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Introduction to Number Theory

Introduction to Number Theory

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

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Introduction to Number Theory

Course No. 1495
Professor Edward B. Burger, Ph.D.
Southwestern University
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4.6 out of 5
40 Reviews
72% of reviewers would recommend this series
Course No. 1495
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Course Overview

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
 |  30 minutes each
  • 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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  • 24 lectures on 4 DVDs
  • 208-page printed course guidebook
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps

What Does The Course Guidebook Include?

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Course Guidebook Details:
  • 208-page printed course guidebook
  • Questions to consider
  • Timeline
  • Bibliography

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Your professor

Edward B. Burger

About Your Professor

Edward B. Burger, Ph.D.
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...
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Introduction to Number Theory is rated 4.6 out of 5 by 39.
Rated 5 out of 5 by from An Introduction to Number Theory I am a retired petroleum engineer (age 78). By the way I am a female. I had a very satisfying career, vs. fast changing and difficult time for the young people. So, I volunteer math tutoring at the local high school (Grade 10 to 12) hopefully helpful to better equip the student with math, particularly joy of math. This course gave me a refreshing insight --> hopefully I can transfer the interesting, beautiful conceps to my students.
Date published: 2017-05-04
Rated 5 out of 5 by from Great course I have a college level mathematics education but even though the math in the course is accessible to anyone with basic high school math, I found this course fascinating and looked forward every morning to watching the next lecture. I would recommend this to anyone interested in mathematics and wants an introduction to number theory.
Date published: 2017-03-02
Rated 5 out of 5 by from Correctly described Now I understand why this course is called an Introduction. Kudos to Prof. Burger for presenting such an engaging overview of what is an enormous field of study. I was so enthused by this course that I bought a copy of Hardy and Wright to study on my own. The book is also an "Introduction," but without the Great Course first, it would be nearly impenetrable. I could follow Prof Burger -- he made much of what he did seem simple, obvious, and fascinating. I can now tell time, but I don't yet know how the clock works. Highly recommended for the curious and the student.
Date published: 2016-11-30
Rated 5 out of 5 by from A Map of Number Theory I ordered about a week ago, and watched the lectures on-line over the past week, finishing this morning. My mathematical training in engineering school included a number of more advanced courses in mathematics, but I was never able to pursue this interest with any degree of rigor. I am now a teacher and professional tutor in mathematics and the physical sciences. I was, in fact, expecting a more rigorous investigation here, but the course served me well anyways. It helps the auditor form a mental map of the landscape of number theory, without delving too much into the details of proofs. Not to say that there were not proofs, at least in a heuristic form, but the prior mathematical knowledge demanded of the listener usually did not go beyond the limits of high school. Those who have not used mathematics since high school or early college will likely struggle with it a bit, whereas those who have been engaged in mathematics in one form or another for a long time will find certain parts tedious, as when basic properties of arithmetic and geometric sequences and series are investigated. But, truth be told, the professor was in a bind. He has two distinct audiences: Those who have little mathematical training, but wish to familiarize themselves with this branch of knowledge as part of their general education and those, such as engineers or scientists, who have a solid foundation in basic mathematics, and are looking to this course as a preparation for further study. Given the divergence of backgrounds and ends of these two groups, I feel that the lecturer did a fine job of holding it all together. It is clear why professor Burger has been honored for his teaching. As a tutor in mathematics, I would recommend this to any high school instructor of mathematics. First, they will be exposed to concepts and techniques that their training likely did not. But, second, they will also benefit from seeing Dr. Burger's teaching. I found the material quite comprehensible, though there were times when the motivation for and importance of some of the theorems escaped me. For instance, I know that the Riemann conjecture is considered to be of great importance to mathematicians, and that many other interesting results depend upon it. Indeed, professor Burger explicitly stated this. However, there was little attempt to show why this is the case. No doubt the matter is of such difficulty that it could not be presented without losing 95% or more of the audience, and that at great cost of lecture time. Nevertheless, it smacked a bit of name-dropping, and did not contribute anything to understanding. There were a few other instances of this, as the lectures gradually took on the aspect of historical study. I think this is largely due to the divergence of interests and backgrounds of his audience. Those without a scientific or technical background were probably completely baffled at some of these points. Perhaps some of these more esoteric studies could be left out in an introductory course, and the time spent investigating simpler notions in more depth. At any rate, it is an excellent course if one is looking for a general introduction to the subject.
Date published: 2014-12-27
Rated 5 out of 5 by from Excellent Survey of Number Theory The Professor on this one is off-the-scale good, as in totally knowledgeable, and a totally excellent teacher. When he handwaves about graphs, for example, his hands reverse the motions so you see them as you would observe a graph rather than how he sees the graph from his personal perspective. He presents a very good survey of number theory concepts, and he selects the most easily understandable examples to show the dramatic approaches and conclusions of number theory. He loves his subject, and you vicariously enjoy it because of his passion and humor. If you remember everything you learned in Junior High School Algebra, you will have enough math to follow this. You also need some High School Geometry, but only a little. The study guide covers the course line-by-line, and gives you reference material, so you can review and learn what you want. My wife started getting a little lost in the second lecture, and she dipped in and out of understanding throughout the course. Still, she was interested, generally speaking. She is okay and high school math, and has been an income tax preparer and is fairly good with understanding numbers. I have a degree in physics, so I've had a lot of math. I've also studied data encryption as a Software Engineer, so I'm pretty familiar with many of the concepts presented, though I never got around to Number Theory or Algebraic Structures classes, though I would have liked to. I followed the whole course, except for about 2 or 3 times when it was a little too fast for me. It mostly pulled together things that I knew about but didn't have a framework to tie together before watching. I will definitely be watching this course again, and may pursue a real university course in it. If you have any interest at all in number theory, you can't go wrong in obtaining and watching this course. It is fabulously good.
Date published: 2014-12-14
Rated 4 out of 5 by from Interesting Introduction I have been watching these lectures in conjunction with discrete mathematics by professor Benjamin. Both courses cover some of the same material and it is interesting to get two presentations on a topic. Overall this course is a lot easier to follow, but lacks the rigour and depth of the discrete mathematics course. If you are new to the area and don't have a very strong math background then I would recommend this as the better course to start with. However, if your math is up to diving straight into the discrete maths course, then you might find this a bit lightweight.
Date published: 2014-04-20
Rated 5 out of 5 by from Magic! This abstract mathematics course enables deeper understanding of algebra, algebraic geometry, Pythagorean triples, irrational numbers, transcendental numbers and more. Prior to this course, many of these were simply terms for me. With "An Introduction to Number Theory," these terms acquire meaning. One needs only to follow the lectures to appreciate the "magic" contained within the structure and order of numbers and numeric progressions. Dr. Burger was recently named president of Southwestern University at Georgetown, Texas. He will inspire students and faculty with his scholarship, dedication to teaching, and imbuing his ten life-lessons with which he closes this course. Bravo, Dr. Burger!
Date published: 2014-03-08
Rated 5 out of 5 by from a pleasant surprise Having spent my technical life utilizing applied math in my engineering career, I was not sure what to expect in this more 'pure' pursuit. In a nutshell, the course was outstanding. First of all, anyone who is comfortable with algebra can handle the majority of concepts presented. The hidden beauty of numerical relations even within such subjects as irrational numbers was intriguing. The professor spends some time describing the ancient basis for many of the concepts, utilizing the work of such gifted contributors as Gauss, Lagrange, etc. All in all, a wide range of people with varying technical backgrounds will find much to enjoy here.
Date published: 2014-01-30
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