100 Best Number Theory Books of All Time
We've researched and ranked the best number theory books in the world, based on recommendations from world experts, sales data, and millions of reader ratings. Learn more
1
"I have discovered a truly marvelous proof, which this margin is too narrow to contain". With these tantalizing words the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations. What came to be known as Fermat's Last Theorem looked simple, yet the finest mathematical minds would be baffled for more than three and a half centuries.Fermat's Last Theorem became the Holy Grail of mathematics. Whole and colorful lives were devoted, and even sacrificed, to finding a solution. Leonhard Euler, the greatest mathematician of the 18th century, had to... more
Sarah-Jayne BlakemoreThe book is great because Simon Singh has this ability to write about the driest and most complex scientific or mathematical concepts and issues, and somehow make them come alive. (Source)
Kirk BorneNew Perspective on Fermat's Last Theorem: https://t.co/YeaHQ6iadB by @granvilleDSC @DataScienceCtrl #abdsc #Mathematics See the best-selling book "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem": https://t.co/dqenmvUw0A by @SLSingh https://t.co/deyMhQTQLU (Source)
2
Douglas Hofstadter's book is concerned directly with the nature of “maps” or links between formal systems. However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more. less
Steve Jurvetson[Steve Jurvetson recommended this book on the podcast "The Tim Ferriss Show".] (Source)
Seth GodinIn the last week, I discovered that at least two of my smart friends hadn't read Godel, Escher, Bach. They have now. You should too. (Source)
Kevin KellyOver the years, I kept finding myself returning to its insights, and each time I would arrive at them at a deeper level. (Source)
3
The Babylonians invented it, the Greeks banned it, the Hindus worshipped it, and the Church used it to fend off heretics. For centuries, the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. Zero follows this number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe and its apotheosis as the mystery of the black hole. Today, zero lies at the heart of one of the biggest scientific controversies of all time, the quest for the theory of everything. Elegant, witty, and enlightening, Zero... more
Alex BellosUnlike Ifrah, Charles Seife is a brilliant popular science writer who has here written the ‘biography’ of zero. And even though he doesn’t talk that much about India, it works well as a handbook to Ifrah’s sections on India. Because Seife talks about how zero is mathematically very close to the idea of infinity, which is another mathematical idea that the Indians thought about differently. Seife... (Source)
Bryan JohnsonChronicles how hard it was for humanity to come up with and hold onto the concept of zero. No zero, no math. No zero, no engineering. No zero, no modern world as we know it... (Source)
4
Written for the one-semester undergraduate number theory course, this text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity. It reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history. less
5
Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent... more
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent... more
6
The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it did not, because science was being stifled by militarism or religious fanaticism. less
Recommended by Alex Bellos, and 1 others. Alex BellosPetr Beckmann was a Czech electrical engineer who lived in Czechoslovakia until he was 39 in 1963, when he went to America as a visiting professor and just stayed there. (Source)
7
In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hypothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic... more
8
Aimed at courses in Elementary Number Theory, this book is for math majors, for mathematics education students, and for Computer Science students. Starting from basic algebra, it takes the reader to mathematical research. It includes numerical examples, analyzed for patterns and used to make conjectures. less
9
Art of Problem Solving Introduction to Counting and Probability Textbook and Solutions Manual 2-Book Set : Learn the basics of counting and probability from former USA Mathematical Olympiad winner David Patrick. Topics covered in the book include permutations, combinations, Pascal's Triangle, basic combinatorial identities, expected value, fundamentals of probability, geometric probability, the Binomial Theorem, and much more. The text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, so the student has a chance to solve them without help... more
10
In 1859, German mathematician Bernhard Riemann presented a paper to the Berlin Academy that would forever change the history of mathematics. The subject was the mystery of prime numbers. At the heart of the presentation was an idea that Riemann had not yet proved but one that baffles mathematicians to this day.
Solving the Riemann Hypothesis could change the way we do business, since prime numbers are the lynchpin for security in banking and e-commerce. It would also have a profound impact on the cutting-edge of science, affecting quantum mechanics, chaos theory, and the future of... more
Solving the Riemann Hypothesis could change the way we do business, since prime numbers are the lynchpin for security in banking and e-commerce. It would also have a profound impact on the cutting-edge of science, affecting quantum mechanics, chaos theory, and the future of... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
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11
Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, Mario Livio tells the tale of a number at the heart of that mystery: phi, or 1.6180339887...This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing... more
Recommended by Kirk Borne, and 1 others. Kirk BorneSome Fun with Gentle Chaos, the Golden Ratio, and Stochastic Number Theory, with Gaming Applications: https://t.co/oQG0y3vA22 #abdsc by @granvilleDSC @DataScienceCtrl #Mathematics #Statistics ————— Learn all about the Golden Ratio in this fantastic book: https://t.co/9QxN9ECpH7 https://t.co/Mt45UZFFHH (Source)
12
This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves. less
13
Green Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's Elements. In keeping with Green Lion's design commitment, diagrams have been placed on every spread for convenient reference while working through the proofs; running heads on every page indicate both Euclid's book number and proposition numbers for that page; and adequate space for notes is allowed between propositions and around diagrams. The all-new index has built into it a glossary of Euclid's Greek terms.
Heath's translation has stood the test of time,... more
Heath's translation has stood the test of time,... more
14
The fifth edition of this classic reference work has been updated to give a reasonably accurate account of the present state of knowledge. less
15
Recommended by Bryan Johnson, and 1 others. Bryan Johnson[Bryan Johnson recommended this book on Twitter.] (Source)
16
"This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages."--MATHEMATICAL REVIEWS less
17
The Universe May Be a Mystery,But It's No Secret Michael Schneider leads us on a spectacular, lavishly illustrated journey along the numbers one through ten to explore the mathematical principles made visible in flowers, shells, crystals, plants, and the human body, expressed in the symbolic language of folk sayings and fairy tales, myth and religion, art and architecture. This is a new view of mathematics, not the one we learned at school but a comprehensive guide to the patterns that recur through the universe and underlie human affairs. A Beginner's Guide to Constructing, the Universe... more
18
Treats the arithmetic theory of elliptic curves in its modern formulation through the use of basic algebraic number theory and algebraic geometry. This book outlines necessary algebro-geometric results and offers an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields. less
19
The international best-seller that makes mathematics a thrilling exploration.
In twelve dreams, Robert, a boy who hates math, meets a Number Devil, who leads him to discover the amazing world of numbers: infinite numbers, prime numbers, Fibonacci numbers, numbers that magically appear in triangles, and numbers that expand without. As we dream with him, we are taken further and further into mathematical theory, where ideas eventually take flight, until everyone - from those who fumble over fractions to those who solve complex equations in their heads - winds up marveling at what... more
In twelve dreams, Robert, a boy who hates math, meets a Number Devil, who leads him to discover the amazing world of numbers: infinite numbers, prime numbers, Fibonacci numbers, numbers that magically appear in triangles, and numbers that expand without. As we dream with him, we are taken further and further into mathematical theory, where ideas eventually take flight, until everyone - from those who fumble over fractions to those who solve complex equations in their heads - winds up marveling at what... more
20
)tPI(}jlOV, e oxov (10CPUljlr1.'CWV Aiux., llpop. . .dsup.. The first part of this volume is based on a course taught at Princeton University in 1961-62; at that time, an excellent set of notes was prepared by David Cantor, and it was originally my intention to make these notes available to the mathematical public with only quite minor changes. Then, among some old papers of mine, I accidentally came across a long-forgotten manuscript by Chevalley, of pre-war vintage (forgotten, that is to say, both by me and by its author) which, to my taste at least, seemed to have aged very well. It... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
- Cutting out the fluff: you focus your time on what's important to know
- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.
21
Between the seemingly impossible tasks of living up to his warrior-father's legend and surmounting his own physical limitations, Miles Vorkosigan faces some truly daunting challenges.
Shortly after his arrival on Beta Colony, Miles unexpectedly finds himself the owner of an obsolete freighter and in more debt than he ever thought possible. Propelled by his manic "forward momentum," the ever-inventive Miles creates a new identity for himself as the commander of his own mercenary fleet to obtain a lucrative cargo; a shipment of weapons destined for a dangerous warzone. less
Shortly after his arrival on Beta Colony, Miles unexpectedly finds himself the owner of an obsolete freighter and in more debt than he ever thought possible. Propelled by his manic "forward momentum," the ever-inventive Miles creates a new identity for himself as the commander of his own mercenary fleet to obtain a lucrative cargo; a shipment of weapons destined for a dangerous warzone. less
22
The Number Sense is an enlightening exploration of the mathematical mind. Describing experiments that show that human infants have a rudimentary number sense, Stanislas Dehaene suggests that this sense is as basic as our perception of color, and that it is wired into the brain. Dehaene shows that it was the invention of symbolic systems of numerals that started us on the climb to higher mathematics. A fascinating look at the crossroads where numbers and neurons intersect, The Number Sense offers an intriguing tour of how the structure of the brain shapes our mathematical abilities, and how... more
Recommended by Peg Tyre, and 1 others. 23
Ideal for a first course in number theory, this lively, engaging text requires only a familiarity with elementary algebra and the properties of real numbers. Author Underwood Dudley, who has written a series of popular mathematics books, maintains that the best way to learn mathematics is by solving problems. In keeping with this philosophy, the text includes nearly 1,000 exercises and problems—some computational and some classical, many original, and some with complete solutions.
The opening chapters offer sound explanations of the basics of elementary number theory and develop the... more
The opening chapters offer sound explanations of the basics of elementary number theory and develop the... more
24
A deep understanding of prime numbers is one of the great challenges in mathematics. In this new edition, fundamental theorems, challenging open problems, and the most recent computational records are presented in a language without secrets. The impressive wealth of material and references will make this book a favorite companion and a source of inspiration to all readers.
Paulo Ribenboim is Professor Emeritus at Queen's University in Canada, Fellow of the Royal Society of Canada, and recipient of the George Polya Award of the Mathematical Association of America. He is the author... more
Paulo Ribenboim is Professor Emeritus at Queen's University in Canada, Fellow of the Royal Society of Canada, and recipient of the George Polya Award of the Mathematical Association of America. He is the author... more
26
Based on a National Magazine Award-winning article, this masterful biography of Hungarian-born Paul Erdos is both a vivid portrait of an eccentric genius and a layman's guide to some of this century's most startling mathematical discoveries. less
27
The tale of a relationship between a young Indian mathematics genius, Ramanujan, and his tutor at Cambridge University, G.H. Hardy, in the years before World War I. Through their eyes the reader is taken on a journey through numbers theory. Ramanujan would regularly telescope 12 steps of logic into two - the effect is said to be like Dr Watson in the train of some argument by Sherlock Holmes. The language of symbols and infinitely large (and small) regions of mathematics should be rendered with clarity for the general reader. less
28
A highly successful presentation of the fundamental concepts of number theory and computer programming
Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced... more
Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced... more
29
Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement. less
30
A wide variety of ready-to-use number talks that help kindergarten through second-grade students learn math concepts in fun and easy ways
Bringing the exciting teaching method of number talks into your classroom has never been easier. Simply choose from the hundreds of great ideas in this book and get going, with no extra time wasted!
From activities on addition and subtraction to fractions and decimals, Classroom-Ready Number Talks for Kindergarten, First and Second Grade Teachers includes:
Grade-level specific strategies
Number talk... more
Bringing the exciting teaching method of number talks into your classroom has never been easier. Simply choose from the hundreds of great ideas in this book and get going, with no extra time wasted!
From activities on addition and subtraction to fractions and decimals, Classroom-Ready Number Talks for Kindergarten, First and Second Grade Teachers includes:
Grade-level specific strategies
Number talk... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
- Cutting out the fluff: you focus your time on what's important to know
- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.
31
Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann Hypothesis, which remains to be one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann Hypothesis. Students with minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible... more
32
Written in 1940 as his mathematical powers were declining, G.H. Hardy's apology offers an engaging account of the thoughts of a man known for his eccentricities as well as his brilliance in mathematics. less
Recommended by Marcus du Sautoy, and 1 others. Marcus du SautoyYes, it really appealed to me when I read it as a kid because I was interested in music, I played the trumpet, I loved doing theatre, and somehow GH Hardy in that book revealed to me how much mathematics is a creative art as much as a useful science. In fact he probably goes further, he really revels in the beauty of the subject and says he’s not particularly interested in the applications. That... (Source)
33
Shows how a young couple turned on to pure mathematics and found total happiness. This title is intended for those who might enjoy an engaging dialogue on abstract mathematical ideas, and those who might wish to experience how new mathematics is created. less
34
The story of [pi] has been told many times, both in scholarly works and in popular books. But its close relative, the number e, has fared less well: despite the central role it plays in mathematics, its history has never before been written for a general audience. The present work fills this gap. Geared to the reader with only a modest background in mathematics, the book describes the story of e from a human as well as a mathematical perspective. In a sense, it is the story of an entire period in the history of mathematics, from the early seventeenth to the late nineteenth century, with the... more
36
Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. The first edition of this volume was respected, both as a textbook and as a source for results, ideas, and references. It helped to spark a growing interest in the mathematical community to bring it back into print. In this second edition, Iwaniec treats the spectral theory of automorphic forms as the study of the space $L (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving... more
37
Number theory, the branch of mathematics that studies the properties of the integers, is a repository of interesting and quite varied problems, sometimes impossibly difficult ones. In this book, the authors have gathered together a collection of problems from various topics in number theory that they find beautiful, intriguing, and from a certain point of view instructive. less
38
This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for solving. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems. By emphasizing examples and applications the... more
40
Like other introductions to number theory, this one includes the usual curtsy to divisibility theory, the bow to congruence, and the little chat with quadratic reciprocity. It also includes proofs of results such as Lagrange's Four Square Theorem, the theorem behind Lucas's test for perfect numbers, the theorem that a regular n-gon is constructible just in case phi(n) is a power of 2, the fact that the circle cannot be squared, Dirichlet's theorem on primes in arithmetic progressions, the Prime Number Theorem, and Rademacher's partition theorem.
We have made the... more
We have made the... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
- Cutting out the fluff: you focus your time on what's important to know
- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.
42
This undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to... more
43
The so-called "Lost Notebook" of S.R. Ramanujan was brought to light in 1976 as part of the Watson bequest, by G.E. Andrews with whose introduction this collection of unpublished manuscripts opens. A major portion of the "Lost Notebook" - really just 90 unpaginated sheets of work on "q"-series and other topics - is reproduced here in facsimile. Letters from Ramanujan to Hardy as well as various other sheets of seemingly related notes are then included, on topics including coefficients in the 1/q3 and 1/q2 problems and the mock theta functions. The next 180 pages consist of unpublished... more
44
This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to give a rapid introduction of how analytic methods are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. less
45
The new edition of this thorough examination of the distribution of prime numbers in arithmetic progressions offers many revisions and corrections as well as a new section recounting recent works in the field. The book covers many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetical progressions and the theorem of Siegel. It also presents a simplified, improved version of the large sieve method. less
46
The fourth edition of Kenneth Rosen's widely used and successful text, Elementary Number Theory and Its Applications, preserves the strengths of the previous editions, while enhancing the book's flexibility and depth of content coverage.The blending of classical theory with modern applications is a hallmark feature of the text. The Fourth Edition builds on this strength with new examples, additional applications and increased cryptology coverage. Up-to-date information on the latest discoveries is included.Elementary Number Theory and Its Applications provides a diverse group of exercises,... more
47
This book presents a historical overview of number theory. It examines texts that span some thirty-six centuries of arithmetical work, from an Old Babylonian tablet to Legendre's Essai sur la Th�orie des Nombres, written in 1798. Coverage employs a historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. The book also takes the reader into the workshops of four major authors of modern number theory: Fermat, Euler, Lagrange and Legendre and presents a detailed and critical examination of their work. less
48
Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane's Online Encyclopedia of Integer Sequences, at the end of several of the sections. less
49
Serre's A Course in Arithmetic is a concentrated, modern introduction to basically three areas of number theory, quadratic forms, Dirichlet's density theorem, and modular forms. The first edition was very well accepted and is now one of the leading introductory texts on the advanced undergraduate or beginning graduate level.
From the reviews: .,." The book is carefully written - in particular very much self-contained. As was the intention of the author, it is easily accessible to graduate or even undergraduate students, yet even the advanced mathematician will enjoy reading it. The last... more
From the reviews: .,." The book is carefully written - in particular very much self-contained. As was the intention of the author, it is easily accessible to graduate or even undergraduate students, yet even the advanced mathematician will enjoy reading it. The last... more
50
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... more
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... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
- Cutting out the fluff: you focus your time on what's important to know
- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.
51
Pure Mathematics for BeginnersPure Mathematics for Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra.The 16 lessons in this book cover basic through intermediate material from each of these 8 topics. In addition, all the proofwriting skills that are essential for advanced study in mathematics are covered and reviewed extensively.Pure Mathematics for Beginners is perfect for
professors teaching an introductory college course in higher... more
professors teaching an introductory college course in higher... more
52
At what point does theory depart the realm of testable hypothesis and come to resemble something like aesthetic speculation, or even theology? The legendary physicist Wolfgang Pauli had a phrase for such ideas: He would describe them as "not even wrong," meaning that they were so incomplete that they could not even be used to make predictions to compare with observations to see whether they were wrong or not. In Peter Woit's view, superstring theory is just such an idea. In Not Even Wrong , he shows that what many physicists call superstring "theory" is not a theory at all. It makes no... more
Recommended by Eric Weinstein, and 1 others. Eric Weinstein[Eric Weinstein recommended this book on Twitter.] (Source)
53
Scientific knowledge grows at a phenomenal pace--but few books have had as lasting an impact or played as important a role in our modern world as The Mathematical Theory of Communication, published originally as a paper on communication theory in the Bell System Technical Journal more than fifty years ago. Republished in book form shortly thereafter, it has since gone through four hardcover and sixteen paperback printings. It is a revolutionary work, astounding in its foresight and contemporaneity. The University of Illinois Press is pleased and honored to issue this... more
54
This volume contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One particularly valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. The book should be a useful resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics... more
55
This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their... more
56
Too often math gets a bad rap, characterized as dry and difficult. But, Alex Bellos says, "math can be inspiring and brilliantly creative. Mathematical thought is one of the great achievements of the human race, and arguably the foundation of all human progress. The world of mathematics is a remarkable place."Bellos has traveled all around the globe and has plunged into history to uncover fascinating stories of mathematical achievement, from the breakthroughs of Euclid, the greatest mathematician of all time, to the creations of the Zen master of origami, one of the hottest areas of... more
58
Number theory, the Queen of Mathematics, is an almost purely theoretical science. Yet it can be the source of endlessly intriguing puzzle problems, as this remarkable book demonstrates. This is the first book to deal exclusively with the recreational aspects of the subject and it is certain to be a delightful surprise to all devotees of the mathematical puzzle, from the rawest beginner to the most practiced expert. Almost every aspect of the theory of numbers that could conceivably be of interest to the layman is dealt with, all from the recreational point of view. Readers will become... more
59
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and... more
60
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book's accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry. less
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
- Cutting out the fluff: you focus your time on what's important to know
- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.
61
Developed from the author's popular graduate-level course, Computational Number Theory presents a complete treatment of number-theoretic algorithms. Avoiding advanced algebra, this self-contained text is designed for advanced undergraduate and beginning graduate students in engineering. It is also suitable for researchers new to the field and practitioners of cryptography in industry.
Requiring no prior experience with number theory or sophisticated algebraic tools, the book covers many computational aspects of number theory and highlights important and... more
Requiring no prior experience with number theory or sophisticated algebraic tools, the book covers many computational aspects of number theory and highlights important and... more
62
. . . both Gauss and lesser mathematicians may be justified in rejoic- ing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. - G. H. Hardy, A Mathematician's Apology, 1940 G. H. Hardy would have been surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography (secret codes). Less than a half-century after Hardy wrote the words quoted above, it... more
63
John D. Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of mathematics: where it comes from, what it is, and where it's going to take us if we follow it to the limit in our search for the ultimate meaning of the universe. Barrow begins by investigating whether math is a purely human invention inspired by our practical needs. Or is it something inherent in nature waiting to be discovered?
In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats... more
In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats... more
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What is the mathematics behind a twitter trend? Does my food really have an equation? And, is there really an algorithm for Love?
Mathematics is inescapable. Wherever you go, whatever you do, however you live your life, mathematics plays a role. From searching for love to donating a kidney, the mathematics governing our world is fascinating, and far reaching. Using interesting anecdotes, simple analogies, and easy explanations, Man vs. Math will distill the complexities of some of the most absorbing mathematics of modern life.
Along the way we will look at why... more
Mathematics is inescapable. Wherever you go, whatever you do, however you live your life, mathematics plays a role. From searching for love to donating a kidney, the mathematics governing our world is fascinating, and far reaching. Using interesting anecdotes, simple analogies, and easy explanations, Man vs. Math will distill the complexities of some of the most absorbing mathematics of modern life.
Along the way we will look at why... more
66
Number theory has a long and distinguished history and the concepts and problems relating to the subject have been instrumental in the foundation of much of mathematics. In this book, Professor Baker describes the rudiments of number theory in a concise, simple and direct manner. Though most of the text is classical in content, he includes many guides to further study which will stimulate the reader to delve into the great wealth of literature devoted to the subject. The book is based on Professor Baker's lectures given at the University of Cambridge and is intended for undergraduate students... more
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Bridges the gap between theoretical and computational aspects of prime numbers
Exercise sections are a goldmine of interesting examples, pointers to the literature and potential research projects
Authors are well-known and highly-regarded in the field less
Exercise sections are a goldmine of interesting examples, pointers to the literature and potential research projects
Authors are well-known and highly-regarded in the field less
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From the reviews of the first edition:
"[This book] can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audience we know of no book on mathematics and its history that covers half as much nonstandard material. Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many." (Mathematical Intelligencer)
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"[This book] can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audience we know of no book on mathematics and its history that covers half as much nonstandard material. Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many." (Mathematical Intelligencer)
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The revised and updated edition includes three completely new chapters on the prediction and control of chaotic systems. It also incorporates new information regarding the solar system and an account of complexity theory. This witty, lucid and engaging book makes the complex mathematics of chaos accessible and entertaining. Presents complex mathematics in an accessible style. Includes three new chapters on prediction in chaotic systems, control of chaotic systems, and on the concept of chaos. Provides a discussion of complexity theory. less
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71
Asking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking "well-posed" questions. The approach taken by the authors in Problems in Algebraic Number Theory is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically... more
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"What makes Gardner so appealing is his ease in exploring deep ideas . . . and making them accessible to the interested but nontechnical reader. This is a special talent and no one has ever displayed it quite as well as he does." — Los Angeles Times
"Absorbing; enlightening; lucid; witty; inventive. An exemplar of science writing at its very best." — American Mathematical Monthly
A substantial revision of Martin Gardner's earlier well-known work on mirror symmetry and asymmetry, The New Ambidextrous Universe takes readers on an extraordinary journey. With... more
"Absorbing; enlightening; lucid; witty; inventive. An exemplar of science writing at its very best." — American Mathematical Monthly
A substantial revision of Martin Gardner's earlier well-known work on mirror symmetry and asymmetry, The New Ambidextrous Universe takes readers on an extraordinary journey. With... more
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These 3 puzzles involve the proof of a basic law governing the world of numbers known to be correct in all tested cases — the problem is to prove that the law is always correct. Includes van der Waerden's theorem on arithmetic progressions, the Landau-Schnirelmann hypothesis and Mann's theorem, and a solution to Waring's problem. less
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The computation of invariants of algebraic number fields such as integral bases, discriminants, prime decompositions, ideal class groups, and unit groups is important both for its own sake and for its numerous applications, for example, to the solution of Diophantine equations. The practical com- pletion of this task (sometimes known as the Dedekind program) has been one of the major achievements of computational number theory in the past ten years, thanks to the efforts of many people. Even though some practical problems still exist, one can consider the subject as solved in a satisfactory... more
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This book grew out of notes from several courses that the first author has taught over the past nine years at the California Institute of Technology, and earlier at the Johns Hopkins University, Cornell University, the University of Chicago, and the University of Crete. Our general aim is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. Our more particular goal is to cover Jolm Tate's visionary thesis, giving virtually all of the necessary analytic details and topological... more
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From the review: "The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further... more
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This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and in mathematical research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of... more
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This concise monograph in probability by Mark Kac, a well-known mathematician, presumes a familiarity with Lebesgue's theory of measure and integration, the elementary theory of Fourier integrals, and the rudiments of number theory. Readers may then follow Dr. Kac's attempt "to rescue statistical independence from the fate of abstract oblivion by showing how in its simplest form it arises in various contexts cutting across different mathematical disciplines."
The treatment begins with an examination of a formula of Vieta that extends to the notion of statistical independence. Subsequent... more
The treatment begins with an examination of a formula of Vieta that extends to the notion of statistical independence. Subsequent... more
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Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve... more
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The majority of students who take courses in number theory are mathematics majors who will not become number theorists. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, in addition to a careful presentation of the standard material usually taught in a first course in elementary number theory, this book includes a chapter on quadratic fields which the author has designed to make students think about some of the obvious concepts they have taken for granted earlier. The book also includes a... more
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81
First printed in 1967, this book has been essential reading for aspiring algebraic number theorists for more than forty years. It contains the lecture notes from an instructional conference held in Brighton in 1965, which was a milestone event that introduced class field theory as a standard tool of mathematics. There are landmark contributions from Serre and Tate. The book is a standard text for taught courses in algebraic number theory. This Second Edition includes a valuable list of errata compiled by mathematicians who have read and used the text over the years. less
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The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, as well as numerous clarifications. less
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In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory. In some ways a sequel to Nahin's An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of... more
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This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the... more
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"A very valuable addition to any mathematical library." — School Science and Math
This book, written by a prominent mathematician and Sterling Professor of Mathematics at Yale, differs from most other books on number theory in two important ways: first, it presents the principal ideas and methods of number theory within a historical and cultural framework, making the subject more tangible and easily grasped. Second, the material requires substantially less mathematical background than many comparable texts. Technical complications and mathematical requirements have been kept to a... more
This book, written by a prominent mathematician and Sterling Professor of Mathematics at Yale, differs from most other books on number theory in two important ways: first, it presents the principal ideas and methods of number theory within a historical and cultural framework, making the subject more tangible and easily grasped. Second, the material requires substantially less mathematical background than many comparable texts. Technical complications and mathematical requirements have been kept to a... more
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Hailed as one of the greatest mathematical results of the twentieth century, the recent proof of Fermat's Last Theorem by Andrew Wiles brought to public attention the enigmatic problem-solver Pierre de Fermat, who centuries ago stated his famous conjecture in a margin of a book, writing that he did not have enough room to show his truly marvelous demonstration. Along with formulating this proposition--xn+yn=zn has no rational solution for n > 2--Fermat, an inventor of analytic geometry, also laid the foundations of differential and integral calculus, established, together with... more
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Cryptography plays a key role in ensuring the privacy and integrity of data and the security of computer networks. Introduction to Modern Cryptography provides a rigorous yet accessible treatment of modern cryptography, with a focus on formal definitions, precise assumptions, and rigorous proofs.
The authors introduce the core principles of modern cryptography, including the modern, computational approach to security that overcomes the limitations of perfect secrecy. An extensive treatment of private-key encryption and message authentication follows. The authors also illustrate... more
The authors introduce the core principles of modern cryptography, including the modern, computational approach to security that overcomes the limitations of perfect secrecy. An extensive treatment of private-key encryption and message authentication follows. The authors also illustrate... more
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Updated in a seventh edition, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers, and also touches on matters of deep mathematical significance. This new edition includes state of the art material on the use of computers in number theory, as well as taking full account of the proving of Fermat's last theorem. less
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91
This undergraduate textbook describes the computational aspects of number theory, such as techniques of factoring. Problems of varying difficulty are used throughout the text to aid comprehension. less
92
The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e. g. the class field theory on which 1 make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collec- tion of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of W eber, Hasse, Hecke, and Hilbert's... more
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Why seemingly unrelated mathematical truths are connected in simple and beautiful equations continues to stump even mathematicians. This recreational math book takes the reader on a fantastic voyage into the world of natural numbers. From the earliest discoveries of the ancient Greeks to various fundamental characteristics of the natural number sequence, Clawson explains fascinating mathematical mysteries in clear and easy prose. He delves into the heart of number theory to see and understand the exquisite relationships among natural numbers, and ends by exploring the ultimate mystery of... more
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This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.
The second... more
The second... more
95
One of the outstanding voices of his generation, David Foster Wallace has won a large and devoted following for the intellectual ambition and bravura style of his fiction and essays. Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity.
Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's... more
Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's... more
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In this engaging and readable book, Dr. K�rner describes a variety of lively topics that continue to intrigue professional mathematicians. The topics range from the design of anchors and the Battle of the Atlantic to the outbreak of cholera in Victorian Soho. The author uses relatively simple terms and ideas, yet explains difficulties and avoids condescension. If you are a mathematician who wants to explain to others how you spend your working days, then seek inspiration here. This book will appeal to everyone interested in the uses of mathematics. less
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The Trachtenberg Speed System of Basic Mathematics is a revolutionary system for calculating and teaching basic math. Children, who had repeatedly failed in arithmetic until their parents sent them to learn this method, were able to perform amazing calculations within seconds. In one demonstration, a ten year old kid when asked to multiply 5132437201 times 452736502785 simply wrote on the blackboard the answer, 2323641669144374104785 in seventy seconds. He never set up the basic and familiar line by line chart, multiplying and adding each row of numbers. Instead, he just wrote down the answer... more
Recommended by Tyler Cowen, and 1 others. Tyler CowenFrom this I learned how powerful the individual human mind could be, and also how much school wasn’t teaching me. It began to occur to me that the mainstream doesn’t necessarily have the best or only methods. That said, non-mainstream approaches still have the responsibility of coming up with the right answer. Query: does it these days ever make sense to actually use this stuff? (Source)
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Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves. New to the Second Edition
Chapters on isogenies and hyperelliptic curves A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational... more
Chapters on isogenies and hyperelliptic curves A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational... more
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This revised and enlarged sixth edition of Proofs from THE BOOK features an entirely new chapter on Van der Waerden's permanent conjecture, as well as additional, highly original and delightful proofs in other chapters.
From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition"
"... It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [...] This book does an invaluable service to... more
From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition"
"... It is almost impossible to write a mathematics book that can be read and enjoyed by people of all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [...] This book does an invaluable service to... more
Don't have time to read the top Number Theory books of all time? Read Shortform summaries.
Shortform summaries help you learn 10x faster by:
- Being comprehensive: you learn the most important points in the book
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- Interactive exercises: apply the book's ideas to your own life with our educators' guidance.