TTC audio - Introduction to Number Theory by Professor Edward B. Burger
Course No.1495 Williams CollegePh.D., The University of Texas at Austin
Publisher:TTC Audio 2012 | ISBN: n/a | Language English | Audio in MP3
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 profession -- 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
Integers
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."
LECTURES
24 Lectures
1Number Theory and Mathematical Research
2Natural Numbers and Their Personalities
3Triangular Numbers and Their Progressions
4Geometric Progressions, Exponential Growth
5Recurrence Sequences
6The Binet Formula and the Towers of Hanoi
7The Classical Theory of Prime Numbers
8Euler's Product Formula and Divisibility
9The Prime Number Theorem and Riemann
10Division Algorithm and Modular Arithmetic
11Cryptography and Fermat's Little Theorem
12The RSA Encryption Scheme
13Fermat's Method of Ascent
14Fermat's Last Theorem
15Factorization and Algebraic Number Theory
16Pythagorean Triples
17An Introduction to Algebraic Geometry
18The Complex Structure of Elliptic Curves
19The Abundance of Irrational Numbers
20Transcending the Algebraic Numbers
21Diophantine Approximation
22Writing Real Numbers as Continued Fractions
23Applications Involving Continued Fractions
24A Journey's End and the Journey Ahead
Edward B. Burger View Other Courses
By This Professor
Edward B. Burger is Professor of Mathematics in the Department of Mathematics and Statistics at Williams College. He graduated summa cum laude from Connecticut College, where he earned a B.A. with distinction in Mathematics. He earned his Ph.D. in Mathematics from The University of Texas at Austin.
Professor Burger is the recipient of many teaching awards and accolades. He was named by Baylor University as the 2010 recipient of the Robert Foster Cherry Award for Great Teaching for his "proven record as an extraordinary teacher and distinguished scholar." Baylor University lauded Dr. Burger as "truly one of our nation's most outstanding, passionate, and creative mathematics professors."
His other teaching awards include the Nelson Bushnell Prize for Scholarship and Teaching from Williams College, the Distinguished Achievement Award for Educational Video Technology from the Association of Educational Publishers, and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America. In 2006 Reader's Digest honored him in its annual "100 Best of America" special issue as "Best Math Teacher."
Professor Burger is the author of more than 40 scholarly papers and books, including Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas, which has been translated into seven languages. He was honored with the Robert W. Hamilton Book Award for his coauthored work with Michael Starbird, The Heart of Mathematics: An invitation to effective thinking. He served for three years as mathematics advisor for the educational series "NUMB3RS," produced by CBS, Paramount Studios, and Texas Instruments.
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