Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.

    It is now possible for cyclists of all abilities to ride a well marked, well marshalled event just about any weekend of the year, usually based around one, two or sometimes as many as ten fearsome hills. For the first time, here is a pocket-sized guide to the 100 greatest climbs in the land, the building blocks for these rides, written by a cyclist for cyclists. From lung busting city centre cobbles to leg breaking windswept mountain passes, this guide locates the roads that have tested riders for generations and worked their way into cycling folklore. Whether you're a leisure cyclist looking for a challenge or an elite athlete trying to break records stick this book in your pocket and head for the hills. To watch a video of Simon Warren in action click here

    It includes three entirely new chapters on the Internet, computer graphics, and multimedia. Thorough yet flexible, Introduction to Computers is appropriate for a full-semester course -- with or without a hands-on lab. The text is available with a student CD-ROM that contains interactive multimedia materials for each chapter. Also available are an Electronic Study Guide on CD-ROM, HyperGraphics package, and an Instructor's Productivity Center on CD-ROM.

    Since the publication of the First Edition over thirty years ago, Div, Grad, Curl, and All That has been widely renowned for its clear and concise coverage of vector calculus, helping science and engineering students gain a thorough understanding of gradient, curl, and Laplacian operators without required knowledge of advanced mathematics.

    There are classical logic puzzles, lateral thinking puzzles, “who am I?” riddles, mathematical brain teasers, word ladders, ditloids, and a large selection of illustrated pen and paper, coins, cups, and toothpicks puzzles (please view the preview of this book for a full listing). This is the first time a collection of such breadth has been compiled and formatted especially for Kindle devices. The puzzles have been carefully organized into 25 chapters, and each question is hyperlinked to its solution, to provide utmost ease of navigation. Alongside the world’s most famous riddles, are some lesser known gems, and some brand new puzzles, in print here for the first time. Our aim was to create a definitive compendium of riddles and puzzles to bring enjoyment to people of all ages. We hope you will enjoy unraveling them as much as we enjoyed creating and editing them. Here are a handful of sample riddles: Outside the Box Riddles: You need to divide a round birthday cake into eight pieces, so each of your guests will have something to eat. How can you do this by making only three straight cuts with a knife, and without moving any of the pieces? The king’s two bodyguards developed an ingenious method for assuring the king’s safety. With the king standing between them, they would face in opposite directions; one looking to the west and the other to the east, but at the same time, and without the use of any reflective surfaces, they would both be able to observe the king clearly. How was this possible? Pure Logic Riddles: There are two glasses. One contains water, and the other contains an equal quantity of wine. A teaspoon of water is removed and mixed into the glass of wine. A teaspoon of the wine-water mixture is then removed and mixed into the glass of water. Which of the mixtures is now purer? The sorcerer’s tower was enchanted in such a way that it was able to build itself. Bricks, slates, tiles, and panes of glass, all flew to it of their own accord and danced into position. The tower doubled in size every day until after 100 days it reached a height that provided fine views over the entire realm. How many days did the tower take to reach half its full height? Lateral Thinking Puzzles: Five men are going to church. It starts to rain, and four of the men begin to run. When they arrive at the church, the four men who ran are soaking wet, whereas the fifth man, who didn’t run, is completely dry. How is this possible? Think Twice Riddles: If you are running a race, and you overtake the person in second place, what place do you move into? Word Riddles: SOS is read the same forwards, backwards, and even upside-down. What four-letter word also shares these properties? Number Puzzlers: How many letters are there in the answer to this question? You have an opportunity to buy a hen. In fact, you have been offered a choice between two quite remarkable animals. One of the hens produces six dozen dozen eggs per month, and the other produces a half dozen dozen. Admittedly, both seem impressive. Does it matter which hen you choose? Traditional Poetic Riddles: Five creatures cross a field of snow; But leave a single track behind Whose loops and bows are soon, I know, Unravelled by the mind. Coins, Cups, and Toothpicks Illustrated Riddles: A coin is dropped into an empty bottle and a cork is then inserted in the neck of the bottle. How is it possible to remove the coin without taking out the cork, or breaking the bottle?

    It's a great project for families and foodies alike, and all you need to start off is this book, a specialist mushroom growing kit, and a small space in which to grow them!In this book I aim to teach you the basics that you need to know to grow mushrooms at home. You'll take a step back in time and learn about the history of mushrooms as food, and why it has taken centuries for home mushroom growing to really catch on. I'll let you know exactly which are the best types of mushrooms for you to start with. If you aren't sure what tools you'll need, don't worry - this book has that covered too. And I'll explain how to make sure that your mushrooms grow quickly and pest free, and the best ways to harvest, preserve and store your crop.But where should you grow your mushrooms? That's a good question! Contained within the pages of this book are tips on how to set up a mushroom patch in your back yard, as well as how to grow them inside. Is it really better to grow them completely in the dark? You are about to find out!So, whether you like shiitakes, portobellos or oyster mushrooms, stop buying them in stores. I'll teach you how to grow your own and keep your entire family well supplied.In as little as a few weeks you can have the perfect fresh mushrooms grown and ready to eat. Yum!This easy to read, beginners book, summarizes the essential information I have learned over the years, and is written to help you decide if mycology is for you, and if so, which route to take.

    For a student just beginning to understand these spiritual laws, the concept of a malleable reality can be difficult to grasp. Well, if physical reality is not what it is… then what really is? How can we begin to shape and create reality in the way we truly desire? If the Universe responds instantaneously to all my desires and intentions, where is my stuff? Why do I have to wait? In his latest work Playing In Time And Space, bestselling author and spiritual explorer Richard Dotts shares the secrets to creating our own physical reality from our current human perspectives. Instead of seeing the physical laws of space and time as restricting us, Dotts shares how anyone can transcend these perceived limitations of space and time by changing their thinking, and manifest right from where they are. As Dotts writes in the opening, “This book is an invitation to play!” Through a series of illuminating mental exercises and “mind games” which you can easily follow along from your comfortable armchair, Dotts shows anyone how they can instantly “bridge the SPACE and TIME gaps” between their current reality and where they ultimately want to be.Follow along as Dotts guides you through simple exercises that allow you to PROVE the existence and infallibility of these Universal Laws for yourself. If you are sick and tired of just merely reading about these principles, now is the time to really use them. These same insights only came to him after years of meditation, and now you too, can access these same great spiritual truths for yourself. For example … * How can you know that the Universe always responds to every single one of your intentions and desires instantaneously, without any delay? Dotts takes you through an inner exercise to understand this. * Why does a buffer of time exist between our intentions and the actual physical manifestation? * Is time merely a persistent illusion as famously stated by Einstein? Here’s how you can prove it to yourself. You may not believe you can BEND TIME at will using these simple statements and exercises. Richard Dotts didn’t think he could do it either… until these two events happened to him… * Are you doing the right thing when you visualize? If you visualize with the intent of ____ out ____, then you’re spending more effort than necessary. However, if you visualize with the specific intent of bridging the gaps of space and time… physical manifestations will happen in your reality very quickly. * How to rid yourself of negative worries and fear thoughts that just pop up throughout the day into your awareness, and why it is crucial to get rid of them for inspired manifestations. (Dotts shares one of the most effective techniques he personally used to rid himself of a lifetime of negative conditioning and chronic worrying.)Anyone who masters space and time would have mastered the art of manifestation!

    Now the journey continues in a new, updated edition that includes all the latest information on mathematical proofs, fractals, prime numbers, and chaos, as well as new material on* the relationship between mathematical knots and DNA* how computers based on quantum logic can significantly speed up the factoring of large composite numbers* the relationship between four-dimensional geometry and physical theories of the nature of matter* the application of cellular automata models to social questions and the peregrinations of virtual ants* a novel mathematical model of quasicrystals based on decagon-shaped tilesBlazing a trail through rows of austere symbols and dense lines of formulae, Peterson explores the central ideas behind the work of professional mathematicians-- how and where their pieces of the mathematical puzzle fit in, the sources of their ideas, their fountains of inspiration, and the images that carry them from one discovery to another.

    It emphasizes forms suitable for students and researchers whose interest lies in solving equations rather than in general theory. Solutions to odd-numbered problems appear at the end. 1957 edition.

    A simple web search picks out a handful of relevant needles from the world's biggest haystack: the billions of pages on the World Wide Web. Uploading a photo to Facebook transmits millions of pieces of information over numerous error-prone network links, yet somehow a perfect copy of the photo arrives intact. Without even knowing it, we use public-key cryptography to transmit secret information like credit card numbers; and we use digital signatures to verify the identity of the websites we visit. How do our computers perform these tasks with such ease? This is the first book to answer that question in language anyone can understand, revealing the extraordinary ideas that power our PCs, laptops, and smartphones. Using vivid examples, John MacCormick explains the fundamental "tricks" behind nine types of computer algorithms, including artificial intelligence (where we learn about the "nearest neighbor trick" and "twenty questions trick"), Google's famous PageRank algorithm (which uses the "random surfer trick"), data compression, error correction, and much more. These revolutionary algorithms have changed our world: this book unlocks their secrets, and lays bare the incredible ideas that our computers use every day.

    

    In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated..

    2 geomorphology. 3 structural and field geology. 4 crystallography. 5 petrology. 6 economic geology. 7 origin mode of occurrence and distribution and economic uses. 8 stratigraphy-palaeontology.

    Prepare to stretch your mind and challenge everything you think you know, as This Book Does Not Exist unveils just how weird a place the world of thought can be. Filled with philosophical and mathematical problems to baffle and delight you, This Book Does Not Exist is packed with thought experiments, real-life examples, and puzzles for you to try. It also introduces you to some of the great names in the field of paradoxes, from the ancient Greeks to Albert Einstein. Divided into eight mind-bending chapters, This Book Does Not Exist takes you on a journey from the counterintuitive to the downright absurd. On the way it will introduce you to topics such as impossible objects, how to expect the unexpected, and the trouble with time travel. This is a book that will change the way you think.

    Yet, while common, oak woodlands are anything but ordinary. In a book rich in illustration and suffused with wonder, author Kate Marianchild combines extensive research and years of personal experience to explore some of the marvelous plants and animals that the oak woodlands nurture. Acorn woodpeckers unite in marriages of up to ten mates and raise their young cooperatively. Ground squirrels roll in rattlesnake skins to hide their scent from hungry snakes. Manzanita's rust-colored, paper-thin bark peels away in time for the summer solstice, exposing sinuous contours that are cool to the touch even on the hottest day. Conveying up-to-the-minute scientific findings with a storyteller's skill, Marianchild introduces us to a host of remarkable creatures in a world close by, a world that rustles, hums, and sings with the sounds of wild things.