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Exploring the Infinite addresses the trend toward a combined transition course and introduction to analysis course. It guides the reader through the processes of abstraction and log- ical argumentation, to make the transition from student of mathematics to practitioner of mathematics. This requires more than knowledge of the definitions of mathematical structures, elementary logic, and standard proof techniques. The student focused on only these will develop little more than the ability to identify a number of proof templates and to apply them in predictable ways to standard problems. This book aims to do something more; it aims to help readers learn to explore mathematical situations, to make conjectures, and only then to apply methods of proof. Practitioners of mathematics must do all of these things. The chapters of this text are divided into two parts. Part I serves as an introduction to proof and abstract mathematics and aims to prepare the reader for advanced course work in all areas of mathematics. It thus includes all the standard material from a transition to proof" course. Part II constitutes an introduction to the basic concepts of analysis, including limits of sequences of real numbers and of functions, infinite series, the structure of the real line, and continuous functions. Features Two part text for the combined transition and analysis course New approach focuses on exploration and creative thought Emphasizes the limit and sequences Introduces programming skills to explore concepts in analysis Emphasis in on developing mathematical thought Exploration problems expand more traditional exercise sets
This book provides a solid image of Escher as he saw himself--not as an 'artist' but as an artisan, a graphic artist with heart and soul, obsessed by contrasts and possessed by a unique creativity.
**Exploring the Infinite Possibilities Unlock the Mysteries of Mathematics** Dive into a world where numbers weave patterns of breathtaking beauty and equations reveal the secrets of the universe. "Exploring the Infinite Possibilities" is a captivating journey through the landscape of mathematics, offering a fresh and insightful perspective on a subject often shrouded in mystery and complexity. Begin your exploration with the aesthetic allure of mathematics, where the harmony of numbers and the elegance of patterns and symmetry spark a sense of wonder. Venture into the rich history of mathematical thought, tracing its evolution from ancient civilizations, through the intellectual fervor of the Renaissance, to the innovations that define modern mathematics today. Discover the boundless nature of infinity, uncover the mysteries of fractals and chaos theory, and delve into the intriguing realm of transfinite numbers. Wander through the natural world, where the Fibonacci sequence and the Golden Ratio manifest in mesmerizing forms and patterns, and explore the symmetrical beauty inherent in biological structures. Unravel the intricacies of mathematical proofs, from historical breakthroughs to contemporary challenges that drive mathematical discovery. Appreciate mathematics as a universal language, bridging the gap between the abstract and the tangible, and see its unifying power in science. From the elegance of Euclidean geometry to the peculiarities of non-Euclidean spaces, geometric concepts open the door to endless possibilities. Explore the hidden symmetries in abstract algebra, the enigmatic nature of prime numbers, and the profound impacts of calculus—the mathematics of change. Venture into the realms of mathematical analysis, probability, and statistics, uncovering the profound insights these fields offer into our world. Engage with the foundations of mathematical logic and embark on a journey through the digital age, where algorithms and machine learning reshape our lives. "Exploring the Infinite Possibilities" is not just a book—it's an inspiring odyssey into a vibrant mathematical universe. Whether you're a curious enthusiast or a seasoned mathematician, this book invites you to continue the great journey of mathematical exploration, inspiring future generations and highlighting the global impact of mathematics.
Infinite Possibilities is the masterwork from teacher, author, and featured speaker Mike Dooley. As the next step beyond his immensely popular Notes from the Universe trilogy, and his follow up, Choose them Wisely, this book contains even more enriching wisdom for living an abundant, joyous life. Mike Dooley knows that we create our own reality, our own fate, and our own luck. We’re beings filled with infinite possibility—just ready to explore how powerful we truly are. Manifesting the magnificence of our dreams isn’t about hard work, but rather about belief and expectation. These principles transcend belief, realizing the truth about our human nature. Your dreams are not accidental, nor inconsequential. And if someone were tell the truth about life, reality, and the powers we all possessed, would it be recognized? Our lives are full of adventures—and not exactly the sky-diving, mountain-climbing variety—but something better. Readers will laugh, applaud, and be inspired by Mike Dooley’s wit and wisdom.
How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.
Real-life psychic, near-death, and paranormal experiences are combined with cutting-edge science and vivid adventure stories in this energetic look at why extreme athletes and mountaineers take the risks that allow them to push the limits of consciousness, and what they encounter there. In the life-or-death world of extreme adventure sports, there is one thing that athletes often keep quiet about: the “forbidden” territory of paranormal experiences. Ranging from fleeting moments of transcendence to full-blown encounters with ghosts and everything in between—visions, near-death experiences, psychic communication—many extreme athletes have experienced these moments of connection with the beyond, but have been reluctant to talk about them. In Explorers of the Infinite, award-winning outdoors journalist and lifelong adventure sports devotee Maria Coffey probes the mystical and paranormal experiences of mountaineers, snowboarders, surfers, and more. She reviews cutting-edge science, and consults the history of philosophy and spirituality to answer the question: Could the state of intense “aliveness” that is the allure of extreme sports for so many actually be a route to a connection with the beyond? Coffey investigates the scientific explanations for mystical phenomena, ranging from simple explanations to theories from consciousness studies and quantum physics, and leaves us wondering where science ends and spirituality begins. An energetic, you-are-there look at the spiritual lives of extreme athletes, Explorers of the Infinite asks why extreme athletes take the risks that allow them to push the limits of consciousness, what they encounter there, and what we can learn from them.
SHORTLISTED FOR THE 2017 ROYAL SOCIETY SCIENCE BOOK PRIZE Even small children know there are infinitely many whole numbers - start counting and you'll never reach the end. But there are also infinitely many decimal numbers between zero and one. Are these two types of infinity the same? Are they larger or smaller than each other? Can we even talk about 'larger' and 'smaller' when we talk about infinity? In Beyond Infinity, international maths sensation Eugenia Cheng reveals the inner workings of infinity. What happens when a new guest arrives at your infinite hotel - but you already have an infinite number of guests? How does infinity give Zeno's tortoise the edge in a paradoxical foot-race with Achilles? And can we really make an infinite number of cookies from a finite amount of cookie dough? Wielding an armoury of inventive, intuitive metaphor, Cheng draws beginners and enthusiasts alike into the heart of this mysterious, powerful concept to reveal fundamental truths about mathematics, all the way from the infinitely large down to the infinitely small.
From the preface of the author: "...I have divided this work into two books; in the first of these I have confined myself to those matters concerning pure analysis. In the second book I have explained those thing which must be known from geometry, since analysis is ordinarily developed in such a way that its application to geometry is shown. In the first book, since all of analysis is concerned with variable quantities and functions of such variables, I have given full treatment to functions. I have also treated the transformation of functions and functions as the sum of infinite series. In addition I have developed functions in infinite series..."
Robinson, the master of fast-paced stories blending horror, science fiction, and thrillers, tackles his most ambitious subject to date: reality itself. An amalgam of the works of J.J. Abrams and Ridley Scott, Infinite is a bold SF novel exploring the vastness of space and a man's desire to exist, find love, and alter the course of his life.
This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel. But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death. Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Aristotle to today's million-dollar reward that awaits whoever cracks Reimann's hypothesis. Filled with idiosyncratic characters from Pythagoras to Euler, Infinite Powers is a compelling human drama that reveals the legacy of calculus on nearly every aspect of modern civilization, including science, politics, ethics, philosophy, and much besides.