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How Numbers Work: Discover the strange and beautiful world of mathematics (New Scientist Instant Expert)
by New ScientistHow Numbers Work is a tour of the mind-blowing but beautiful realm of numbers and the mathematical rules that connect them.Think of a number between one and ten. No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it?How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself. ABOUT THE SERIESNew Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.(P)2018 Hodder & Stoughton Limited
How Pi Can Save Your Life: Using Math to Survive Plane Crashes, Zombie Attacks, Alien Encounters, and Other Improbable Real-World Situations
by Chris WaringDiscover how you can guard a priceless painting in the Louvre, pilot a space shuttle back to Earth, and survive other ridiculous-yet-possible scenarios using the all-encompassing power of mathematical equations.Whether you paid much attention in math class or not, the inescapable truth is that real life is full of equations. You use differentiation when driving from point A to B and apply basic geometry when you&’re crossing the road between traffic, even if you don&’t realize it. But what if you were plummeting to your death inside a plane with no engine and you needed to know what size parachute to make from your cabinmate&’s sari in order to jump and survive? How Pi Can Save Your Life tackles some frankly ridiculous scenarios with essential, bullet-proof equations that you'll probably never apply in real life. But before you get into the life-saving mathematics, this book will start you off with the essential math that will be the key to taking on the wild scenarios later on in this book. Using only the power of equations, you will be able to: Communicate with an alien civilization Save your town from a zombie apocalypse Contain a major oil spill Excavate a fossil that could be a major scientific discovery Perform Hollywood stunts in a blockbuster action film And more! How Pi Can Save Your Life will take your basic understanding of mathematical equations to the next level, teaching you that the real key to surviving in an unpredictable world is understanding the natural forces and relationships that rule everything we do.
How Qualitative Data Analysis Happens: Moving Beyond “Themes Emerged”
by Áine M. Humble M. Elise RadinaHow Qualitative Data Analysis Happens: Moving Beyond “Themes Emerged”, offers an in-depth look into how qualitative social science researchers studying a wide range of human experiences and dynamics approach their data analyses. This expanded edition consists of 13 new chapters from a broad range of disciplines (and an added conclusion) that document the stories about how qualitative data analysis occurred.Chapters for this expanded edition represent a diversity of disciplines (e.g., criminology, family science, education, health, nutrition, sociology, sport psychology) that focus on the human experience and describe a diversity of methodological approaches. These chapters may be used to introduce readers to newer or innovative ways of analysing data. It moves beyond the usual vague statement of “themes emerged from the data” to show readers how researchers actively and consciously arrive at their themes and conclusions, revealing the complexity and time involved in making sense of thousands of pages of interview data, multiple data sources, and diverse types of data. The various authors provide detailed narratives into how they analysed their data from previous publications. The methodologies range from arts-based research, autoethnography, community-based participatory research, ethnography, grounded theory, to narrative analysis. The volume allows readers to be seemingly “in the room” with these international scholars (representing Canada, the US, Austria, Germany, the UK, and the Philippines) and getting their own hands vicariously dirty with the data.This expanded edition also includes a conclusion chapter, in which the authors reflect on commonalities across the chapters. Supplemental figures, images, and screenshots, which are referred to in the chapters, are included in an accompanying eResource (that can be accessed at www.routledge.com/ 9781032183213), as well as links to the previously published work on which the chapters are based. This book is an invaluable resource for experienced and novice qualitative researchers throughout the social sciences, as well as undergraduate and postgraduate students in the field.
How Qualitative Data Analysis Happens: Moving Beyond "Themes Emerged"
by Áine Humble Elise RadinaWinner of the 2020 Anselm Strauss Award for Qualitative Family Research, National Council on Family Relations. How is qualitative data actually collected, analyzed, and accomplished? Real stories of How Qualitative Data Analysis Occurs: Moving Beyond "Themes Emerged" offers an in-depth look into how qualitative social science researchers studying family issues and dynamics approach their data analyses. It moves beyond the usual vague statement of "themes emerged from the data" to show readers how researchers actively and consciously arrive at their themes and conclusions, revealing the complexity and time involved in making sense of thousands of pages of interview data, multiple data sources, and diverse types of data. How Qualitative Data Analysis Occurs focuses on a diversity of topics in family research across the life course. The various authors provide detailed narratives into how they analyzed their data from previous publications, and what methodologies they used, ranging from arts-based research, autoethnography, community-based participatory research, ethnography, grounded theory, to narrative analysis. Supplemental figures, images, and screenshots which are referred to in the chapters, are included in an accompanying eResource, as well as links to the previously published work on which the chapters are based. This book is an invaluable resource for experienced and novice qualitative researchers throughout the social sciences.
How Round Is Your Circle?: Where Engineering and Mathematics Meet
by John Bryant Chris SangwinHow do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. How Round Is Your Circle? invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun. John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, How Round Is Your Circle? reveals some of the hidden complexities in everyday things.
How Slow Is a Sloth?: Measure the Rainforest (Nature Numbers)
by Jill EsbaumIn Nature Numbers, math is beautiful, recognizable, and all around us! Highly engaging pictures of animals and nature scenes, along with cool chalk illustrations, are used to introduce basic math concepts and encourage kids to see a world of numbers all around them.K-2 math concepts include measuring. This book explores measuring elements of the rainforest with amazing nature pictures and chalk illustrations!
How Small Social Systems Work: From Soccer Teams to Jazz Trios and Families (The Frontiers Collection)
by Yair NeumanMost of us are intuitively familiar with small social systems, such as families and soccer teams. Surprisingly, though, most of us are unaware of how complex these systems are or of the fact that they have a unique character distinguishing them from both populations and individuals. The current manuscript, which emerged from high-level scientific publications on the subject, aims to bridge this gap in our understanding of small social systems. The book aims to explain, illustrate, and model the unique and fascinating nature of small (social) systems by relying on deep scientific foundations and by using examples from sport, movies, music, and the martial arts. To support its friendly exposition of challenging scientific ideas, the book also discusses entertaining questions such as (1) why inviting your mother-in-law to dinner might be a challenging event, for reasons you have never considered; (2) why soccer teams should be messy in order to win; (3) why Nazis are deeply wrong in their understanding of the importance of entropy; and (4) why “panda fighters” failed in the UFC (Ultimate Fighting Championship)."How Small Systems Work is a welcome book, which sheds light on a branch of mathematics overlooked by scholars: how networks store information. Focusing on small systems, the book asks fundamental questions, providing the tools (and the examples) for answering them –with fun. Neuman analyses, with plenty of humor, the dynamics of a family of cats, the pleasure of listening to jazz, and the science behind football championships, while uncovering hidden gems in the history of cinema”Dr. Mario Alemi, author of “The Amazing Journey of Reason: from DNA to Artificial Intelligence”
How the Brain Learns Mathematics
by Dr David A. SousaTo reach all your math students, use your brain—and theirs, too! This updated bestseller takes readers to the next level with new brain-friendly strategies backed by the latest research and even more ways to seamlessly incorporate what you learn about your students’ developing minds into your math classroom. Discover the cognitive mechanisms for learning math, explore factors that contribute to learning difficulties, and follow a four-step teaching model that relates classroom experience to real-world applications. Features include: New strategies for motivating adolescents Integration of the arts into mathematics instruction New information on how technology affects attention and memory Expanded sections on number sense and ELL instruction More than 160 new references
How the World Really Works: The Science Behind How We Got Here and Where We're Going
by Vaclav SmilINSTANT NEW YORK TIMES BESTSELLER&“A new masterpiece from one of my favorite authors… [How The World Really Works] is a compelling and highly readable book that leaves readers with the fundamental grounding needed to help solve the world&’s toughest challenges.&”—Bill Gates &“Provocative but perceptive . . . You can agree or disagree with Smil—accept or doubt his &‘just the facts&’ posture—but you probably shouldn&’t ignore him.&”—The Washington PostAn essential analysis of the modern science and technology that makes our twenty-first century lives possible—a scientist's investigation into what science really does, and does not, accomplish.We have never had so much information at our fingertips and yet most of us don&’t know how the world really works. This book explains seven of the most fundamental realities governing our survival and prosperity. From energy and food production, through our material world and its globalization, to risks, our environment and its future, How the World Really Works offers a much-needed reality check—because before we can tackle problems effectively, we must understand the facts. In this ambitious and thought-provoking book we see, for example, that globalization isn&’t inevitable—the foolishness of allowing 70 per cent of the world&’s rubber gloves to be made in just one factory became glaringly obvious in 2020—and that our societies have been steadily increasing their dependence on fossil fuels, such that any promises of decarbonization by 2050 are a fairy tale. For example, each greenhouse-grown supermarket-bought tomato has the equivalent of five tablespoons of diesel embedded in its production, and we have no way of producing steel, cement or plastics at required scales without huge carbon emissions. Ultimately, Smil answers the most profound question of our age: are we irrevocably doomed or is a brighter utopia ahead? Compelling, data-rich and revisionist, this wonderfully broad, interdisciplinary guide finds faults with both extremes. Looking at the world through this quantitative lens reveals hidden truths that change the way we see our past, present and uncertain future.
How Things Work: The Computer Science Edition (How Things Work Series)
by Charles F. BowmanIt’s axiomatic to state that people fear what they do not understand, and this is especially true when it comes to technology. However, despite their prevalence, computers remain shrouded in mystery, and many users feel apprehensive when interacting with them. Smartphones have only exacerbated the issue. Indeed, most users of these devices leverage only a small fraction of the power they hold in their hands. How Things Work: The Computer Science Edition is a roadmap for readers who want to overcome their technophobia and harness the full power of everyday technology. Beginning with the basics, the book demystifies the mysterious world of computer science, explains its fundamental concepts in simple terms, and answers the questions many users feel too intimidated to ask. By the end of the book, readers will understand how computers and smart devices function and, more important, how they can make these devices work for them. To complete the picture, the book also introduces readers to the darker side of modern technology: security and privacy concerns, identity theft, and threats from the Dark Web.
How to Account for Sustainability: A Simple Guide to Measuring and Managing (Doshorts Ser.)
by Laura MusikanskiLearn how to measure, manage and account for sustainability in your business in clear, simple and feasible steps.This book takes you from concept to innovation and back to action items for all aspects of sustainability. Each chapter has four sections: (1) a specific description of sustainability challenges, (2) an example of a business making a profit by sustainability problem, (3) an exercise challenging the reader to identify business solutions and (4) clear, simple takeaways.The book is structured around the world’s most accepted guidelines for sustainability reporting, the Global Reporting Initiative.
How to Ace Calculus: The Streetwise Guide
by Joel Hass Colin Adams Abigail ThompsonWritten by three gifted-and funny-teachers, How to Ace Calculus provides humorous and readable explanations of the key topics of calculus without the technical details and fine print that would be found in a more formal text. Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams-all the tricks of the trade that will make learning the material of first-semester calculus a piece of cake. Funny, irreverent, and flexible, How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun.
How to Ace the Rest of Calculus: The Streetwise Guide (How to Ace)
by Colin Adams Abigail Thompson Joel HassThe sequel to How to Ace Calculus, How to Ace the Rest of Calculus provides humorous and highly readable explanations of the key topics of second and third semester calculus-such as sequences and series, polor coordinates, and multivariable calculus-without the technical details and fine print that would be found in a formal text.
How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics
by Eugenia ChengWhat is math? How exactly does it work? And what do three siblings trying to share a cake have to do with it? In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard. Of course, it's not all about cooking; we'll also run the New York and Chicago marathons, take a closer look at St. Paul's Cathedral, pay visits to Cinderella and Lewis Carroll, and even get to the bottom of why we think of a tomato as a vegetable. At the heart of it all is Cheng's work on category theory, a cutting-edge "mathematics of mathematics," that is about figuring out how math works. This is not the math of our high school classes: seen through category theory, mathematics becomes less about numbers and formulas and more about how we know, believe, and understand anything, including whether our brother took too much cake.Many of us think that math is hard, but, as Cheng makes clear, math is actually designed to make difficult things easier. Combined with her infectious enthusiasm for cooking and a true zest for life, Cheng's perspective on math becomes this singular book: a funny, lively, and clear journey through a vast territory no popular book on math has explored before. How to Bake Pi offers a whole new way to think about a field all of us think we know; it will both dazzle the constant reader of popular mathematics and amuse and enlighten even the most hardened math-phobe.So, what is math? Let's look for the answer in the kitchen.
How to Be a Math Genius: Your Brilliant Brain and How to Train It (DK Train Your Brain)
by DKRefine your math skills and be well on your way to becoming a numbers whizz!Learn about the intriguing and wonderful world of mathematics, how your brain works to interpret numbers and shapes, and how to improve your math skills in this informative math book. This educational fact-filled guide to math will have you beaming with newfound knowledge. It includes: • Fun things to do, from brainteasers to puzzles • Clear, engaging text that demystifies math • Tips and techniques to help you boost their brainpower • Brand new biographies of pioneering mathematicians, such as Grace Hopper and Katherine Johnson • Lively illustrations that bring the topic to life and make the subject matter engaging for kids • Creative math exercises and activities put kids&’ skills to the test This informative educational book allows young readers to understand the basic ideas behind math. It not only teaches new math skills, but also provides them with greater confidence in their own ability to handle numbers and mathematical problems. How To Be A Math Genius puts the ideas into context to help children understand why math is useful and even exciting! Uncover the amazing sphere of algebra, puzzling primes, super sequences and special shapes. Challenge yourself with thrilling quizzes, solve dynamic puzzles and crack complex codes and inspirational geometrical illusions. Learn about the big names and even bigger brains who made mathematical history such as Pythagoras, Grace Hopper and Alan Turing. The books in the Train Your Brain series make complex subjects fun, accessible and exciting, and are perfect for any child! Journey through science subjects at home and have fun with clever activities! Other books in the series include Train Your Brain: How to be a Genius.
How to Be a Math Wizard (Careers for Kids)
by DKBring math to life with this exciting new math activity book for kids.With 30 activities and crafts that children can play their way through, this kids' book takes the fear out of math--and gives you the skills to become a math expert. Children will find out that being a mathematician isn't just about scrawling numbers on a dusty chalkboard--it's part of our everyday lives! How to Be a Math Wizard teaches kids to think like math pros as they ask mathematical questions and get hands-on with key math concepts. From calculation and numbers; to space, shape, and measure; all core curriculum math topics are covered in creative and engaging ways. This book invites kids to investigate math in a playful, hands-on way, using things from around the house: find out about perimeter by designing and building a mini house, practice multiplication through a game of bingo, sweeten probability through chocolate, and uncover the mystery of coordinates through a treasure map.If you like solving problems, making things, and learning facts, then this is the book for you, mathematician!
How to be a Quantitative Ecologist: The 'A to R' of Green Mathematics and Statistics
by Jason MatthiopoulosEcological research is becoming increasingly quantitative, yet students often opt out of courses in mathematics and statistics, unwittingly limiting their ability to carry out research in the future. This textbook provides a practical introduction to quantitative ecology for students and practitioners who have realised that they need this opportunity. The text is addressed to readers who haven't used mathematics since school, who were perhaps more confused than enlightened by their undergraduate lectures in statistics and who have never used a computer for much more than word processing and data entry. From this starting point, it slowly but surely instils an understanding of mathematics, statistics and programming, sufficient for initiating research in ecology. The book's practical value is enhanced by extensive use of biological examples and the computer language R for graphics, programming and data analysis. Key Features: Provides a complete introduction to mathematics statistics and computing for ecologists. Presents a wealth of ecological examples demonstrating the applied relevance of abstract mathematical concepts, showing how a little technique can go a long way in answering interesting ecological questions. Covers elementary topics, including the rules of algebra, logarithms, geometry, calculus, descriptive statistics, probability, hypothesis testing and linear regression. Explores more advanced topics including fractals, non-linear dynamical systems, likelihood and Bayesian estimation, generalised linear, mixed and additive models, and multivariate statistics. R boxes provide step-by-step recipes for implementing the graphical and numerical techniques outlined in each section. How to be a Quantitative Ecologist provides a comprehensive introduction to mathematics, statistics and computing and is the ideal textbook for late undergraduate and postgraduate courses in environmental biology.
How to Be Good at Math Workbook Grades 2-3 (DK How to Be Good at)
by DKUnleash your child&’s inner math genius and help them master math for Grades 2 and 3!Whether you enjoy math or not, it&’s an essential subject to understand. See how everything adds up with this fully illustrated home-study guide.Get inspired by numbers and see how mathematical explanations come to life with this engaging math book for kids! It includes: • Full color, with a clear layout. • Clear instructions that are easy for children to follow by themselves. • Answers that are given at the back of the book. • Practice questions and practical exercises to help expand your child&’s knowledge of the subject. Make math manageableHow to be Good at Math Grade 2-3 keeps the math simple and easy to understand! It comes packed with eye-catching illustrations and easy-to-follow instructions to teach kids everything they need to know about math. This brilliant visual math workbook is ideal for reinforcing classroom teaching. It helps kids understand what they&’ve learned in school and gives them extra math revision practice before an important test!Perfect for kids ages 7-9, this colorful math practice book covers all the key areas of the school curriculum for this level. It includes fractions, multiplication, divisio, measurement, geometry, coordinates, data handling and graphs. And there are answers at the back to check that you're on the right path.This engaging and clear workbook accompanies How to be Good at Math Grade 4-6, which covers ages 9-11 (Grades 4, 5, and 6).Discover How to be Good in other subjectsDK&’s successful How to be Good at... workbook series provides your child with the tools to learn how to look at the world around them and figure out how it works. There are more books to discover! Learn all about the influence of science and technology in the modern age with How to Be Good at Science, Technology, and Engineering.
How to Be Good at Math Workbook, Grades 4-6: The simplest–ever visual workbook (DK How to Be Good at)
by DKHelp your child learn and master grade 4, 5 and 6 math in no timeWhether you&’re good at math or not, it&’s an essential subject to understand. Luckily, you don&’t have to be a math genius to follow along with this fully illustrated home-study guide!Get inspired by numbers and see how mathematical explanations come to life with this engaging math book for kids! It includes: • Full color pages, with clear, and easy to comprehend layouts • Clear instructions that are easy for children to follow by themselves. • Answer guides at the back of the book • Practice questions and practical exercises to help expand your child&’s knowledge of the subject. Make math manageablePacked with eye-catching illustrations and easy-to-follow instructions, How to be Good at Math Grade 4-6 continues to keep the math simple and easy to understand for kids! This brilliant visual math workbook teaches them everything they need to know about math. Ideal for reinforcing classroom teaching, it helps kids understand what they&’ve learned in school and gives them extra math revision practice before an important test!Perfect for children ages 9-11, this colorful math practice book covers all the key areas of the school curriculum for this level. It includes working with fractions and decimal numbers, percentages, long multiplication and division, measurement, geometry, coordinates, statistics, probability and basic algebra. And there are answers at the back to check that you&’re on the right path.This engaging and clear workbook accompanies How to be Good at Math Grade 2-3, which covers ages 7-9 (Grades 2 and 3).Help your child get better in other topicsDK&’s successful How to be Good at... workbook series provides your child with the tools to learn how to look at the world around them and figure out how it works. There are more books to discover! Learn all about the influence of science and technology in the modern age with How to Be Good at Science, Technology, and Engineering.
How to Build a Modern Tontine: Algorithms, Scripts and Tips (Future of Business and Finance)
by Moshe Arye MilevskyThis open access book introduces the modern tontine and its applications in retirement and decumulation. Personal financial management in the later stages of life presents unique challenges, and renowned retirement planning expert Dr. Milevsky proposes the modern tontine as a solution. With the goal of guiding professionals and retirees in more efficient decumulation, the book demonstrates how to build a modern tontine. It is technically oriented, employing a cookbook format, featuring R code, and examining retirement planning through a statistical lens. This how-to guide, which is a sequel to his 2020 book “Retirement Income Recipes in R”, will be invaluable for retirement planning professionals and advisors, as well as for PhD scholars in retirement planning, quantitative finance, and related fields.This book is open access.
How to Calculate Quickly: Full Course in Speed Arithmetic (Dover Books on Mathematics)
by Henry StickerDo you want to double or triple the speed with which you calculate? How to Calculate Quickly is a tried and true method for helping you in the mathematics of daily life -- addition, subtraction, multiplication, division, and fractions. The author can awaken for you a faculty which is surprisingly dormant in accountants, engineers, scientists, businesspeople, and others who work with figures. This is "number sense" -- or the ability to recognize relations between numbers considered as whole quantities. Lack of this number sense makes it entirely possible for a scientist to be proficient in higher mathematics, but to bog down in the arithmetic of everyday life. This book teaches the necessary mathematical techniques that schools neglect to teach: Horizontal addition, left to right multiplication and division, etc. You will learn a method of multiplication so rapid that you'll be able to do products in not much more time than it would take to write the problem down on paper. This is not a collection of tricks that work in only a very few special cases, but a serious, capably planned course of basic mathematics for self-instruction. It contains over 9,000 short problems and their solutions for you to work during spare moments. Five or ten minutes spent daily on this book will, within ten weeks, give you a number sense that will double or triple your calculation speed.
How to Conduct Your Own Survey
by Priscilla Salant Don A. DillmanA nuts-and-bolts guide to conducting your own professional-quality surveys without paying professional fees. How can you gauge public support for a cause or test the market for a product or service? What are the best methods for validating opinions for use in apaper or dissertation? A well-documented survey is the answer. But what if you don't have thousands of dollars to commission one? N oproblem. How to Conduct Your Own Survey gives you everything you need to do it yourself! Without any prior training, you can learn expert techniques for conducting accurate, low-cost surveys. In step-by-step, down-to-earth language, Priscilla Salant and Don A. Dillman give you the tools you need to: Determine which type of survey is best for you Estimate the cost of your survey Conduct mail, telephone, and face-to-face surveys Draw accurate samples Write effective questionnaires Compile and report results Avoid common survey errors Find reliable outside assistance And much more
How to Count
by Robert A. BeelerProviding a self-contained resource for upper undergraduate courses in combinatorics, this text emphasizes computation, problem solving, and proof technique. In particular, the book places special emphasis the Principle of Inclusion and Exclusion and the Multiplication Principle. To this end, exercise sets are included at the end of every section, ranging from simple computations (evaluate a formula for a given set of values) to more advanced proofs. The exercises are designed to test students' understanding of new material, while reinforcing a working mastery of the key concepts previously developed in the book. Intuitive descriptions for many abstract techniques are included. Students often struggle with certain topics, such as generating functions, and this intuitive approach to the problem is helpful in their understanding. When possible, the book introduces concepts using combinatorial methods (as opposed to induction or algebra) to prove identities. Students are also asked to prove identities using combinatorial methods as part of their exercises. These methods have several advantages over induction or algebra.
How to Count to Infinity (Little Ways to Live a Big Life #1)
by Marcus Du SautoyDo something amazing and learn a new skill thanks to the Little Ways to Live a Big Life books! Birds do it, bees do it, even educated fleas do it... Not falling in love, but counting. Animals and humans have been using numbers to navigate their way through the jungle of life ever since we all evolved on this planet. But this book will help you to do something that humans have only recently understood how to do: to count to regions that no animal has ever reached. By the end of this book you'll be able to count to infinity...and beyond.On our way to infinity we'll discover how the ancient Babylonians used their bodies to count to 60 (which gave us 60 minutes in the hour), how the number zero was only discovered in the 7th century by Indian mathematicians contemplating the void, why in China going into the red meant your numbers had gone negative and why numbers might be our best language for communicating with alien life.But for millennia contemplating infinity has sent even the greatest minds into a spin. Then at the end of the nineteenth century mathematicians discovered a way to think about infinity that revealed that it is a number that we can count. Not only that. They found that there are an infinite number of infinities, some bigger than others. Just using the finite neurons in your brain and the finite pages in this book, you'll have your mind blown discovering the secret of how to count to infinity.
How to Count to Infinity
by Marcus Du SautoyDo something amazing and learn a new skill thanks to the Little Ways to Live a Big Life books! Birds do it, bees do it, even educated fleas do it... Not falling in love, but counting. Animals and humans have been using numbers to navigate their way through the jungle of life ever since we all evolved on this planet. But this book will help you to do something that humans have only recently understood how to do: to count to regions that no animal has ever reached. By the end of this book you'll be able to count to infinity...and beyond.On our way to infinity we'll discover how the ancient Babylonians used their bodies to count to 60 (which gave us 60 minutes in the hour), how the number zero was only discovered in the 7th century by Indian mathematicians contemplating the void, why in China going into the red meant your numbers had gone negative and why numbers might be our best language for communicating with alien life.But for millennia contemplating infinity has sent even the greatest minds into a spin. Then at the end of the nineteenth century mathematicians discovered a way to think about infinity that revealed that it is a number that we can count. Not only that. They found that there are an infinite number of infinities, some bigger than others. Just using the finite neurons in your brain and the finite pages in this book, you'll have your mind blown discovering the secret of how to count to infinity.