Matematikens grunder

*A Financial Times and Waterstones Book of the Year 2025* Many artists are unaware of the mathematics that bubble beneath their craft, while some consciously use it for inspiration. Our instincts might tell us that these two subjects are incompatible forces with nothing in common, but what if we’re wrong? Marcus du Sautoy, acclaimed mathematician and Simonyi Professor for the Public Understanding of Science at the University of Oxford, looks to art, music, design and literature to uncover the key mathematical structures that underpin both human creativity and the natural world.Blueprints takes us from the earliest stone circles to the modernist architecture of Le Corbusier, from Bach’s circular compositions to Radiohead’s disruptive soundscapes, and from Shakespeare’s hidden numerical clues to the Dada artists who embraced randomness. Instead of polar opposites we find a complementary relationship that spans a vast historical and geographic landscape.Whether we are searching for meaning in an abstract painting or deciphering poetry, there are blueprints everywhere: prime numbers, symmetry, fractals and the weirder worlds of Hamiltonian cycles and hyperbolic geometry. Nature similarly exploits these structures to achieve the wonders of our universe.In this innovative and delightfully bold exploration of creativity, Marcus explains how we make art, why a creative mindset is vital for discovering new mathematics and how a fundamental connection to the natural world intrinsically links these two subjects.‘Blueprints is an extraordinary book which shows us how mathematics and art are connected through structures. Du Sautoy shows us how to bridge the divide of science and the humanities and proves that we can only face and solve the big challenges of the twenty-first century if we go beyond the fear of pooling knowledge’ Hans Ulrich Obrist, Artistic Director, Serpentine Galleries

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.

Test your powers of logic with 200 puzzles from The Times.Just like a mathematical crossword, these puzzles are ideal for fans of Su Doku who relish grappling with figures. Contains four levels of difficulty, combining arithmetic with an intersecting layout: Mild, Medium, Tricky and Tough.Kakuro originated in the United States in the 1950s under the name of Cross Sums. A logic puzzle, the aim is to complete a grid using digits from 1 to 9, with the digits in each row/column adding up to the clue for that block. It’s similar to Su Doku in the sense that digits cannot be repeated in a row/column.Also known as: Croco Puzzle, Cross Sums, CrossSam, Kasan Kurosu, Kreuzsummenrätsel,Tashizan Cross, ZahlenschwedeAre you up to the challenge?

Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life - from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.

There is an explosion of interest in dynamical systems in the mathematical community as well as in many areas of science. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Scientists and engineers realize the power and the beauty of the geometric and qualitative techniques. These techniques apply to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics.Computer graphics have allowed us to view the dynamical behavior geometrically. The appearance of incredibly beautiful and intricate objects such as the Mandelbrot set, the Julia set, and other fractals have really piqued interest in the field.This is text is aimed primarily at advanced undergraduate and beginning graduate students. Throughout, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory.The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in the past century and its importance continues to expand. There is no question that the field is becoming more and more important in a variety of scientific disciplines.New to this edition:•Greatly expanded coverage complex dynamics now in Chapter 2•The third chapter is now devoted to higher dimensional dynamical systems.•Chapters 2 and 3 are independent of one another.•New exercises have been added throughout.

For thousands of years, mathematicians have used the timeless art of logic to see the world more clearly. In The Art of Logic, Royal Society Science Book Prize nominee Eugenia Cheng shows how anyone can think like a mathematician - and see, argue and think better.Learn how to simplify complex decisions without over-simplifying them. Discover the power of analogies and the dangers of false equivalences. Find out how people construct misleading arguments, and how we can argue back.Eugenia Cheng teaches us how to find clarity without losing nuance, taking a careful scalpel to the complexities of politics, privilege, sexism and dozens of other real-world situations. Her Art of Logic is a practical and inspiring guide to decoding the modern world.

A marvelous compendium of mathematical symbols and their fascinating historiesGalileo famously wrote that the book of nature is written in mathematical language. The Language of Mathematics is a wide-ranging and beautifully illustrated collection of short, colorful histories of the most commonly used symbols in mathematics, providing readers with an engaging introduction to the origins, evolution, and conceptual meaning of each one.In dozens of lively and informative entries, Raúl Rojas shows how today’s mathematics stands on the shoulders of giants, mathematicians from around the world who developed mathematical notation through centuries of collective effort. He tells the stories of such figures as al-Khwārizmī, René Descartes, Joseph-Louis Lagrange, Carl Friedrich Gauss, Augustin-Louis Cauchy, Karl Weierstrass, Sofia Kovalevskaya, David Hilbert, and Kenneth Iverson. Topics range from numbers and variables to sets and functions, constants, and combinatorics. Rojas describes the mathematical problems associated with different symbols and reveals how mathematical notation has sometimes been an accidental process. The entries are self-contained and can be read in any order, each one examining one or two symbols, their history, and the variants they may have had over time.An essential companion for math enthusiasts, The Language of Mathematics shows how mathematics is a living and evolving entity, forever searching for the best symbolism to express relationships between abstract concepts and to convey meaning.

Introduction to Mathematical Proofs helps students develop the necessary skills to write clear, correct, and concise proofs.Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs.This new edition includes more than 125 new exercises in sections titled More Challenging Exercises. Also, numerous examples illustrate in detail how to write proofs and show how to solve problems. These examples can serve as models for students to emulate when solving exercises. Several biographical sketches and historical comments have been included to enrich and enliven the text. Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It prepares them to succeed in more advanced mathematics courses, such as abstract algebra and analysis.

'Wooldridge is like the teacher you wish you'd had' GUARDIAN'This book is simply brilliant . . . I learned so much' SIR DAVID SPIEGELHALTER, UNIVERSITY OF CAMBRIDGE'Fresh, clever and playful' MATT RIDLEY, AUTHOR OF THE ORIGINS OF VIRTUE'Michael explains the complex, strategic structures of our world with masterful beauty'MILIND TAMBE, HARVARD UNIVERSITY'Effortlessly covers a broad swath of core ideas in game theory, with a sprinkle of British humour'EDITH ELKIND, NORTHWESTERN UNIVERSITY'A wonderful book' YOAV SHOHAM, STANFORD UNIVERSITY______A fascinating deep-dive into game theory, its history and its practical applications by award-winning expert on computer science and artificial intelligence.Most of us have some idea about game theory. Concepts like zero-sum games and the prisoner's dilemma crop up in films, novels and casual conversation, often as shorthand for cut-throat competition or inevitable betrayal. But game theory, the science behind these ideas, is widely misunderstood. All too often it's seen solely as the science of conflict and greed. In fact, this field has a lot to teach us about how to make a better world. It shows us just what it takes to enable cooperation and mutual benefit.This essential primer presents twenty one 'life lessons from game theory', which illustrate the key ideas in the field, and which are packed with real world examples: from overfishing in the Atlantic; to the Cuban Missile Crisis; to security measures at international airports; to advertising wars between major brands. This is a hopeful and practical book that highlights the value of game theory and provides a course correction for its integration into the real world.

One of the most illuminating, useful and exciting books ever published in the mathematical fieldTaking only a modicum of knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus.His illuminating explanation is addressed to the person who wants to understand the place of mathematics in modern civilization but who has been intimidated by its supposed difficulty.Mathematics is the language of size, shape, and order – a language Hogben shows one can both master and enjoy.

'Wooldridge is like the teacher you wish you'd had'GUARDIAN'This book is simply brilliant . . . I learned so much'SIR DAVID SPIEGELHALTER, UNIVERSITY OF CAMBRIDGE'Fresh, clever and playful'MATT RIDLEY, AUTHOR OF THE ORIGINS OF VIRTUE'Michael explains the complex, strategic structures of our world with masterful beauty'MILIND TAMBE, HARVARD UNIVERSITY'Effortlessly covers a broad swath of core ideas in game theory, with a sprinkle of British humour'EDITH ELKIND, NORTHWESTERN UNIVERSITY'A wonderful book'YOAV SHOHAM, STANFORD UNIVERSITY______A fascinating deep-dive into game theory, its history and its practical applications by award-winning expert on computer science and artificial intelligence.Most of us have some idea about game theory. Concepts like zero-sum games and the prisoner's dilemma crop up in films, novels and casual conversation, often as shorthand for cut-throat competition or inevitable betrayal. But game theory, the science behind these ideas, is widely misunderstood. All too often it's seen solely as the science of conflict and greed. In fact, this field has a lot to teach us about how to make a better world. It shows us just what it takes to enable cooperation and mutual benefit.This essential primer presents twenty one 'life lessons from game theory', which illustrate the key ideas in the field, and which are packed with real world examples: from overfishing in the Atlantic; to the Cuban Missile Crisis; to security measures at international airports; to advertising wars between major brands. This is a hopeful and practical book that highlights the value of game theory and provides a course correction for its integration into the real world.

Test your powers of logic with 200 puzzles from The TimesJust like a mathematical crossword, these puzzles are ideal for fans of Su Doku who relish grappling with figures. Contains four levels of difficulty, combining arithmetic with an intersecting layout: Mild, Medium, Tricky and Tough.Kakuro originated in the United States in the 1950s under the name of Cross Sums. A logic puzzle, the aim is to complete a grid using digits from 1 to 9, with the digits in each row/column adding up to the clue for that block. It’s similar to Su Doku in the sense that digits cannot be repeated in a row/column.Also known as: Croco Puzzle, Cross Sums, CrossSam, Kasan Kurosu, Kreuzsummenrätsel,Tashizan Cross, ZahlenschwedeAre you up to the challenge?

This easy-to-understand textbook introduces the mathematical language and problem-solving tools essential to anyone wishing to enter the world of computer and information sciences. Specifically designed for the student who is intimidated by mathematics, the book offers a concise treatment in an engaging style.The thoroughly revised third edition features a new chapter on relevance-sensitivity in logical reasoning and many additional explanations on points that students find puzzling, including the rationale for various shorthand ways of speaking and ‘abuses of language’ that are convenient but can give rise to misunderstandings. Solutions are now also provided for all exercises.Topics and features: presents an intuitive approach, emphasizing how finite mathematics supplies a valuable language for thinking about computation; discusses sets and the mathematical objects built with them, such as relations and functions, as well as recursion and induction; introduces core topics of mathematics, including combinatorics and finite probability, along with the structures known as trees; examines propositional and quantificational logic, how to build complex proofs from simple ones, and how to ensure relevance in logic; addresses questions that students find puzzling but may have difficulty articulating, through entertaining conversations between Alice and the Mad Hatter; provides an extensive set of solved exercises throughout the text.This clearly-written textbook offers invaluable guidance to students beginning an undergraduate degree in computer science. The coverage is also suitable for courses on formal methods offered to those studying mathematics, philosophy, linguistics, economics, and political science. Assuming only minimal mathematical background, it is ideal for both the classroom and independent study.

Derived from courses the author taught at Harvard and Johns Hopkins, this original book introduces the concepts of category theory ― categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctive, monads ― and other topics, revisiting a broad range of mathematical examples from the categorical perspective. Basic set theory and logic are the only prerequisites.

'Fascinating and immensely readable' Ian Stewart 'A charming tour through the realm of the very, very, very numerous, from the ancient world through to the distant future' Jordan Ellenberg, author of How Not to Be WrongWhat if, every time you wanted to write down 1,000,000, you had to draw a picture of a god? And what if that number were the biggest you had a symbol for? In ancient Egypt, those were the rules: anything bigger broke maths. Writing down some numbers is still beyond us today: try it with all the zeroes in a googolplex, or an outrageous alien number like FISH 7. Even harnessing every particle in the universe, you wouldn't come close. But that hasn't stopped us from hunting down these mind-bendingly big numbers and studying them.In Huge Numbers, mathematician and Numberphile presenter Richard Elwes shows how counting has shaped human thought. Whether recorded with notches carved on a tally stick, beads on an abacus, or electrical signals carrying binary code, it allows us to test the limits of mathematics over and over, breaking it down and putting it back together again.Come on a fascinating tour that spans continents and millennia, from the Mayan calendar to today's chatbots. You'll see that huge numbers are everywhere, expanding our horizons and powering our modern world.

'Fascinating and immensely readable' Ian Stewart 'A charming tour through the realm of the very, very, very numerous, from the ancient world through to the distant future' Jordan Ellenberg, author of How Not to Be WrongWhat if, every time you wanted to write down 1,000,000, you had to draw a picture of a god? And what if that number were the biggest you had a symbol for? In ancient Egypt, those were the rules: anything bigger broke maths. Writing down some numbers is still beyond us today: try it with all the zeroes in a googolplex, or an outrageous alien number like FISH 7. Even harnessing every particle in the universe, you wouldn't come close. But that hasn't stopped us from hunting down these mind-bendingly big numbers and studying them.In Huge Numbers, mathematician and Numberphile presenter Richard Elwes shows how counting has shaped human thought. Whether recorded with notches carved on a tally stick, beads on an abacus, or electrical signals carrying binary code, it allows us to test the limits of mathematics over and over, breaking it down and putting it back together again.Come on a fascinating tour that spans continents and millennia, from the Mayan calendar to today's chatbots. You'll see that huge numbers are everywhere, expanding our horizons and powering our modern world.

Combining stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic, this new textbook for first courses in mathematical logic was written by the subject's creative master. Raymond Smullyan offers clear, incremental presentations of difficult logic concepts with creative explanations and unique problems related to proofs, propositional logic and first-order logic, undecidability, recursion theory, and other topics.

The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry.

*A Financial Times and Waterstones Book of the Year 2025* Many artists are unaware of the mathematics that bubble beneath their craft, while some consciously use it for inspiration. Our instincts might tell us that these two subjects are incompatible forces with nothing in common, but what if we’re wrong? Marcus du Sautoy, acclaimed mathematician and Simonyi Professor for the Public Understanding of Science at the University of Oxford, looks to art, music, design and literature to uncover the key mathematical structures that underpin both human creativity and the natural world.Blueprints takes us from the earliest stone circles to the modernist architecture of Le Corbusier, from Bach’s circular compositions to Radiohead’s disruptive soundscapes, and from Shakespeare’s hidden numerical clues to the Dada artists who embraced randomness. Instead of polar opposites we find a complementary relationship that spans a vast historical and geographic landscape.Whether we are searching for meaning in an abstract painting or deciphering poetry, there are blueprints everywhere: prime numbers, symmetry, fractals and the weirder worlds of Hamiltonian cycles and hyperbolic geometry. Nature similarly exploits these structures to achieve the wonders of our universe.In this innovative and delightfully bold exploration of creativity, Marcus explains how we make art, why a creative mindset is vital for discovering new mathematics and how a fundamental connection to the natural world intrinsically links these two subjects.‘Blueprints is an extraordinary book which shows us how mathematics and art are connected through structures. Du Sautoy shows us how to bridge the divide of science and the humanities and proves that we can only face and solve the big challenges of the twenty-first century if we go beyond the fear of pooling knowledge’ Hans Ulrich Obrist, Artistic Director, Serpentine Galleries

This classic by one of the 20th century's most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. Topics include the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters.

Test your powers of logic with these challenging Su Doku puzzles.Specially compiled to provide the most deadly Su Doku challenge, this is the only volume for Su Doku enthusiasts who need a puzzle that really tests their mettle.Prepare yourself for the toughest Su Doku challenge there is. These diabolically difficult Ultimate Killer Su Doku puzzles will really put your brainpower to the test as you ‘warm up’ with the 100 Deadly Killer puzzles before steeling yourself to take on the 100 Extra Deadly Su Dokus. Are you ready for the challenge?Not for the faint-hearted. The puzzles use the same 9x9 grid as a regular Su Doku, but have an extra mathematical element that multiplies the challenge. The aim is not only to complete every row, column and cube so that it contains the digits 1 to 9, but also to make sure that the outlined sections, called cages, add up to the number given in each cage.Warning: Not suitable for amateur puzzlers!