# Discrete Mathematics

Course No. 1456
Professor Arthur T. Benjamin, Ph.D.
Harvey Mudd College
4.7 out of 5
55 Reviews
92% of reviewers would recommend this product
Course No. 1456
Video Streaming Included Free

## Course Overview

Welcome to Discrete Mathematics, a subject that is off the beaten track that most of us followed in school but that has vital applications in computer science, cryptography, engineering, and problem solving of all types.

Most of the mathematics taught after elementary school is aimed at preparing students for one subject—calculus, which is the mathematics of how things grow and change continuously, like waves in the water or clouds in the sky. Discrete mathematics, on the other hand, deals with quantities that can be broken into neat little pieces, like pixels on a computer screen, the letters or numbers in a password, or directions on how to drive from one place to another.

While continuous mathematics resembles an old-fashioned analog clock, whose second hand sweeps continuously across a dial, discrete mathematics is like a digital watch, whose numbers proceed one second at a time. As a result, discrete mathematics achieves fascinating mathematical results using relatively simple means, such as counting.

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.

Problems, Proofs, and Applications

Discrete mathematics covers a wide range of subjects, and Professor Benjamin delves into three of its most important fields, presenting a generous selection of problems, proofs, and applications in the following areas:

• Combinatorics: How many ways are there to rearrange the letters of Mississippi? What is the probability of being dealt a full house in poker? Central to these and many other problems in combinatorics (the mathematics of counting) is Pascal's triangle, whose numbers contain some amazingly beautiful patterns.
• Number theory: The study of the whole numbers (0, 1, 2, 3, ...) leads to some intriguing puzzles: Can every number be factored into prime numbers in exactly one way? Why do the digits of a multiple of 9 always sum to a multiple of 9? Moreover, how do such questions produce a host of useful applications, such as strategies for keeping a password secret?
• Graph theory: Dealing with more diverse graphs than those that plot data on x and y axes, graph theory focuses on the relationship between objects in the most abstract sense. By simply connecting dots with lines, graph theorists create networks that model everything from how computers store and communicate information to transportation grids to even potential marriage partners.

Learn to Think Mathematically

Professor Benjamin describes discrete mathematics as "relevant and elegant"—qualities that are evident in the practical power and intellectual beauty of the material that you study in this course. No matter what your mathematical background, Discrete Mathematics will enlighten and entertain you, offering an ideal point of entry for thinking mathematically.

In discrete math, proofs are easier and more intuitive than in continuous math, meaning that you can get a real sense of what mathematicians are doing when they prove something, and why proofs are an immensely satisfying and even aesthetic experience.

The applications featured in this course are no less absorbing and include cases such as these:

• Internet security: Financial transactions can take place securely over the Internet, thanks to public key cryptography—a seemingly miraculous technique that relies on the relative ease of generating 1000-digit prime numbers and the near impossibility of factoring a number composed of them. Professor Benjamin walks you through the details and offers a proof for why it works.
• Information retrieval: A type of graph called a tree is ideal for organizing a retrieval structure for lists, such as words in a dictionary. As the number of items increases, the tree technique becomes vastly more efficient than a simple sequential search of the list. Trees also provide a model for understanding how cell phone networks function.
• ISBN error detection: The International Standard Book Number on the back of every book encodes a wealth of information, but the last digit is very special—a "check digit" designed to guard against errors in transcription. Learn how modular arithmetic, also known as clock arithmetic, lies at the heart of this clever system.

Professor Benjamin believes that, too often, mathematics is taught as nothing more than a collection of facts or techniques to be mastered without any real understanding. But instead of relying on formulas and the rote manipulation of symbols to solve problems, he explains the logic behind every step of his reasoning, taking you to a deeper level of understanding that he calls "the real joy and mastery of mathematics."

Dr. Benjamin is unusually well qualified to guide you to this more insightful level, having been honored repeatedly by the Mathematical Association of America for his outstanding teaching. And for those who wish to take their studies even further, he has included additional problems, with solutions, in the guidebook that accompanies the course.

With these rich and rewarding lectures, Professor Benjamin equips you with logical thinking skills that will serve you well in your daily life—as well as in any future math courses you may take.

24 lectures
|  Average 31 minutes each
• 1
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. x
• 2
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. x
• 3
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. x
• 4
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. x
• 5
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. x
• 6
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? x
• 7
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. x
• 8
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. x
• 9
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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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. x
• 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." x

## Lecture Titles

Clone Content from Your Professor tab

## What's Included

### What Does Each Format Include?

##### Instant Video Includes:
• FREE video streaming of the course from our website and mobile apps
##### DVD Includes:
• 24 lectures on 4 DVDs
• 160-page printed course guidebook
• FREE video streaming of the course from our website and mobile apps

### What Does The Course Guidebook Include?

##### Course Guidebook Details:
• 160-page printed course guidebook
• diagrams & equations
• Questions to consider

### Enjoy This Course On-the-Go with Our Mobile Apps!*

• Android Devices
• Kindle Fire Tablet + Firephone
*Courses can be streamed from anywhere you have an internet connection. Standard carrier data rates may apply in areas that do not have wifi connections pursuant to your carrier contract.

Arthur T. Benjamin, Ph.D.
Harvey Mudd College
Dr. Arthur T. Benjamin is Professor of Mathematics at Harvey Mudd College. He earned a Ph.D. in Mathematical Sciences from Johns Hopkins University in 1989. Professor Benjamin's teaching has been honored repeatedly by the Mathematical Association of America (MAA). In 2000, he received the MAA Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics. The MAA also named...
Also By This Professor

## Reviews

Discrete Mathematics is rated 4.6 out of 5 by 55.
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
Rated 4 out of 5 by from Fond memories and passing the torch About thirty years ago, I took a Discrete Math course and it was the best and most interesting courses that I took. I saw this course in the catalog recently and checked the contents against my college textbook (which I have on a shelf at work). It is very similar and has a few extra lessons that were also included in another Great Courses series that I took a couple of years ago. I was impressed enoough that I bought a copy for a graduation bonus gift for my grandson, who starts college in the Fall and could use some interesting ideas to kepp his mind active during Summer. It this price, I am thinking of getting a review copy for myself.Ratings based on catalog description comparison to textbook and never having had a less than Good Great Courses instructor.
Date published: 2019-06-04
Rated 3 out of 5 by from Based only on the first 3 lectures This review is based on watching the first 3 lectures so things may get better as I watch more. 1) This is the first class which I purchased which I had to buy the accompanying "book". And the book really does not go into any more detail than the lecture does so kinda frustrating that I paid \$15 bucks for it 2) I have taken classes in statistics so I am familiar with Combinatorics from a different angle. But the lecture on the 12 fold way is very difficult to wrap my head around since it is gone through very quickly. If I was in an actual class lecture and did not have the ability to "rewatch" the lecture like with the videos, I would have walked out of that lecture thinking "Did I actually learn anything here"
Date published: 2019-05-22
Rated 4 out of 5 by from Great flow of ideas! I am enjoying the method of presenting the different topics.
Date published: 2019-04-19
Rated 5 out of 5 by from Excellent program I'm 81.5 years old it will help to refresh my brain
Date published: 2018-02-02
Rated 5 out of 5 by from Beyond Fantastic Prof Benjamin does an amazing job here. Except for the 12-fold lecture being a little early, I'd have to say everything is terrifically well-paced and well-explained. He does a great job mki
Date published: 2018-01-31
Rated 1 out of 5 by from Discreet Math Hard to understand. I like the professor. I have several other courses of his, but this one was out in left field. I am a doctor, but was lost from the first class.
Date published: 2018-01-07
Rated 5 out of 5 by from A Wonderful Course There is a surprising amount of detail to keep track of in a subject like Discreet Mathematics. I have actually taught this course at the university level and I now wish I had had this program at my disposal when I did. For clarity of explanation and connecting the dots between the discreet sub-subjects of this material, you can't get much better than Prof. Benjamin. He is, indeed, a Mathemagician!
Date published: 2017-12-03