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Introduction to Number Theory

Introduction to Number Theory

Professor Edward B. Burger Ph.D.
Southwestern University
Course No.  1495
Course No.  1495
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Course Overview

About This Course

24 lectures  |  30 minutes per lecture

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.

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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
  • 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."

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24 Lectures
  • 1
    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. x
  • 2
    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. x
  • 3
    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. x
  • 4
    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. x
  • 5
    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. x
  • 6
    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. x
  • 7
    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. x
  • 8
    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. x
  • 9
    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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x

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Edward B. Burger
Ph.D. Edward B. Burger
Southwestern University

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 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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