I just wish I had known all of this twelve years ago...'When you speak to the likes of Dylan Wiliam, Doug Lemov, Daisy Christodoulou, Kris Boulton and the Bjorks, you are bound to learn a thing or two. But when he started his Mr Barton Maths Podcast, Craig Barton wasn't expecting to have his whole outlook on teaching and learning turned upside down. How I Wish I'd Taught Maths is the story of an experienced and successful maths teacher's journey into the world of research, and what it looks like in the classroom.Along the way we meet practical, easy-to-implement strategies including Supercharged Worked Examples, Silent Teacher, SSDD problems, low-stakes quizzes, diagnostic questions, Purposeful Practice, self-explanations, harnessing the power of the hypercorrection effect, how to (and how not to) teach problem-solving and much more. No matter your experience, teaching style or favourite number, every maths teacher will find something to think about in this book.

     In MATH WITH BAD DRAWINGS, Ben Orlin answers math's three big questions: Why do I need to learn this? When am I ever going to use it? Why is it so hard? The answers come in various forms-cartoons, drawings, jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone.Eschewing the tired old curriculum that begins in the wading pool of addition and subtraction and progresses to the shark infested waters of calculus (AKA the Great Weed Out Course), Orlin instead shows us how to think like a mathematician by teaching us a new game of Tic-Tac-Toe, how to understand an economic crisis by rolling a pair of dice, and the mathematical reason why you should never buy a second lottery ticket. Every example in the book is illustrated with his trademark "bad drawings," which convey both his humor and his message with perfect pitch and clarity. Organized by unconventional but compelling topics such as "Statistics: The Fine Art of Honest Lying," "Design: The Geometry of Stuff That Works," and "Probability: The Mathematics of Maybe," MATH WITH BAD DRAWINGS is a perfect read for fans of illustrated popular science.

    In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the fifth-grade level through visualization, play, and investigation.During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message--that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual mathematics tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that:There is no such thing as a math person - anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.

    Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.

    He prepared this comprehensive workbook (with full solutions to every problem) to share his strategies for mastering calculus. This workbook covers a variety of essential calculus skills, including: derivatives of polynomials, trig functions, exponentials, and logarithms the chain rule, product rule, and quotient rule second derivatives how to find the extreme values of a function limits, including l’Hopital’s rule antiderivatives of polynomials, trig functions, exponentials, and logarithms definite and indefinite integrals techniques of integration, including substitution, trig sub, and integration by parts multiple integrals The goal of this workbook isn’t to cover every possible topic from calculus, but to focus on the most essential skills needed to apply calculus to other subjects, such as physics or engineering.

    In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the third-grade level through visualization, play, and investigation.During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message--that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that:There is no such thing as a math person - anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.

    With his trademark clarity and humor, Holt probes the mysteries of quantum mechanics, the quest for the foundations of mathematics, and the nature of logic and truth. Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot. Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U.S. Constitution contained a terrible contradiction--and whether the universe truly has a future.

    Maths is about patterns, and our universe is extraordinarily patterned. With enthusiasm and wonder, Eddie is here to help us discover these patterns.With engaging clarity and entertaining anecdotes, Eddie demonstrates the intricacy of maths in all the things we love - from music in our iPods to our credit cards. Filled with humour and heart, this book will fascinate, entertain and illuminate the maths that surrounds us.

    But what does it mean to be smart? Do smart people think conventionally? Schopenhauer wrote, “Talent hits a target no one else can hit; Genius hits a target no one else can see.” Geniuses can make connections others cannot even conceive. So, if you imagine you’re one of the clever ones, what targets are you aiming at? If you’re aiming at an invisible target – invisible to the masses – the first thing they will say to you is, “There’s nothing there. You’re a fool.” They will troll you. They will subject you to all the muck they can throw at you. That’s the way they roll. Amongst the worst trolls are scientific empiricists who aim for a target everyone can see, the surest sign of anti-genius. Geniuses never wait for public approval. They don’t expect to be understood. If the masses had a clue what a genius was talking about, the person would not be a genius. The highest geniuses will never be popular. They will never be successful. Their ideas will be ignored or trashed. They will get no support. They will be permanent outsiders. That’s the price of peerless genius. As Nietzsche said, “The higher we soar the smaller we appear to those who cannot fly.” The true geniuses have soared so high, they have reached the vanishing point. Machiavelli said, “It must be considered that there is nothing more difficult to carry out nor more doubtful of success nor more dangerous to handle than to initiate a new order of things; for the reformer has enemies in all those who profit by the old order, and only lukewarm defenders in all those who would profit by the new order; this lukewarmness arising partly from the incredulity of mankind who does not truly believe in anything new until they actually have experience of it.” A genius is always in the business of initiating a new order of things. Geniuses are those that change the paradigm, but the supporters of the existing paradigm don’t want the paradigm to be changed, so are forever opposed to geniuses. The powers-that-be are never going to surrender their power, so will never be on the side of geniuses. They usually want to discredit and marginalize geniuses, and even kill them off. How can a genius persuade the masses when the masses have no clue what the genius is saying, no interest in what the genius is saying, and are fully invested in whatever paradigm the genius is seeking to overthrow? The last people of use to the genius are the masses. In Ibsen’s play "An Enemy of the People", the protagonist Dr. Stockmann said, “The most insidious enemy of truth and freedom among us is the solid majority. Yes, the damned, solid, liberal majority – that’s it! Now you know.” Geniuses are never democrats. Democrats are always the opposition. The democrats of Athens executed Socrates for his troubles. What’s the most radical idea possible, the one least likely to be understood by humanity, the one that is so far beyond humanity’s level of intelligence that whoever champions it is likely to be regarded as insane? Go on, if you’re smart you ought to be conceive of a connection so unobvious, so invisible to the masses, so improbable in relation to any other idea, that it will be almost universally rejected. That idea – despised, mocked and reviled by practically everyone – will be the No.1 candidate for the answer to existence. If you’re not one of the masses, if you’re supremely open-minded, then this book will present you with the proof of the case that existence is 100% mathematical, and that this is the sole way for reality to be intelligible and have a definitive answer.

    Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 100 illustrations. The writing is relaxed and includes periodic historical notes, poor attempts at humor, and occasional diversions into other interesting areas of mathematics.The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis.The author believes most textbooks are extremely overpriced and endeavors to help change this.

    Authors Ruth Parker and Cathy Humphreys return with  Digging Deeper: Making Number Talks Matter Even More, Grades 3-10 , taking the readers into classrooms where their Number Talks routines are taught.In this comprehensive sequel to their best-selling book, Making Number Talks Matter, Parker and Humphreys apply their 15 minute lessons to older grade levels to inspire and initiate math talks. Through vignettes in the book, you'll meet other teachers learning how to listen closely to students and how to prompt them into figuring out solutions to problems. You will learn how to make on-the-spot decisions, continually advancing and deepening the conversation.  Digging Deeper  includes:Sample Problems:  Digging Deeper  is filled with a range of Number Talks problems, 10-15 minute warm-up routines that lend themselves to mental math and comparison of strategiesNavigating Rough Spots: Learn how to create a safe environment for tricky, problematic, or challenging student discussions that can arise when talking through problems and sharing ideasResponding to Mistakes: Ways to handle misconceptions and mathematical errors that come up during the course of Number Talk conversations Digging Deeper  is filled with teaching tips for using wait time between problems more efficiently, honoring student contributions while still correcting errors, and teaching concepts while nudging independent thinking. Through daily practice and open conversation, you can make Number Talks matter more.

    He prepared this comprehensive workbook to share his strategies for mastering fractions. With 20 chapters and 250 pages, this workbook covers a variety of essential fractions skills, including:* fractions, decimals, and percentages* how to find a common denominator* repeating decimals* adding, subtracting, multiplying, and dividing fractions* converting between mixed numbers and improper fractions* solving problems with ratios or proportions* word problems* long division with remainders* reducing fractions* finding reciprocals* what a fraction means visuallyThis workbook includes a pretest and a posttest to help assess student learning. Comparing pretest and posttest scores helps to measure how much students have learned by using this workbook. Teachers, parents, and even students may find this helpful.

    Crafted specifically for undergraduates who are studying the subject for the first time, and requiring minimal prerequisites, the book focuses on the essential fundamentals of computer science theory and features a practical approach that uses real computer programs (Python and Java) and encourages active experimentation. It is also ideal for self-study and reference.The book covers the standard topics in the theory of computation, including Turing machines and finite automata, universal computation, nondeterminism, Turing and Karp reductions, undecidability, time-complexity classes such as P and NP, and NP-completeness, including the Cook-Levin Theorem. But the book also provides a broader view of computer science and its historical development, with discussions of Turing's original 1936 computing machines, the connections between undecidability and G�del's incompleteness theorem, and Karp's famous set of twenty-one NP-complete problems.Throughout, the book recasts traditional computer science concepts by considering how computer programs are used to solve real problems. Standard theorems are stated and proven with full mathematical rigor, but motivation and understanding are enhanced by considering concrete implementations. The book's examples and other content allow readers to view demonstrations of--and to experiment with--a wide selection of the topics it covers. The result is an ideal text for an introduction to the theory of computation.An accessible and rigorous introduction to the essential fundamentals of computer science theory, written specifically for undergraduates taking introduction to the theory of computationFeatures a practical, interactive approach using real computer programs (Python in the text, with forthcoming Java alternatives online) to enhance motivation and understandingGives equal emphasis to computability and complexityIncludes special topics that demonstrate the profound nature of key ideas in the theory of computationLecture slides and Python programs are available at whatcanbecomputed.com

    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 forprofessors teaching an introductory college course in higher mathematicshigh school teachers working with advanced math studentsstudents wishing to see the type of mathematics they would be exposed to as a math major.The material in this pure math book includes:16 lessons in 8 subject areas.A problem set after each lesson arranged by difficulty level.A complete solution guide is included as a downloadable PDF file.Pure Math Book Table Of Contents (Selected) Here's a selection from the table of contents: Introduction Lesson 1 - Logic: Statements and TruthLesson 2 - Set Theory: Sets and SubsetsLesson 3 - Abstract Algebra: Semigroups, Monoids, and GroupsLesson 4 - Number Theory: Ring of IntegersLesson 5 - Real Analysis: The Complete Ordered Field of RealsLesson 6 - Topology: The Topology of RLesson 7 - Complex Analysis: The field of Complex NumbersLesson 8 - Linear Algebra: Vector SpacesLesson 9 - Logic: Logical ArgumentsLesson 10 - Set Theory: Relations and FunctionsLesson 11 - Abstract Algebra: Structures and HomomorphismsLesson 12 - Number Theory: Primes, GCD, and LCMLesson 13 - Real Analysis: Limits and ContinuityLesson 14 - Topology: Spaces and HomeomorphismsLesson 15 - Complex Analysis: Complex Valued FunctionsLesson 16 - Linear Algebra: Linear Transformations

    The book gives step-by-step guidance through the process of statistical analysis and provides multiple examples of how statistical techniques can be used to analyse and visualise linguistic data. It also includes a useful selection of discussion questions and exercises which you can use to check your understanding. The book comes with a Companion website, which provides additional materials (answers to exercises, datasets, advanced materials, teaching slides etc.) and Lancaster Stats Tools online (http: //corpora.lancs.ac.uk/stats), a free click-and-analyse statistical tool for easy calculation of the statistical measures discussed in the book

    The contributions to this growth have come from all over the world and India has been at the forefront of mathematical progress, with its discovery of zero and the modern number system. History of Mathematics brings to life this journey through the evolution of the art and science of Mathematics, both in India and the western world, with a fascinating blend of fact and anecdote.

    Lecture two reviews the connection between the second law of complexity and the interior of black holes. The final lecture is about the thermodynamics of complexity, and 'uncomplexity' as a resource for doing computational work.

    This book, complete with exercises and answers, forms a course which will take you from beginner or intermediate level to being a confident mathematician. This book includes: simple step-by-step explanations, to help you grasp new topics or those that have previously confused you; practice questions throughout, to help you embed your learning and improve your confidence; and end of chapter summaries, to help you remember the key points you've learned--all in one great-value book, so you don't need any separate workbooks or course books.Chapters include: number, angles, fractions, two-dimensional shapes, decimals, statistics, directed numbers, graphs, measurement, perimeter and area, algebraic, three-dimensional shapes, ratio and proportion, pythagoras' theorem and trigonometry, indices and standard form.

    Requiring no prior knowledge of the subject, it covers the aspects of linear algebra - vectors, matrices, and least squares - that are needed for engineering applications, discussing examples across data science, machine learning and artificial intelligence, signal and image processing, tomography, navigation, control, and finance. The numerous practical exercises throughout allow students to test their understanding and translate their knowledge into solving real-world problems, with lecture slides, additional computational exercises in Julia and MATLAB(R), and data sets accompanying the book online. Suitable for both one-semester and one-quarter courses, as well as self-study, this self-contained text provides beginning students with the foundation they need to progress to more advanced study.

    A introduction to using Lean as an interactive theorem prover.

    The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data.

    

    You'll learn about the central objects and theorems of mathematics, covering graphs, calculus, linear algebra, eigenvalues, optimization, and more. You'll also be immersed in the often unspoken cultural attitudes of mathematics, learning both how to read and write proofs while understanding why mathematics is the way it is. Between each technical chapter is an essay describing a different aspect of mathematical culture, and discussions of the insights and meta-insights that constitute mathematical intuition.As you learn, we'll use new mathematical ideas to create wondrous programs, from cryptographic schemes to neural networks to hyperbolic tessellations. Each chapter also contains a set of exercises that have you actively explore mathematical topics on your own. By the end of the book, you will be able to learn mathematics on your own. In short, this book will teach you to engage with mathematics.

    Lagrangian and Hamiltonian dynamics is the continuation of Newton's classical physics into new formalisms, each highlighting novel aspects of mechanics thatgradually build in complexity to form the basis for almost all of theoretical physics. Lagrangian and Hamiltonian dynamics also acts as a gateway to more abstract concepts routed in differential geometry and field theories and can be used to introduce these subject areas to newcomers.Journeying in a self-contained manner from the very basics, through the fundamentals and onwards to the cutting edge of the subject, along the way the reader is supported by all the necessary background mathematics, fully worked examples, thoughtful and vibrant illustrations as well as an informalnarrative and numerous fresh, modern and inter-disciplinary applications.The book contains some unusual topics for a classical mechanics textbook. Most notable examples include the 'classical wavefunction', Koopman-von Neumann theory, classical density functional theories, the 'vakonomic' variational principle for non-holonomic constraints, the Gibbs-Appell equations, classical path integrals, Nambu brackets and the full framing of mechanics in the language of differential geometry.

    If a child aspires to be a scientist or an engineer or an architect, they need to first understand math. Little Leonardo's Fascinating World of Math introduces many mathematical concepts in a context showing their interconnectedness with so many of the things in our everyday lives.This new series of books featuring illustrations by popular artist Greg Paprocki is the perfect way to encourage your brilliant child's interest in all facets of the educational STEAM curriculum: Science, Technology, Engineering, the Arts, and Mathematics. Your child will be enticed by the fun and surprising details revealed in Paprocki's cool and colorful artwork on every page.Greg Paprocki works full-time as an illustrator and book designer. He has illustrated several Curious George books, as well as The Big Book of Superheroes for Gibbs Smith. He began his career as an advertising art director after studying fine art and graphic design at the University of Nebraska. He lives in Omaha, Nebraska.Bob Cooper has been an editor for more than 20 years, working with everything from comic books to legal briefs. He lives near Salt Lake City.

    

    Starting from an experimental perspective, it provides a unified view of these fields that reflects the very rapid advances being made. This new edition has a number of improvements and has been updated to describe the recent discovery of gravitational waves and astrophysical neutrinos, which started the new era of multimessenger astrophysics; it also includes new results on the Higgs particle. Astroparticle and particle physics share a common problem: we still don't have a description of the main ingredients of the Universe from the point of view of its energy budget. Addressing these fascinating issues, and offering a balanced introduction to particle and astroparticle physics that requires only a basic understanding of quantum and classical physics, this book is a valuable resource, particularly for advanced undergraduate students and for those embarking on graduate courses. It includes exercises that offer readers practical insights can be used equally well as a self-study book, a reference and a textbook.

    You'll build on simple programming techniques like loops and conditionals to create your own custom functions. You'll come away with a toolkit of strategies for analyzing and visualizing data of all sorts using R and RStudio.Purchase of the print book includes a free eBook in PDF, Kindle, and ePub formats from Manning Publications.About the TechnologySpreadsheets are powerful tools for many tasks, but if you need to interpret, interrogate, and present data, they can feel like the wrong tools for the task. That's when R programming is the way to go. The R programming language provides a comfortable environment to properly handle all types of data. And within the open source RStudio development suite, you have at your fingertips easy-to-use ways to simplify complex manipulations and create reproducible processes for analysis and reporting.About the BookWith Beyond Spreadsheets with R you'll learn how to go from raw data to meaningful insights using R and RStudio. Each carefully crafted chapter covers a unique way to wrangle data, from understanding individual values to interacting with complex collections of data, including data you scrape from the web. You'll build on simple programming techniques like loops and conditionals to create your own custom functions. You'll come away with a toolkit of strategies for analyzing and visualizing data of all sorts.What's insideHow to start programming with R and RStudioUnderstanding and implementing important R structures and operatorsInstalling and working with R packagesTidying, refining, and plotting your dataAbout the ReaderIf you're comfortable writing formulas in Excel, you're ready for this book.About the AuthorDr Jonathan Carroll is a data science consultant providing R programming services. He holds a PhD in theoretical physics.Table of ContentsIntroducing data and the R languageGetting to know R data typesMaking new data valuesUnderstanding the tools you'll use: FunctionsCombining data valuesSelecting data valuesDoing things with lots of dataDoing things conditionally: Control structuresVisualizing data: PlottingDoing more with your data with extensions

    This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

    Presented by Nate Silver’s FiveThirtyEight, this first-ever collection features the column’s most popular problems, along with six never-before-published puzzles, all exploring the math implicit in everyday occurences. The simplest require a mere flash of insight, while the toughest involve deep applications of analysis and probability theory. Designed to appeal to a range of skill levels, The Riddler is a must-have for any math or puzzle enthusiast.

    Understanding Topology aims to change that.The perfect introductory topology textbook, Understanding Topology requires only a knowledge of calculus and a general familiarity with set theory and logic. Equally approachable and rigorous, the book's clear organization, worked examples, and concise writing style support a thorough understanding of basic topological principles. Professor Shaun V. Ault's unique emphasis on fascinating applications, from mapping DNA to determining the shape of the universe, will engage students in a way traditional topology textbooks do not.This groundbreaking new text: - presents Euclidean, abstract, and basic algebraic topology- explains metric topology, vector spaces and dynamics, point-set topology, surfaces, knot theory, graphs and map coloring, the fundamental group, and homology- includes worked example problems, solutions, and optional advanced sections for independent projectsFollowing a path that will work with any standard syllabus, the book is arranged to help students reach that "Aha!" moment, encouraging readers to use their intuition through local-to-global analysis and emphasizing topological invariants to lay the groundwork for algebraic topology.

    The features of this book are as follows: 1. It covers all the 285 exercises with detailed and completed solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. 2. There are 55 illustrations and 3 tables for explaining the mathematical concepts or ideas used behind the questions or theorems. 3. Hyperlinks of equations, formulas, references and websites are provided according to modern standard. 4. Sections in each chapter are added so as to increase the readability of the exercises. 5. Different colors are used frequently in order to highlight or explain problems, lemmas, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. 6. Necessary lemmas with proofs and references are provided because some questions require additional mathematical concepts which are not covered by Rudin. 7. Three appendices are included which further explain and supplement some theories in Chapters 10 and 11.

    

    Following this text from start to finish will provide the careful reader with the tools, vocabulary andconceptual foundation necessary to continue in further self-study and begin to explore current areas of active research in dynamical systems.

    Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression. In studying this book, students will encounter:the interconnections between set theory and mathematical statements and proofs;the fundamental axioms of the natural, integer, and real numbers;rigorous ε-N and ε-δ definitions;convergence and properties of an infinite series, product, or continued fraction;series, product, and continued fraction formul� for the various elementary functions and constants.Instructors will appreciate this engaging perspective, showcasing the beauty of these fundamental results.

    It emphasizes applications and simulation utilizing Python.The hands-on approach of Dynamical Systems with Applications using Python has minimal prerequisites, only requiring familiarity with ordinary differential equations. It will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a broad range of disciplines such as population dynamics, biology, chemistry, computing, economics, nonlinear optics, neural networks, and physics.

    This book teaches beginning algebra concepts in a fun and funny way. Also available in paperback.

    While pure mathematics is mostly interested in abstract structures, applied mathematics sits at the interface between this abstract world and the world inwhich we live. This area of mathematics takes its nourishment from society and science and, in turn, provides a unified way to understand problems arising in diverse fields.This Very Short Introduction presents a compact yet comprehensive view of the field of applied mathematics, and explores its relationships with (pure) mathematics, science, and engineering. Explaining the nature of applied mathematics, Alain Goriely discusses its early achievements in physics andengineering, and its development as a separate field after World War II. Using historical examples, current applications, and challenges, Goriely illustrates the particular role that mathematics plays in the modern sciences today and its far-reaching potential.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, andenthusiasm to make interesting and challenging topics highly readable.

    

    The book covers all the key areas of algebra including elementary operations, linear equations, formulae, simultaneous equations, quadratic equations, logarithms, variation, laws and sequences.Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.Chapter 1: The meaning of algebraChapter 2: Elementary operations in algebraChapter 3: Brackets and operations with themChapter 4: Positive and negative numbersChapter 5: Equations and expressionsChapter 6: Linear equationsChapter 7: FormulaeChapter 8: Simultaneous equationsChapter 9: Linear inequalitiesChapter 10: Straight-line graphs; coordinatesChapter 11: Using inequalities to define regionsChapter 12: Multiplying algebraical expressions Chapter 13: FactorsChapter 14: FractionsChapter 15: Graphs of quadratic functionsChapter 16: Quadratic equationsChapter 17: IndicesChapter 18: LogarithmsChapter 19: Ratio and proportionChapter 20: VariationChapter 21: The determination of lawsChapter 22: Rational and irrational numbers and surdsChapter 23: Arithmetical and geometric sequences

    Instead of trying to come up with a new number talk every day, simply select one of the hundreds of great offerings provided in this book.With chapters on addition, subtraction, multiplication, division, fractions and decimals, Classroom-Ready Number Talks for 3rd, 4th and 5th Grade Teachers includes: • Grade-level specific strategies• Number talk how-tos• Visual and numerical examples• Scaffolding suggestions• Common core alignments• Questions to build understandingWith these ready-to-use number talks, you’ll reduce time spent lesson planning and enjoy more time discussing math with your students. It’s sure to create a more engaging environment in your classroom and increase student comprehension of math concepts and how numbers function in the world around them.

    Its two major streams—the south Indian (Carnatic) and the north Indian (Hindustani)—share common origins and were part of a single tradition until the thirteenth century. L. Subramaniam, one of the best-known classical musicians of our age, and the late and highly celebrated vocalist Viji Subramaniam examine the origins, history, structure and spirit of this system. Classical Music of India discusses in depth the melodic and rhythmic aspects of both Carnatic and Hindustani, briefly comparing these concepts with the classical music of Europe. There is a lucid comparison of the two Indian forms, an examination of instruments integral to them and a re-examination of fundamental concepts. An expanded, revised and updated reissue of the classic Euphony, this work is a definitive appraisal of Indian classical music. It is an essential addition to the library of any serious music lover, whether steeped in the music of India or new to it.

    Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic.Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it.By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

    First, it will probe deeper into the politics, history, and contemporary practices of the geological sciences as a way of thinking, representing, and communicating the geos. This will open up the history of the earth sciences as a science that has been fundamentally imbricated with politics and that its politics has been one of making the geological sensible. Second, it will consider in detail geologies that are volatile and vulnerable and that because of this are subject to practices of governance. Finally, it will multiply the tradition of geological thought in the sciences by considering subaltern, amodern, vernacular, and counter traditions of geological practice and science and its political resonances. This volume will consider these three frameworks through essays historical, ethnographic and conceptual, mindful of the richness of empirical detail and the innovative consequences of looking at the intersections of geology and politics.