An Introduction to Number Theory [TTC Video]
02 July 2015, 10:04
Course No 1495 | AVI, XviD, 1206 kbps, 624x480 | English, MP3, 128 kbps, 2 Ch | 24x30 mins | 6.63GB
How could an ancient king be tricked into giving his servant more than 671 billion tons of rice? It's all due to a simple but powerful calculation involving the sum of geometric progression -- an important concept in number theory and just one of the fascinating concepts you'll encounter in An Introduction to Number Theory. Taught by veteran Teaching Company instructor Edward B. Burger, this 24-lecture course offers an exciting adventure into the world of numbers.
An Introduction to Number Theory is a great introduction to the field for anyone who loves numbers, is fascinated by math, and wants to go further into the relationships among these mysterious objects.
What Is Number Theory?
Called "the queen of mathematics" by the legendary mathematician Carl Friedrich Gauss, number theory is one of the oldest and largest branches of pure mathematics. Practitioners of number theory delve deep into the structure and nature of numbers, and explore the remarkable, often beautiful relationships among them.
In this course, you'll cover all the fundamentals of this exciting discipline and explore the many different types of numbers:
- Natural numbers
- Prime numbers
- Negative and irrational numbers
- Algebraic numbers
- Imaginary numbers
- Transcendental numbers
But in An Introduction to Number Theory, you'll also develop a deeper understanding of their nature and the relationships among them, and gain insights into the many branches of number theory, including:
- Elementary number theory, an exploration of the fundamental patterns involving the natural numbers
- Analytic number theory, which uses the techniques of calculus to establish its results, and focuses on the prime numbers and their patterns
- Algebraic number theory, which relies on arithmetic—the operations of addition, subtraction, multiplication, division—to find solutions to equations
- Algebraic geometry, a combination of algebra and geometry which reveals the important connection between solutions to certain equations and points on certain curves.
An Intriguing Landscape of Calculation
Using brain-teasing problems and fascinating anecdotes, Professor Burger offers deep insights into the complex and beautiful patterns that structure the world of numbers. For example, you'll encounter:
- The sieve of Eratosthenes: A simple but brilliant method for "sifting out" all the prime numbers within any sequence of natural numbers
- The golden ratio: A mysterious and powerful number that recurs throughout the natural and scientific worlds, and is believed by some to hold the key to aesthetic beauty
- The method of modular arithmetic: An alternative way of counting that focuses on remainders rather than quotients to open up new possibilities for the manipulation of numbers
- The relationship between algebra and geometry: Exactly how does an algebraic formula correspond to a line or curve? What can we understand about these two mathematical entities?
Number Theory in Everyday Life
You'll quickly see that number theory—though complex and intellectually challenging—is no "ivory tower" endeavor. Throughout the course, Professor Burger answers intriguing questions about how the work of number theorists touches our daily lives:
- How do modern computers "talk" to each other through vast remote networks?
- What technologies lay behind crucial security procedures, such as the encryption used when you pay with a credit card on an e-commerce website?
- How do the puzzling numbers found on everyday consumer products—the UPC labels, or bar codes—allow you to move quickly and easily through the check-out stand at the local market?
- Why does the chromatic musical scale include 12 notes?
- What is the meaning of the mysterious bank routing numbers on checks? How do they provide a way to identify specific bank accounts?
Mathematical Proofs: The Agony and the Ecstasy
You'll also experience the exhilarating but often heartbreaking process of mathematical proof. In creating a proof, a mathematician is at once akin to the lawyer and the artist, constructing logical structures that deduce newly discovered truths from previously accepted premises.
While some mathematical problems readily yield proofs, others prove to be more intractable. In this course, you'll hear about the trials, tribulations, and triumphs of great minds as they attempted to solve some of the most vexing "open questions" that have tickled the curiosity of mathematicians for centuries.
Consider, for example, the quest of Andrew Wiles. Since childhood, he dedicated himself to finding a proof to one of the most notorious open questions in mathematics, Fermat's last theorem. Professor Burger follows Wiles's frustrating disappointments and miraculous breakthrough in proving this elusive theorem. It's a story that demonstrates the discipline, mental acuity, and remarkable creativity required to establish new areas of knowledge.
And you'll learn about some of the open questions that remain, including the Collatz conjecture, a tricky puzzle for which the great 20th-century Hungarian mathematician Paul Erdös offered to pay $500 to anyone who could provide a solution.
Are You Ready for a Challenge?
These are the kinds of mathematical puzzles that make An Introduction to Number Theory a treat for anyone who loves numbers. While the course requires confidence with basic math concepts, Professor Burger provides clear and effective guidance for students at all levels.
With each tantalizing problem, Professor Burger begins with an overview of the high-level concepts. Next, he provides a step-by-step explanation of the formulas and calculations that lay at the heart of each conundrum. Through clear explanations, entertaining anecdotes, and enlightening demonstrations, Professor Burger makes this intriguing field of study accessible for anyone who appreciates the fascinating nature of numbers.
And while the course is demanding, you'll soon find that An Introduction to Number Theory offers rewards that are equal to its tests. Through the study of number theory, you'll experience an enlightened perspective on the world around us. As Professor Burger says of number theory: "Great secrets and structures lie close by, to be uncovered only if we open our imagination to all possibilities."
- Number Theory and Mathematical Research
- Natural Numbers and Their Personalities
- Triangular Numbers and Their Progressions
- Geometric Progressions, Exponential Growth
- Recurrence Sequences
- The Binet Formula and the Towers of Hanoi
- The Classical Theory of Prime Numbers
- Euler's Product Formula and Divisibility
- The Prime Number Theorem and Riemann
- Division Algorithm and Modular Arithmetic
- Cryptography and Fermat's Little Theorem
- The RSA Encryption Scheme
- Fermat's Method of Ascent
- Fermat's Last Theorem
- Factorization and Algebraic Number Theory
- Pythagorean Triples
- An Introduction to Algebraic Geometry
- The Complex Structure of Elliptic Curves
- The Abundance of Irrational Numbers
- Transcending the Algebraic Numbers
- Diophantine Approximation
- Writing Real Numbers as Continued Fractions
- Applications Involving Continued Fractions
- A Journey's End and the Journey Ahead
The Higgs Boson and Beyond [TTC Video]
01 July 2015, 12:03
2015 | Course No 1205 | M4V, AVC, 2000 kbps, 640x360 | AAC, 128 kbps, 2 Ch | 12x30 mins | + PDF Guidebook | 5.87GB
The search for, and ultimate discovery of, the Higgs boson is a triumph of modern physics—a global, half-century effort whose outcome would make or break the vaunted Standard Model of particle physics. The hunt for the Higgs was the subject of wide media attention due to the cost of the project, the complexity of the experiment, and the importance of its result. And, when it was announced with great fanfare in 2012 that physicists has succeeded in creating and identifying this all-important new particle, the discovery was justly celebrated around the world.
And yet, virtually no one who happily read that news could tell you what, exactly, the Higgs boson was, and why its discovery was so important that we had to spend 10 billion dollars and build the single largest and most complex device in the history of mankind in order to find it. When you truly understand the details, this story ranks as one of the most fascinating, important, and thrilling in the history of modern physics.
Now, in The Higgs Boson and Beyond, award-winning theoretical physicist Sean Carroll returns to The Great Courses to guide you through the details of this incredible story. A brilliant researcher working at the very forefront of physics, as well as a gifted speaker who excels in explaining difficult and esoteric scientific concepts to the public, Professor Carroll is perfectly positioned to tell this story. In this 12-lecture masterpiece of scientific reporting, you'll learn everything you need to know to fully grasp the significance of this scientific discovery, including:
- the basics of quantum mechanics;
- the four forces that comprise the Standard Model of particle physics;
- how these forces are transmitted by fields and particles; and
- the importance of symmetry in physics.
In addition, Professor Carroll offers an in-depth view of the Large Hadron Collider—the largest machine ever built, and the device responsible for finally revealing the concept of the Higgs boson as reality. By the end, you’ll understand how the Higgs boson verifies the final piece in the Standard Model of particle physics, and how its discovery validates and deepens our understanding of the universe.
A Discovery of Mass Importance
It’s hard to overstate the importance of the Higgs field to the structure of our world and the universe itself. First postulated to exist in 1964 by Peter Higgs, the Higgs field and the particle associated with it explain one of the most fundamental concepts of reality itself—how elementary particles get mass. In addition, the discovery of the Higgs boson has importance for a huge number of unanswered questions and exciting avenues of research in modern physics, including:
- insight into the nature of the universe and its ultimate fate;
- the existence of scalar fields—fields with a zero "spin" that, prior to the discovery of the Higgs field, were only theoretical in nature;
- insight into cosmic inflation—a theory that describes exponential expansion of space during the first few moments of the universe;
- new understandings of how symmetry and symmetry breaking works; and
- new approaches to understanding dark matter and possible extra dimensions.
In addition, you'll come to understand how the Large Hadron Collider, which was built specifically for the task of creating a Higgs boson, has immense value in future experiments in physics.
A Feast for the Eyes and the Mind
Professor Carroll's Higgs Boson and Beyond is a feast for the eyes as well as the mind. Produced on a virtual set, the course utilized purpose-built graphics and 3-D models as an incredible aid to understanding the material, allowing somewhat abstract and unusual concepts to be clearly rendered before your eyes. This is an experience like no other, a presentation beyond anything that you could experience in a traditional classroom.
A theoretical physicist of significant standing in the scientific community, Professor Carroll followed the hunt for the Higgs boson with rapt attention and was present at CERN on the day they announced the discovery. Between his access to the researchers working on the project, and his expertise in modern physics, he has a uniquely clear view of the subject. He also has an exceptional ability to explain complex scientific concepts in a way that makes them clear and comprehensible to any motivated learner. All of these factors make Professor Carroll the perfect guide to this complex and rewarding story.
The Higgs Boson and Beyond will expand your understanding of the universe by taking you on a tour of its most fundamental components. With the insightful guidance of Professor Carroll, you’ll soon be able to understand one of the most important discoveries of our age.
- The Importance of the Higgs Boson
- Quantum Field Theory
- Atoms to Particles
- The Power of Symmetry
- The Higgs Field
- Mass and Energy
- Colliding Particles
- Particle Accelerators and Detectors
- The Large Hadron Collider
- Capturing the Higgs Boson
- Beyond the Standard Model of Particle Physics
- Frontiers: Higgs in Space
Foundations of Eastern Civilization [TTC Video]
19 June 2015, 06:29
Course No 3630 | MP4, AVC, 1168 kbps, 640x360 | English, MP3, 128 kbps, 2 Ch | 48x30 mins | + PDF Guidebook | 13.23GB
So much of our historical knowledge is rooted in Western civilization, from the stories of ancient Greece and Rome to the intellectual developments of the Renaissance. But this history tells only part of the story of our global world. Eastern civilization has its own fascinating story, with consequences that matter deeply to our lives today.
How did Eastern civilization—particularly that of China, Korea, Japan, and the nations of Southeast Asia—develop? What do we know about the history, politics, governments, art, science, and technology of these countries? And how does the story of Eastern civilization play out in today’s world of business, politics, and international exchange?
Foundations of Eastern Civilization takes you on a grand journey to explore the big accomplishments of Eastern civilization, from the material economy of day-to-day life to the political and religious philosophies that would bind these cultures together for thousands of years. Over the course of 48 ambitious lectures, Professor Craig G. Benjamin of Grand Valley State University introduces you to the many people, achievements, and ideas that came out of Eastern civilization and played a role in creating the modern world.
In this course, you will travel across continents and over the ages to arrive at a full understanding of the Eastern world.
- See how climate and geography allowed powerful civilizations to emerge in certain regions.
- Discover the origins of the yin and yang cosmology, the Mandate of Heaven political philosophy, and Confucianism and Daoism.
- Trace the spread of ideas between East and West, especially along the Silk Roads.
- Explore the rise and fall of empires—some famous and others largely unknown.
- Survey the role of Eastern civilization in the 20th and 21st centuries, and see what the future may hold for the “Asian tiger” economies.
“To truly understand the modern world, it is essential to know something about the many extraordinary contributions Eastern civilization has made,” Professor Benjamin says. “Simply put, it is not enough to know just the ‘Western’ half of the story any more—both Eastern and Western are critical to understanding our present and our future.”
Foundations of Eastern Civilization offers you just that—the chance to fill in the other half of the story. You may be surprised to realize that all of us have been students of Eastern civilization, even if we have not been aware of it. Filled with captivating stories and surprising details, this course is an excellent overview of one of the most dynamic regions in the world.
Immerse Yourself in a Rich Cultural History
This course covers an impressive amount of ground, from the emergence of early cultures 10,000 years ago to the booming economies of the 21st century. China is at the hub of Eastern civilization, and when you complete this course, you’ll come away with a comprehensive understanding of its intriguing history:
- Uncover the Xia dynasty, which was long thought to be legendary but for which there is now some archaeological evidence.
- Study the mysterious “oracle bones” and the development of Chinese writing in the Shang dynasty.
- Examine the development of different administrative structures, educational programs, and civil service exams.
- Delve into the remarkable agricultural and industrial revolutions that occurred during the Song dynasty.
- Learn about China’s 19th-century difficulties, including opium wars, humiliating trade agreements with the British, peasant uprisings, and, eventually, the revolution that ended the dynastic system.
Along the way, you’ll meet some of the most extraordinary people in Chinese history: emperors and empresses, soldiers and envoys, administrative eunuchs, philosophers, and more. You’ll also consider the myriad inventions and innovations that drove the Chinese economy—including gunpowder, paper, the porcelain industry, and paper money.
While China is home to some of the great moments in world history, it is far from the only significant nation in the East. Professor Benjamin takes you on several extended forays to examine a wealth of other cultures:
- Discover the many dynasties of Korea, the “land of the morning calm.”
- Explore the extraordinary history of Japan, including a deep examination into the era of medieval shoguns and samurai warriors.
- Find out about the Mongols, who had the largest continuous empire in world history.
- Venture into India to witness the rise of Buddhism and other Indus civilization religions.
- Witness the amazing spread of Islam throughout Southeast Asia, as well as the impact of Christian missionaries.
- Unpack many of the 20th century’s most significant wars, including the Japanese aggression that culminated in World War II and the cold war conflicts in Korea and Vietnam.
Discover the Ancient Roots of Eastern Society
What unifies the foundations of Eastern civilization? With all the many cultures and nations and peoples—some of whom are little known outside highly specialized circles—what can we say about these societies as a whole?
By going back to the beginnings of Eastern civilization, Professor Benjamin shows you the groundwork for today’s global village. You’ll be surprised to find out just how far back some of the modern-day divides go. For instance, the conflicts between northern and southern Korea originated hundreds of years ago and have their roots in the peninsula’s geography.
Eastern civilization today is grounded in ancient history in a number of ways, one of the most interesting of which is the way Eastern nations think about human nature, government, and economics. Whereas the Western nations tend to take an individualist approach to society—with ideas originating in ancient Greece and Rome and expanded on during the Enlightenment—Eastern nations still tend to take a collectivist tack.
This collectivist approach has its roots in the Warring States Era at the end of the Zhou dynasty, when philosophers reflected on human nature and the best way to organize society.
- Confucius and his followers created a model of ethical leadership based on education and moral behavior.
- Daoists withdrew from society and looked to harmony in the cosmos and the natural world.
- Legalists imposed gruesome punishments to enforce the rule of law.
Each of these philosophies had different notions of human nature and laid out a different path to forming an orderly state. These philosophies provide an important foundation for Eastern thought, and their approaches to government are completely different from our conceptions in the West. Yet in today’s interconnected world, it’s more important than ever to understand the cultural foundations of countries with which we interact, do business, and negotiate global politics.
Witness a Dynamic Cultural Exchange
During the Han dynasty, the Silk Roads connected East and West and enabled a surprising amount of cross-cultural interaction and exchange. The West received goods and information from the East—including silk and spices—but the East also learned about the West, that other civilizations existed beyond the mountains, deserts, and nomadic tribes of Central Asia. Professor Benjamin takes you on a voyage along the Silk Roads and introduces you to many of the unsung heroes of history:
- The Xiongnu
- The Yuezhi
- The Kushans
- The Parthians
- The Mongols
You’ll also meet the Chinese ambassador Zhang Qian, whose breathtaking escapades blazed a trail for the Silk Roads. You’ll travel the caravan routes, consider what it would have been like to stop at one of the many “caravanserai”—the inns where merchants would stop along the trade routes—and study the Kushan Empire, a little-known and little-studied “lost civilization” of important middlemen in what is now Afghanistan.
In addition to the Silk Roads, you’ll explore the vibrant cross-cultural exchange within the East itself. China heavily influenced Korea, Japan, and Southeast Asia, yet these nations also evolved as independent, distinct cultures. How did these countries develop? What was their relationship to China? How did China influence them, and how did they influence China? Enjoy an Inside Look at a Fascinating Civilization
Foundations of Eastern Civilization is a sweeping course, taking you across time and space. But after providing the broad strokes, Professor Benjamin zooms in on specifics to give you a flavor for the texture of daily life. You’ll learn about massive building projects such as the Great Wall of China. You’ll encounter the great art and architecture, the poetry and literature, and the many other artifacts from the East:
- Ancient burial tombs in China
- Chulmun pottery from ancient Korea
- Calligraphy, poetry, and novels from the great Tang dynasty
- The oldest surviving printed document in world history
- Famous Japanese novels
Throughout all of these lectures, Professor Benjamin is a lively guide and a dazzling storyteller, taking you inside the great cities where riches abound—jewels, silks, and great works of art. He shares several stories from his visits to these locations, and many of his personal photographs add a charming touch to the course. Indeed, his enthusiasm for the subject and his remarkable style of lecturing will open up an entirely new world for you as he unfolds the story of Eastern civilization.
- Journey to the East
- Yin and Yang—The Geography of China
- Early China and the Mysterious Xia
- The Coming of the Shang
- The Shang and Writing for the Gods
- The Zhou and the Mandate of Heaven
- Great Ideas of the Zhou—Confucianism
- Great Ideas of the Zhou—Later Confucianism
- Great Ideas of the Zhou—Daoism
- Great Ideas of the Zhou—Legalism
- The Qin and the First Emperor of China
- Contact with the West—The Early Han
- Triumph and Tragedy—The Later Han
- Silk Roads—In the Footsteps of Nomads
- Silk Roads—The Envoy Zhang Qian
- Silk Roads—Perils of Camels and Caravans
- Silk Roads—Rome and Roads from the West
- Silk Roads—The Lost Kushan Empire
- Origins of Buddhism
- The Age of Disunity
- The Great Taizong and the Rise of the Tang
- Changan and the Glittering Tang
- Korea—Mysterious Beginnings
- Korea—The Land of Morning Calm
- Korea—The Unified Silla
- Korea—The Koryo
- Japan—Geography and Early Cultures
- Japan—Treasures of the Tomb Period
- Japan—Nara and the Great Eastern Temple
- Japan—The World of the Heian
- Southeast Asia—Vietnam
- Southeast Asia—Indian and Islamic Influences
- The Industrial Revolution of the Song
- Intellectual and Cultural Life of the Song
- The Mongols Conquer the World
- Shaking the Foundation—Mongols in the East
- The Rise of the Ming
- Great Treasure Fleets of the Ming
- The Qing—Nomads Return from the North
- The Qing—The Last Emperor of China
- Korea Choson—Rise of the Yangban
- Korea Choson—The Last Dynasty
- Medieval Japan—Samurai and Shoguns
- Tokugawa and Meiji Japan
- The People’s Republic of China
- Isolation and Cold War Conflicts
- The Rise of the East Asian Tigers
- The Enduring Ideas of Eastern Civilization