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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.
 
 
Rated 5 out of 5 by from If only schools taught this way I watched this course through Wondrium but will probably buy it. Dr. Burger is very good, I wish that all math teachers had his capability and taught number theory as an adjunct to the regular mathematical curriculum as it provides a useful view in theoretical math as a tool. In fact, this explains where certain things come from, such as where 22/7 comes from that was used as a close approximation to pi. Number theory may seem esoteric, but Dr. Burger explains well that which underlies much of the tools that engineering and science relies on. Students need to understand this and maybe, if presented in this way, would actually develop a good understanding of math as a practical tool.
Date published: 2024-04-11
Rated 5 out of 5 by from Very good...but not easy Great class from Professor Burger; clarity, interest and sense of humor characterize his lectures. But I could not always follow some of them as he delved into more advanced mathematics.
Date published: 2023-12-02
Rated 5 out of 5 by from Great Ending I think he summed up the value of the process of discovery very well. One idea leads to a better idea if you have an open mind and look beyond the artificial constraints we impose on ourselves. An inspiring course.
Date published: 2023-11-15
Rated 5 out of 5 by from Fantastic course Well organized, interesting and amusing. A math course that actually contains math,.presented by a mathematician who has made solid meaningful advancements in his field. I would recommend all of his courses on this site.
Date published: 2023-10-23
Rated 4 out of 5 by from Good course. Time for an update I will begin by confessing that number theory is not at the top of the list of things I like. I listened to this course as a form of therapy to help me get over this aversion, and it was partly successful. On the whole, a worthwhile experience. The course was prepared in 2008, so it is about due for an update. Here are some suggestions for things I enjoyed and would like to see more of: cryptography sequences and progressions connections with geometry factorization On the other hand, my appetite for learning about infinity is not infinite. Seeing the proofs was worthwhile to a degree, because it demonstrated tricks of the trade for manipulating functions. More tricks would be good, but prove something else maybe. Trying to present the proof of Fermat’s last theorem without actually proving it was not an unqualified success. Also, the example applications were sometimes a bit strained. One might say the conversion of km to miles using the fibonacci sequence went too far, since it is a pure accident.
Date published: 2023-09-05
Rated 5 out of 5 by from Fun, Engaging, Empowering! I am an older Engineering person and can honestly say I haven't enjoyed the study of math as much as this course since I was in school in the '80s. Professor Burger's style is very accommodating, slow enough but not overly so. I like it and his examples a great deal! Also appreciate his sense of humor. Is there any way we can get his previous lecture series available on Wondrium? It would be greatly appreciated!
Date published: 2023-08-28
Rated 5 out of 5 by from Great introduction to a fascinating subjectd I have had an interest in mathematics and number theory for some time but never had the opportunity to delve into it in any detail. This course has been a great introduction and overview of number theory and the presentation is very clear and precise. Prof. Burger is an enthusiastic presenter and his love for the subject clearly comes through in making his presentations enjoyable and informative. The range of topics Prof. Burger is broad and all fascinating. Even the more abstract topics are well presented and understandable.' This course will be of value to anyone with an interest in mathematics.
Date published: 2022-01-04
Rated 5 out of 5 by from It is amazing course I am an old customer for the Greatest Courses and I keep returning to shop for more and they keep delivering. Excellent material. I am very please. Thank you very much
Date published: 2021-12-24
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Number Theory and Mathematical Research

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
Natural Numbers and Their Personalities

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
Triangular Numbers and Their Progressions

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
Geometric Progressions, Exponential Growth

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
Recurrence Sequences

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
The Binet Formula and the Towers of Hanoi

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
The Classical Theory of Prime Numbers

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
Euler's Product Formula and Divisibility

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
The Prime Number Theorem and Riemann

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
Division Algorithm and Modular Arithmetic

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
Cryptography and Fermat's Little Theorem

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
The RSA Encryption Scheme

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
Fermat's Method of Ascent

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
Fermat's Last Theorem

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
Factorization and Algebraic Number Theory

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
Pythagorean Triples

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
An Introduction to Algebraic Geometry

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
The Complex Structure of Elliptic Curves

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
The Abundance of Irrational Numbers

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
Transcending the Algebraic Numbers

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
Diophantine Approximation

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
Writing Real Numbers as Continued Fractions

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
Applications Involving Continued Fractions

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
A Journey's End and the Journey Ahead

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

Overview Course No. 1495

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.

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."

About

Edward B. Burger
Edward B. Burger

For the truly wise individual, learning never ends.

INSTITUTION

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 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.

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