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Discrete Mathematics

Discrete Mathematics

Course No.  1456
Course No.  1456
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Course Overview

About This Course

24 lectures  |  31 minutes per lecture

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.

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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.

Deepen Your Understanding of Mathematics

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.

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24 Lectures
  • 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
    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. 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

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Arthur T. Benjamin
Ph.D. Arthur T. Benjamin
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 Professor Benjamin the 2006-2008 George Pólya Lecturer. In 2012, Princeton Review profiled him in The Best 300 Professors. He is a professional magician, whose techniques are explained in his book Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks. Professor Benjamin also served for five years as coeditor of Math Horizons magazine. An avid games player, Dr. Benjamin is a past winner of the American Backgammon Tour and has written more than 15 papers on the mathematics of games and puzzles. Professor Benjamin has appeared on dozens of television and radio programs and has been featured in publications, including Scientific American, People, and The New York Times. In 2005, Reader's Digest called him America's Best Math Whiz.

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Reviews

Rated 5 out of 5 by 31 reviewers.
Rated 5 out of 5 by Excellent, but intense! This is certainly not one of those overview courses that you can watch and absorb with little effort. It's the real deal, discrete maths for people who want to explore and use the techniques. I had to watch some of the videos more than once to follow some of the proofs. The presenter is a joy to watch, his enthusiasm is contagious! Great course for those with a reasonably strong math background. April 20, 2014
Rated 5 out of 5 by Great lecturer A great job of lecturing. I will have to listen to it several times to take it all in but Prof. Benjamin is a joy to listen to. I wish I had had math teachers half as good. January 21, 2014
Rated 5 out of 5 by High-level maths with gifted teacher! An advanced, specialised course for those with a strong grounding in mathematics, especially algebra. Definitely not for everyone! Professor Benjamin presents and explains the intricacies of discrete mathematics in his usual energetic, joyful, entertaining manner ~~ with good sprinkles of humour! It's pretty obvious he lives and loves numbers! To illustrate his points, he uses examples which are readily familiar to all of us, such as flavours of ice cream and the probabilities of being dealt certain hands at card games. He expounds his proofs with careful clarity. Oh yes ~~ he also loves prestidigitation, so he introduces a touch of magic into some discussions. This young lecturer is a gem, and while I readily admit that this level of maths is certainly hard for me to try to follow, I happily recommend the course to those at this maths level who wish a challenging series of lectures by a gifted teacher. Each lecture gives a number of challenging questions, with answers supplied at the end of the guidebook which also contains a timeline, glossary, and bibliography. It is indeed a 5-star course for its target audience. January 31, 2013
Rated 5 out of 5 by Fun - (for the math savvy) As a retired engineer that specialized in structural dynamics and spectral analysis, I thought it fun to deal in the math world without decimal points! Even in areas that I felt "comfortable" with, there certainly was much to learn. I especially liked the prime number discussions and foray into encryption. The lessons on graph theory were completely new to me and well presented. I recommend this course to anyone with a better-than-average math background (and even appreciated the "bad" puns from the talented professor!) Good job. January 26, 2013
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