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, startling, and often beautiful relationships that exist among them. Gain deep insights into the complex and beautiful patterns that structure the world of numbers, the branches of study that reveal these patterns, and the processes by which great thinkers establish new truths through dazzling mathematical proofs.
An Introduction to Number Theory
Dig into one of the oldest and largest branches of pure mathematics with this captivating course on the structure and nature of numbers.
Overview
About
Dr. Edward B. Burger is President of Southwestern University in Georgetown, Texas. Previously, he was Francis Christopher Oakley Third Century Professor of Mathematics at Williams College. He graduated summa cum laude from Connecticut College, where he earned a BA with distinction in Mathematics. He earned his PhD 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.
01: Number Theory and Mathematical Research
In this opening lecture, we take our first steps into this ever-growing area of intellectual pursuit and see how it fits within the larger mathematical landscape.
31 min
02: Natural Numbers and Their Personalities
The journey begins with the numbers we have always counted upon—the natural numbers 1, 2, 3, 4, and so forth.
32 min
03: Triangular Numbers and Their Progressions
Using an example involving billiard balls and equilateral triangles, Professor Burger demonstrates the fundamental mathematical concept of arithmetic progressions and introduces a famous collection of numbers: the triangular numbers.
28 min
04: Geometric Progressions, Exponential Growth
Professor Burger introduces the concept of the geometric progression, a process by which a list of numbers is generated through repeated multiplication. Later, we consider various real-world examples of geometric progressions, from the 12-note musical scale to the take-home prize money of a game-show winner.
32 min
05: Recurrence Sequences
The famous Fibonacci numbers make their debut in this study of number patterns called recurrence sequences. Professor Burger explores the structure and patterns hidden within these sequences and derives one of the most controversial numbers in human history: the golden ratio.
30 min
06: The Binet Formula and the Towers of Hanoi
Is it possible to find a formula that will produce any specific number within a recurrence sequence without generating all the numbers in the list? To tackle this challenge, Professor Burger reveals the famous Binet formula for the Fibonacci numbers.
30 min
07: The Classical Theory of Prime Numbers
The 2,000-year-old struggle to understand the prime numbers started in ancient Greece with important contributions by Euclid and Eratosthenes. Today, we can view primes as the atoms of the natural numbers—those that cannot be split into smaller pieces. Here, we'll take a first look at these numerical atoms.
31 min
08: Euler's Product Formula and Divisibility
As we look more closely at the prime numbers, we encounter the great 18th-century Swiss mathematician Leonhard Euler, who proffered a crucial formula about these enigmatic numbers that ultimately gave rise to modern analytic number theory.
31 min
09: The Prime Number Theorem and Riemann
Can we estimate how many primes there are up to a certain size? In this lecture, we tackle this question and explore one of the most famous unsolved problems in mathematics: the notorious Riemann hypothesis, an "open question" whose answer is worth $1 million in prize money.
33 min
10: Division Algorithm and Modular Arithmetic
How can clocks help us do calculations? In this lecture, we learn how cyclical patterns similar to those used in telling time open up a whole new world of calculation, one that we encounter every time we make an appointment, read a clock, or purchase an item using a scanned UPC bar code.
32 min
11: Cryptography and Fermat's Little Theorem
After examining the history of cryptography—code making—we combine ideas from the theory of prime numbers and modular arithmetic to develop an extremely important application: "public" key cryptography.
31 min
12: The RSA Encryption Scheme
We continue our consideration of cryptography and examine how Fermat's 350-year-old theorem about primes applies to the modern technological world, as seen in modern banking and credit card encryption.
31 min
13: Fermat's Method of Ascent
When most people think of mathematics, they think of equations that are to be "solved for x." Here we study a very broad class of equations known as Diophantine equations and an important technique for solving them. We also encounter one of the most widely recognized equations, x2 + y2 = z2, the cornerstone of the Pythagorean theorem.
30 min
14: Fermat's Last Theorem
One of the most famous and romantic stories in number theory is the legendary tale of Fermat's last theorem. Professor Burger explicates this most mysterious of proposed "theorems" and describes how the greatest mathematical minds of the 18th and 19th centuries failed again and again in their attempts to provide a proof.
29 min
15: Factorization and Algebraic Number Theory
This lecture returns to a fundamental mathematical fact—that every natural number greater than 1 can be factored uniquely into a product of prime numbers—and pauses to imagine a world of numbers that does not exhibit the property of unique factorization.
31 min
16: Pythagorean Triples
In this lecture, Professor Burger returns to Pythagoras and his landmark theorem to identify an important series of numbers: the Pythagorean triples. After recounting an ingenious proof of this theorem, Professor Burger explores the structure of triples.
29 min
17: An Introduction to Algebraic Geometry
The shapes studied in geometry—circles, ellipses, parabolas, and hyperbolas—can also be described by quadratic (second-degree) equations from algebra. The fact that we can study these objects both geometrically and algebraically forms the foundation for algebraic geometry.
29 min
18: The Complex Structure of Elliptic Curves
Here we study a particularly graceful shape, the elliptic curve, and learn that it can be viewed as contour curves describing the surface of—of all things—a doughnut. This delicious insight leads to many important theorems and conjectures, and leads to the dramatic conclusion of the story of Fermat's last theorem.
30 min
19: The Abundance of Irrational Numbers
Ancient mathematicians recognized only rational numbers, which can be expressed neatly as fractions. But the overwhelming majority of numbers are irrational. Here, we'll meet these new characters, including the most famous irrational numbers, p, e, and the mysterious g.
32 min
20: Transcending the Algebraic Numbers
We move next to the exotic and enigmatic transcendental numbers, which were discovered only in 1844. We return briefly to a consideration of irrationality and the moment of inspiration that led to their discovery by mathematician Joseph Liouville. We even get a glimpse of Professor Burger's original contributions to the field.
31 min
21: Diophantine Approximation
In this lecture, Professor Burger explores a technique for generating a list of rational numbers that are extremely close to the given real number. This technique, called Diophantine approximation, has interesting consequences, including new insights into the motion of billiard balls and planets.
30 min
22: Writing Real Numbers as Continued Fractions
Real numbers are often expressed as endless decimals. Here we study an algorithm for writing real numbers as an intriguing repeated fraction-within-a-fraction expansion. Along the way, we encounter new insights about the hidden structure within the real numbers.
32 min
23: Applications Involving Continued Fractions
This lecture returns to the consideration of continued fractions and examines what happens when we truncate the continued fraction of a real number. The result involves two of our old friends—the Fibonacci numbers and the golden ratio—and finally explains why the musical scale consists of 12 notes.
32 min
24: A Journey's End and the Journey Ahead
In this final lecture, we take a step back to view the entire panorama of number theory and celebrate some of the synergistic moments when seemingly unrelated ideas came together to tell a unified story of number.
31 min
