay reader (by trade I am a physician), quite an amount of attention, concentration, and perseverance is needed to comprehend the many captivating results in it. However, equations are not numbered so cross-references are inconvenient and at times hard, particularly when one has not really remembered and mastered them all. The last chapter on Riemann Hypothesis is too dense – interested readers should go to John Derbyshire's classic on the same topic. So, overall, I will give this little book three stars. It is very fascinating but a bit too effortful for me.
August 14, 2020
The best number theory intro I have read. I wish it was a little longer. 😁
December 28, 2022
This book is set up PERFECTLY for anybody interested in number theory, and is great for any university maths student that is about to take a number theory module.
To start off, Wilson introduced the famous mathematicians that you never stop hearing the names of in the maths world and puts a face to the name which so many people don’t know what they look like.
He also asks 10 questions- and satisfyingly answers them all at the end of the book with theorems that he introduced throughout.
Not only that, but he talks about the millennia math problems. If you are interested in math at all you cannot go about without knowing about these! So for an introduction it’s amazing that he mentions them
As well, he introduces many theorems (for example, Chinese remainder theorem) which appear in any number theory class in a great way, always putting examples so you can see how the theorems are useful and play out.
To start off, Wilson introduced the famous mathematicians that you never stop hearing the names of in the maths world and puts a face to the name which so many people don’t know what they look like.
He also asks 10 questions- and satisfyingly answers them all at the end of the book with theorems that he introduced throughout.
Not only that, but he talks about the millennia math problems. If you are interested in math at all you cannot go about without knowing about these! So for an introduction it’s amazing that he mentions them
As well, he introduces many theorems (for example, Chinese remainder theorem) which appear in any number theory class in a great way, always putting examples so you can see how the theorems are useful and play out.
This entire review has been hidden because of spoilers.
September 30, 2021
An interesting little introduction to the fundamentals of numbers - actually understood it!
June 30, 2025
Clear, concise, and surprisingly fun — a great introduction to number theory for curious readers.
Robin Wilson’s Introduction to Number Theory offers an engaging and approachable tour through one of the oldest and most fascinating branches of mathematics. Whether you're a student brushing up or just a math enthusiast, this book lays out the essentials with clarity and charm.
Some of the highlights include: Perfect Numbers: Numbers equal to the sum of their proper divisors, like 6 and 28. Prime Numbers: The building blocks of arithmetic—divisible only by 1 and themselves. Squares and Cubes: Powers of integers like 4 (=2²) and 27 (=3³). Triangular Numbers: Sums of consecutive integers (1+2+3...) and their link to geometry. Pythagorean Theorem: Right triangle rule: a² + b² = c². LCM and GCD: Finding the smallest multiple and greatest divisor of two numbers. Euclid’s Algorithm: A timeless method to compute the GCD. Divisor Tests: Simple tricks to check if a number is divisible by 2, 3, 9, 11, and so on. Casting Out Nines: A neat arithmetic check using mod 9. Infinitude of Primes: Euclid’s classic proof that there are infinitely many primes. Prime Varieties: Euler’s primes, Mersenne primes (2ᵖ − 1), and Fermat primes (2²ⁿ + 1). Mills' Constant: The mysterious constant A such that ⌊A³ⁿ⌋ is always prime. Modular Arithmetic: Clocks (mod 12), days of the week (mod 7), and solving congruences. Calendars: How the Gregorian calendar cycles repeat every 400 years—and weekdays every 28.
Wilson weaves mathematical history and logic into short, digestible chapters. You don’t need a math degree to follow along, but if you love numbers, you'll find plenty to enjoy.
Recommended for: High school and college students, puzzle lovers, and anyone who wants to see why math has fascinated thinkers for millennia.
May 24, 2021
I don't find number theory particularly interesting. It seems to me to a great extent a recreational activity, setting and solving arbitrary puzzles involving integers that usually have no practical application except by accident. But the author deserves credit for setting out the basics for the most part in a readable way, though I found a few explanations hard to follow.
Here's one of the problems discussed, adapted from one by Leonardo Fibonacci in 1202: "If I can buy partridges for 3 cents, pigeons for 2 cents, and 2 sparrows for a cent, and if I spend 30 cents on buying 30 birds, how many birds of each kind must I buy?" The answer given in the book is 3 partridges, 5 pigeons, and 22 sparrows. But of course there's another answer that has been overlooked.
Here's one of the problems discussed, adapted from one by Leonardo Fibonacci in 1202: "If I can buy partridges for 3 cents, pigeons for 2 cents, and 2 sparrows for a cent, and if I spend 30 cents on buying 30 birds, how many birds of each kind must I buy?" The answer given in the book is 3 partridges, 5 pigeons, and 22 sparrows. But of course there's another answer that has been overlooked.
August 29, 2020
Pretty dry, and I think it would be easy for a layperson to get in way over their head very quickly with this book. Seemed very formally written to me - useful for math people, bad for people that aren't math people. I teach math for a living so it wasn't a deal-breaker for me but this wasn't the book I had hoped for - a book I could recommend to people interested in number theory that didn't have a strong, formal background in mathematics beforehand.
January 28, 2023
I am sure this was well written and adequately covered the topic. I am just giving this text three starts to warn future readers that are not mathematically gifted to 1) know that there will be a lot of "number talk" and less "theory talk" 2) do not get the audio book, what information I could have gleaned from this text was lost because I was listening to it and could not track with the frequent equations and lists of numbers with out seeing them.
July 20, 2023
More or less a mathematical showroom. Loosely connected sections. I was not able to see the grand narrative. The book is very dry. If this what you are looking for, you'd still be better of buying a study book.
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July 25, 2024very short and very quick but cool! lots of stuff about prime numbers and not as much explanation as i would like but cool introduction :)
June 1, 2025
well written. I read most chapters and understood a lot. Sometimes I found the examples however too trivial.
June 22, 2024
Very good introduction to the field, giving an excellent overview over it's various areas of knowledge. The author explains in depth fundamental operations and concepts as they relate to the counting numbers, such as multiplication/division, powers, primes, clocks and calendars. - Leading on from that, the reader will learn:
*about important hypotheses and theorems;
*how to determine if a certain integer is prime or composite;
*how to apply modular arithmetics to solving famous riddles and problems (such as dividing up a treasure trove or finding out the weekday of a calendar date many years in the future);
*and many other things!
My only criticism is with the last chapters of this volume; instead of repeating previous results from earlier chapters, the author should have expanded more on the Riemann Hypothesis in the end. For example, Wilson could have delved deeper into the remarkable connection between the non-trivial zeros and the number of primes up to x, as expressed by Riemann's exact version of the prime number theorem.
Regardless of this minor issue, I strongly recommend this book to anyone genuinely interested in mathematics who wants to gain some foundational insight into the amazing properties and intriguing behaviour of the (deceptively) familiar counting numbers.
*about important hypotheses and theorems;
*how to determine if a certain integer is prime or composite;
*how to apply modular arithmetics to solving famous riddles and problems (such as dividing up a treasure trove or finding out the weekday of a calendar date many years in the future);
*and many other things!
My only criticism is with the last chapters of this volume; instead of repeating previous results from earlier chapters, the author should have expanded more on the Riemann Hypothesis in the end. For example, Wilson could have delved deeper into the remarkable connection between the non-trivial zeros and the number of primes up to x, as expressed by Riemann's exact version of the prime number theorem.
Regardless of this minor issue, I strongly recommend this book to anyone genuinely interested in mathematics who wants to gain some foundational insight into the amazing properties and intriguing behaviour of the (deceptively) familiar counting numbers.
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December 11, 2022I forgot I finished this awhile ago oopz
Displaying 1 - 14 of 14 reviews