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Discrete Mathematics

Explore this modern realm of digital math with this information that is mathematically rigorous and yet entertaining and accessible.
Discrete Mathematics is rated 4.6 out of 5 by 63.
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Rated 5 out of 5 by from Very enjoyable class! This class is useful and really fun! The Professor knows how to convey complex concepts in a very simple and elegant way!
Date published: 2024-03-10
Rated 5 out of 5 by from Very Interesting rich and clearly presented course This is a great course on the Discrete mathematics, covering many important topics and presented by an excellent sequence of lectures. I recommend this course for anyone interested in mathematical education. I found one error (typo) on the minute 12:00 of lecture 4 (Pascal triangle) , but if is easy to detect and correct if you follow the course steps.
Date published: 2022-04-08
Rated 4 out of 5 by from Challenging Had this course a couple of weeks and not really got into it yet. I can tell it's going to be a challenge, and it's fairly fast paced. There are concepts I am not familiar with, but hopefully I will be by the end.
Date published: 2022-01-30
Rated 5 out of 5 by from Top Drawer This is simply an outstanding course. I knew a bit about number theory and combinatorics, but the graph theory sequence was new to me and very exciting. Prof. Benjamin’s delivery is outstanding and there is humor to boot. HWF, Mesa AZ
Date published: 2021-07-18
Rated 5 out of 5 by from Great Foundation for Exploring Mathematics. My goal was to pursue statistics and computer science, specifically algorithms in computer science. I took Dr. Benjamin's course as a mathematical foundation for my pursuit into to these areas. This was an outstanding and very engaging course that expanded my interest in mathematics in general. Dr. Benjamin is a very engaging professor that makes an often difficult subject approachable. In fact it stimulated my interest in mathematics as a hobby, something I thought I would never do. I highly recommend this course for anyone interested in mathematics or anyone looking for an avenue of intellectual pursuits. You won't be disappointed.
Date published: 2021-04-12
Rated 5 out of 5 by from Passion for the Big, "discrete" picture I was truly refreshed by professor Benjamin's passion for his craft. He connects all the dots and and drives home the conceptual foundations that will remain with me even when the details of the formulas are long forgotten.
Date published: 2020-07-26
Rated 4 out of 5 by from A too fast primer. I am half way through this 24 lesson DVD. It is well presented but the content is packed and I have to see many lessons more than once or twice to grasp the conclusions. I would have been able to grasp the conclusions of a proof if I could have seen the steps in more detail as the instructor uses results "learned" two or three lessons before. The other alternative would be to include some homework examples to be worked offline while the DVD player was on Pause. I will finish the course at least once and start again and try to find a text that addresses the same material for backup. Part of the problem may be that the instructor is trying to fit too much into a compressed timeline. Perhaps there should be 50% more time in some lessons dedicated to proofs.
Date published: 2020-06-13
Rated 1 out of 5 by from Real Mudd, Harvey! I love the Great Courses. The Great Courses gave me the opportunity to study the things that I had wanted to study in college so many years ago. For this I am truly grateful. The Great Courses filled a hole in my broken heart and healed a gaping wound to my intellect. I purchased this course several years ago and have started watching it at least a dozen times. For a long time I thought that my lack of understanding was my fault. But I was wrong. Recently I found other sources of math instruction on the internet and through those sources have developed a pretty good understanding of the topics covered in the course. In fact, I found that the math I studied in high school 60 years ago clearly explained to me what this prof purports to be teaching. To state it clearly, my experience with this course, this professor, UN-Taught me something I already knew and thoroughly understood.
Date published: 2019-08-01
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Explore this modern realm of digital math in Discrete Mathematics, 24 mind-expanding lectures by veteran Teaching Company Professor Arthur T. Benjamin, an award-winning educator and "mathemagician" who has designed a course that is mathematically rigorous and yet entertaining and accessible to anyone with a basic knowledge of high school algebra.


Arthur T. Benjamin

As a professor, I have always wanted to bring math to the masses. The Great Courses has helped make that dream come true.


Harvey Mudd College

Arthur T. Benjamin is the Smallwood Family Professor of Mathematics at Harvey Mudd College. He earned a PhD in Mathematical Sciences from Johns Hopkins University. His teaching has been honored by the Mathematical Association of America, and he was named to The Princeton Review’s list of the Best 300 Professors. He has also served as president of the Fibonacci Association. A professional magician, he is the author of the book The Magic of Math, a New York Times bestseller. He has appeared on numerous television and radio programs and has been featured in Scientific American and The New York Times.

By This Professor

The Joy of Mathematics
The Secrets of Mental Math
The Mathematics of Games and Puzzles: From Cards to Sudoku
Math and Magic
Fibonacci Numbers and the Golden Ratio
What Is Discrete Mathematics?

01: What Is Discrete Mathematics?

In this introductory lecture, Professor Benjamin introduces you to the entertaining and accessible field of discrete mathematics. Survey the main topics you'll cover in the upcoming lectures—including combinatorics, number theory, and graph theory—and discover why this subject is off the beaten track of the continuous mathematics you studied in high school.

33 min
Basic Concepts of Combinatorics

02: Basic Concepts of Combinatorics

Combinatorics is the mathematics of counting, which is a more subtle exercise than it may seem, since the question "how many"? has at least four interpretations. Investigate factorials as well as the binomial coefficient, "n choose k," which shows the number of ways that "k" things can be chosen from "n" objects.

34 min
The 12-Fold Way of Combinatorics

03: The 12-Fold Way of Combinatorics

As an overview of combinatorial concepts, explore 12 different interpretations of counting by asking how many ways x pieces of candy can be distributed among b bags. The answers depend on such factors as whether the candies and bags are distinguishable, and how many candies are allowed in each bag.

31 min
Pascal's Triangle and the Binomial Theorem

04: Pascal's Triangle and the Binomial Theorem

Devised to calculate the payout in games of chance, Pascal's triangle is filled with beautiful mathematical patterns, all based on the binomial coefficient, "n choose k." Professor Benjamin demonstrates some of the triangle's amazing properties.

33 min
Advanced Combinatorics—Multichoosing

05: Advanced Combinatorics—Multichoosing

How many ways can you choose three scoops of ice cream from 31 flavors, assuming that flavors are allowed to be repeated? Using the method of "stars and bars," you find 5,456 possibilities if the order of flavors does not matter. The technique also works for counting endgame positions in backgammon.

32 min
The Principle of Inclusion—Exclusion

06: The Principle of Inclusion—Exclusion

Learn how the principle of inclusion-exclusion allows you to solve problems such as these: What is the probability that a five-card poker hand has at least one card in each suit? If homework papers are randomly distributed among students for grading, what are the chances that no student gets his or her own homework back?

33 min
Proofs—Inductive, Geometric, Combinatorial

07: Proofs—Inductive, Geometric, Combinatorial

Proofs by induction are a fundamental tool in any discrete mathematician's toolkit. This lecture guides you through several inductive proofs and then introduces geometric proof, also known as proof without words, and combinatorial proof. You see how all three techniques can prove properties of Pascal's triangle and Fibonacci numbers.

31 min
Linear Recurrences and Fibonacci Numbers

08: Linear Recurrences and Fibonacci Numbers

Investigate some interesting properties of Fibonacci numbers, which are defined using the concept of linear recurrence. In the 13th century, the Italian mathematician Leonardo of Pisa, called Fibonacci, used this sequence to solve a problem of idealized reproduction in rabbits.

33 min
Gateway to Number Theory—Divisibility

09: Gateway to Number Theory—Divisibility

Starting the section of the course on number theory, explore some key properties of numbers, beginning with what you know intuitively and working toward surprising properties such as Bezout's theorem. You also prove several important theorems relating to divisibility and prime factorization.

33 min
The Structure of Numbers

10: The Structure of Numbers

Study the building blocks of integers and how numbers can be created additively or multiplicatively. For example, every integer can be expressed as the sum of distinct powers of 2 in a unique way. Similarly, every integer is the product of a unique set of prime numbers.

34 min
Two Principles—Pigeonholes and Parity

11: Two Principles—Pigeonholes and Parity

Explore fascinating examples of two ideas: the pigeonhole principle, which can be used to prove that a mathematical situation is inevitable, such as that there must be a power of 3 that ends in the digits 001; and the parity principle, which is useful for proving that certain outcomes are impossible.

31 min
Modular Arithmetic—The Math of Remainders

12: Modular Arithmetic—The Math of Remainders

Introducing the important tool of modular arithmetic, Professor Benjamin uses the example of a clock to show how practically everyone is already adept with mod 12 arithmetic. Among the technique's many applications are the ISBN codes found on books, which use mod 11 for error detection.

32 min
Enormous Exponents and Card Shuffling

13: Enormous Exponents and Card Shuffling

Exploring more applications of modular arithmetic, examine the Chinese remainder theorem, used in ancient China as a fast way to count large numbers of troops. Also learn about password protection, the mathematics behind the "perfect shuffle," and the "seed planting" technique for raising big numbers to big powers.

31 min

14: Fermat's "Little" Theorem and Prime Testing

Use modular arithmetic to investigate more properties of prime numbers, leading to a practical way to test if an integer is prime. At the same time, meet two important figures in the history of number theory: Pierre de Fermat and Leonhard Euler.

33 min
Open Secrets—Public Key Cryptography

15: Open Secrets—Public Key Cryptography

The idea behind public key cryptography sounds impossible: The key for encoding a secret message is publicized for all to know, yet only the recipient can reverse the procedure. Learn how this approach, widely used over the Internet, relies on Euler's theorem in number theory.

34 min
The Birth of Graph Theory

16: The Birth of Graph Theory

This lecture introduces the last major section of the course, graph theory, covering the basic definitions, notations, and theorems. The first theorem of graph theory is yet another contribution by Euler, and you see how it applies to the popular puzzle of drawing a given shape without lifting the pencil or retracing any edge.

29 min
Ways to Walk—Matrices and Markov Chains

17: Ways to Walk—Matrices and Markov Chains

Use matrices to answer the question, How many ways are there to "walk" from one vertex to another in a given graph? This exercise leads to a discussion of random walks on graphs and the technique used by many search engines to rank web pages.

28 min
Social Networks and Stable Marriages

18: Social Networks and Stable Marriages

Apply graph theory to social networks, investigating such issues as the handshake theorem, Ramsey's theorem, and the stable marriage theorem, which proves that in any equal collection of eligible men and women, at least one pairing exists for each person so that no extramarital affairs will take place.

29 min
Tournaments and King Chickens

19: Tournaments and King Chickens

Discover some interesting properties of tournaments that arise in sports and other competitions. Represented as a graph, a tournament must contain a Hamiltonian path that visits each vertex once; and at least one "king chicken" competitor who has either beaten every opponent or beaten someone who beat that opponent.

31 min
Weighted Graphs and Minimum Spanning Trees

20: Weighted Graphs and Minimum Spanning Trees

When you call someone on a cell phone, you can think of yourself as a leaf on a giant "tree"—a connected graph with no cycles. Trees have a very simple yet powerful structure that make them useful for organizing all sorts of information.

31 min
Planarity—When Can a Graph Be Untangled?

21: Planarity—When Can a Graph Be Untangled?

Professor Benjamin introduces the concept of a planar graph, which is a graph that can be drawn on a sheet of paper in such a way that none of its edges cross. Then, encounter the two simplest nonplanar graphs, at least one of which must be contained within "any" nonplanar graph.

30 min
Coloring Graphs and Maps

22: Coloring Graphs and Maps

According to the four-color theorem, any map can be colored in such a way that no adjacent regions are assigned the same color and, at most, four colors suffice. Learn how this problem went unsolved for centuries and has only been proved recently with computer assistance.

33 min
Shortest Paths and Algorithm Complexity

23: Shortest Paths and Algorithm Complexity

Examine more problems in graph theory, including the shortest path problem, the traveling salesman problem, and the Hamiltonian cycle problem. Some problems can be solved efficiently, while others are so hard that no simple solution has yet been found.

33 min
The Magic of Discrete Mathematics

24: The Magic of Discrete Mathematics

In his final lecture, Professor Benjamin reviews areas where combinatorics, number theory, and graph theory overlap. Then he looks ahead at topics that build on the course's solid foundation in discrete mathematics. He closes with a flourish of mathematical magic, including the "four-ace surprise."

33 min