# Discrete Mathematics

Course No. 1456
Professor Arthur T. Benjamin, Ph.D.
Harvey Mudd College
4.7 out of 5
57 Reviews
92% of reviewers would recommend this product
Course No. 1456
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## 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.

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.

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

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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...
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## Reviews

Discrete Mathematics is rated 4.6 out of 5 by 57.
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
Rated 4 out of 5 by from Viewing interrupted I have only seen 2 lessons. Complete knee replacement has interrupted viewing. The prof is very clear in his explanation. His jokes are bad. Once i get out of rehab, I will continue.
Date published: 2017-02-24
Rated 5 out of 5 by from Discrete Math: Excellent Course Discrete Math course is excellent. Professor Benjamin's enthusiasm for the material is contagious and his presentation is both very clear and interesting. He is also funny on occasion and always presents the material in down to earth understandable ways. It is a course that can be “mind twisting” and requires effort to absorb, but is well worth the effort.
Date published: 2017-01-22
Rated 4 out of 5 by from a good brain stretcher i have many or most of the great courses math series. i am finding this a bit daunting because a lot of new to me vocabulary and teacher stays almost entirely in numbers, nouns and formulas. he does not tie things to concepts apart from the numbers themselves very much. he also speaks extremely fast. i recognize a number of things from the other courses. i will replay the series over and over until i am comfortable or more so with speaking almost entirely in a kind of short hand. the course is not what i expected but i chose it for stretching my brain and it is going to do that and, yes.....i would recommend it to others interested in math or interested in some of the applications like secure numbers on bank codes etc. an important field of thinking much of which is new ideas for me thank you
Date published: 2016-12-05
Rated 5 out of 5 by from What a superb lecturer I have been studying Physics and Mathematics now for over 50 years and not too old to actually be able to teach it. The most striking aspect of this course is the lecturer. - Professor Benjamin in so enthusiastic, clearly spoken (even a Brit can understand him!), not rigidly following a teleprompter, and even has a few jokes up his sleeve. This factor makes this course outstanding!!. Now I can't claim to understand more than bits of the maths he is teaching but he is such a wonderful presenter that I feel carried along - I intend to go back and make notes next time. The maths is not straightforward but I would have been disappointed with The Great Courses if it had been easy - its certainly mind stretching. All in all a wonderful experience, well worth the money. I would love to have Professor Benjamin as a dinner guest - he would thrill all my guests. Well done The Great Courses/ .
Date published: 2016-08-24
Rated 5 out of 5 by from Filled with outstanding elaborations. Thumb Up!! Having graduated from university with degree of computer science for almost a decade, Mathematic never been my favorite subject, I’d learned all that in the way of dull if not wars :). I am always having a thought to revisit to certain subjects which I’d learned earlier that shown rusted today. Dr. Arthur Benjamin undoubtedly brought the subject into the level of brand new and it fulfilled my learning objectives, much like his mental mathematics courses, it filled with outstanding elaborations. A big thumb up!! I wish there will be a HD remaster. :)
Date published: 2016-08-07
Rated 5 out of 5 by from The 12-fold way of combinatorics and the Candy Man This is a superb course on mathematics. The three main topics: combinatorics, number theory and graph theory are explained in a very nice and congruent way. I was impressed by the lecture entitled the 12-fold way of combinatorics where Professor Benjamin summarized in about 30 minutes the essence of the main problems in combinatorics in very didactic way. I would like to have a course like that on Group Theory by Professor Benjamin.
Date published: 2016-07-30
Rated 5 out of 5 by from Computer science math, you can count on it! Computer math is combinatorics, recurrence relations, number theory, graph theory, matrix operations, proofs (for algorithm construction), complexity, and generic algorithms (for a start). The architecture and design and development, of computer systems and programs, is strongly enhanced by exactly this mathematics offering. I should know, I taught this stuff (before the Rosen text), and I wish I'd had material resources this organized and complete. Be forewarned however, Benjamin rates Knuth's "Concrete Mathematics" as "Intermediate difficulty". Oh my, the horror! Benjamin's offering covers very similar terrain, providing a deep overview beyond the needs of the dilettante. I wish Benjamin had included the same web Resources found in "Joy of Math", but discretely/ concretely skewed. One resource recommended was the "Algorithms and Complexity" text by Wilf, which I obtained. In any case, Rosen, Knuth, and Wilf are my recommended "next steps", especially for problems to solve and academic work to do.
Date published: 2016-07-03
Rated 3 out of 5 by from This course is more challenging because it is more advanced and geared towards students majoring in mathematics in college, but professor Benjamin does a good job of breaking it down into small manageable parts and makes the material easier to understand. This course is useful in computer science and cryptography/code breaking.
Date published: 2016-03-01
Rated 5 out of 5 by from Fantastic Arthur Benjamin rea!ly knows his stuff!! He presents every thing in a well organized and entertaining manner. I highly recommend.
Date published: 2016-01-21
Rated 5 out of 5 by from Tough but rewarding There are some excellent detailed reviews here, so I don't have much to add. I just want to emphasize the clarity of Prof Benjamin's exposition. He is one of the clearest Great Courses presenters I've yet heard. Cancel that. He is one of the clearest presenters anywhere I've yet heard, on any subject. Added to his passionate-without-histrionics manner and you have a perfect guide. Warning: even with Prof Benjamin cracking jokes this is not "math-lite" - this course is demanding but there are excellent graphics, a very useful course book (with full solutions) and, most importantly, the replay button! There are lots of math goodies in here and even if you have some previous familiarity with the topic I'm sure you will pick up something new. If you aren't familiar with discrete math then be prepared for a challenging but eye-opening course.
Date published: 2015-11-21
Rated 5 out of 5 by from Super Course, Super Instructor This was truly an amazing course. Dr. Benjamin is an incredibly gifted educator: a very dynamic, engaging, and clear speaker. I have a Computer Science and math background, so much of the material is relevant and useful to me. The examples were really well-chosen. I enjoyed the whole course. I had to replay some of the subtopics to understand them, but if I just re-viewed a challenging topic, it made sense. This course went above and beyond my expectations. I highly recommend this course and this instructor. Let's clone him!
Date published: 2015-11-14
Rated 5 out of 5 by from Just Amazing I would like to start by saying, I am no math major. I work as a computer systems engineer (pretty much self-taught), and I am always looking for integrating mathematics into my software. My main goal in watching lectures on mathematics is for the pure love and passion for math. While watching Professor Benjamin, I could see his passion for mathematics, and it is inspirational. He really knows what he's talking about; he isn't just reciting from some PowerPoint. The subject of Discrete Math is not for the weak of heart. I had to watch many of the videos multiple times to fully grasp what was going on. The professor made the material interesting, while still conveying the complexities of discrete math. I would definitely recommend this course. I'm actually going back through this course now, just as a refresher. For anyone out there who is undecided whether to purchase this course or not, buy it; and, be prepared to learn. Eddiejackson.net
Date published: 2015-05-05
Rated 5 out of 5 by from Solid resource for students et al College math students will find this course valuable as either a reinforcement of combinatorics, number theory, graph theory and proof skills or as a solid introduction for those whom calculus and algebra had been their primary focus. For the average learner this course will be a serious challenge but thanks to a well designed format reinforced by excellent examples and delivered by a gifted Professor the rewards are proportional to the amount of effort one wishes to put into the material. In any case this is one course which when reviewed multiple times always reaches a new level of understanding and appreciation for the lesson. Professor Benjamin employs all of his considerable entertainment skills in producing an enjoyable examination of a very difficult group of topics. I am not sure I would have been able to tackle them otherwise. I am not only appreciative but I am looking forward to more courses from this very talented intructor.
Date published: 2015-04-03
Rated 1 out of 5 by from Simpler and clearer explanations elsewhere Simpler and clearer explanations (in my opinion,) for some of the formulas given in the first lectures can be found elsewhere. I returned the course before completing it. I received a refund from The Great Courses on the same day that I requested it. I was dissatisfied with this particular course, but very satisfied with the customer service of The Great Courses and how they handled my refund request.
Date published: 2014-11-09
Rated 5 out of 5 by from Great prof, hard course Professor Benjamin is as bright as a button and a pleasure to spend time with, which is a good thing, because the course was way over my head, and more often than not it seemed as though he was speaking in tongues. Discrete Mathematics is not for the math-phobic or algebra-deficient. But Professor Benjamin is so buoyant and charming, with such mirth in his eyes, that he seduces you into staying the course even if you realize, as I did, that you will not be acquiring Discrete Mathematics in this lifetime. For those who are truly comfortable with algebra, and willing to stop each lesson multiple times to work through the proofs, I think it would be possible to learn this material. For the rest of us, we can marvel at his cheerful brilliance and maybe, if we are diligent, pick up a magic trick or two.
Date published: 2014-09-29
Rated 5 out of 5 by from 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.
Date published: 2014-04-20
Rated 5 out of 5 by from 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.
Date published: 2014-01-22
Rated 5 out of 5 by from 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.
Date published: 2013-01-31
Rated 5 out of 5 by from 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.
Date published: 2013-01-26
Rated 5 out of 5 by from A Mental Workout I listened to this course while working out -- I am not sure if the physical workout was as challenging as the mental workout provided by Prof. Benjamin. Prof. Benjamin did a nice job reviewing the fundamentals of discrete math (combinatorics, number theory, and graph theory). He was "super enthusiastic," and I enjoyed his numerous mathematical puns. He provided a useful bibliography and selected the course material well. Although I sometimes became lost in the numerous mathematical proofs, I most enjoyed the real world applications of math. These applications included (among others): calculating the probability of various poker hands, various decision-making algorithms, and, even, magic tricks. I would recommend this course to those who are up to a challenging mathematical course complete with a number of proofs.
Date published: 2012-07-30
Rated 5 out of 5 by from Gentle, But Not Wimpy Prof. Benjamin presents the material in a very friendly, lively (even joyous), and easily understandable way. The lectures are accessible to just about anyone. But make no mistake - this is the real deal. By the end of this course (assuming you don't just passively listen, but take pencil to paper and work through it honestly as you go), you'll be ready to dig into any textbook to further explore the topics on your own. In addition to the beauty of the mathematics, this material also happens to be particularly relevant for algorithm/software people, as it hits just about all the key fundamental points. Excellent course.
Date published: 2012-07-28
Rated 4 out of 5 by from Great course with typos Professor Benjamin is an enthusiastic presenter of the mathematical material. The concepts are clearly presented and discussed. The public key inscription lecture was of particular interest. However there are typographical errors in a few on-screen slides and many in the accompanying book. This does slowdown the learning. Although most errors are easily identified, a course on mathematics desires a carefully editing so as to not to confuse a student unfamiliar with the material.
Date published: 2012-07-14
Rated 5 out of 5 by from Intellectual stimulation at its best! Discrete Mathematics is intellectual stimulation at its best! Professor Benjamin's passion for the subject, his clear presentation style and "punny" wit make for a delightful treat. I did not spend the time required to fully grasp the concepts of combinatorics, number theory and graph theory, but I did gain sufficient knowledge such that I have a new appreciaton for the number of possibilities resulting from choices, identification of prime numbers and factors (as they apply to cryptography), and finding the shortest path among many to a designated end-point. For me lectures 20-22 were tedious, but lecture 23 (Shortest Paths and Algorithm Complexity) made relevant all lectures on graph theory. As a physician (reproductive endocrinologist) I realized that physicians are trained in combinatorics and graph theory (but not through discrete mathematics) to make a differential diagnosis from the number of variables that must be consisdered. Despite my failure to maximize the content of the course, I completed it with a new appreciation for the variety of ways that discrete mathematics impacts my daily life. I highly recommend this course for anyone who wants to further their knowledge of mathematics, and enjoy the presentation skills of an excellent professor.
Date published: 2012-06-22
Rated 5 out of 5 by from Grand Slam!!! Prof. Benjamin has done it again -- upbeat , entertaining and mathematically satisfying. One lecture alone helped me to review my combinatorics in preparation for my National Board Certified Exams. This is my second course from Prof. Benjamin. A nice balance -- but thorough treatment -- of combinatorics, number & graph theory. Highly recommended!!
Date published: 2011-06-06
Rated 5 out of 5 by from Very clear presentation, challenging material Really challenging. I wanted to get a more in-depth treatment of number theory but this delivers considerably more than that. The lectures demand close attention and repeated viewing (for me anyway) and any lapse on the viewer's part generally leaves him scratching his head and saying "what just happened here?" The story of the emergence of public key cryptography from something as ancient and abstract as number theory is alone worth the price of the lectures. Graph theory as well, leading up to the four-color map theorem and related matters. Remarkable course, highly recommended.
Date published: 2011-02-10
Rated 5 out of 5 by from Discrete Mathematics What a pleasure to learn something new and more thoroughly than I thought I could. I had a great time listening and watching Dr, Benjamin present the material in a simple and delightful manner. Puns included. Stan
Date published: 2011-01-30
Rated 5 out of 5 by from Excellent and Rewarding The most difficult Teaching Co. course that I've taken but very rewarding. For those of us with a background in continuous math. this course offers a welcome expansion of math knowledge. Dr. Benjamin obviously knows the material and presents it in an upbeat and interesting manner., I recommend it.
Date published: 2010-09-18
Rated 5 out of 5 by from Beyond Excellent To say that this course is excellent would be an understatement. I rate this course Excellent Plus. It is a pleasure to listen to Prof. Benjamin. His presentation is lively and clear, and full of helpful teaching aides that facilitate understanding of the material. The problems at the end of each lecture reinforced my understanding of the lecture. Referring to the carefully selected bibliography helped me delve deeper into the presented material. I am waiting for the next Teaching Company course by Prof. Benjamin.
Date published: 2010-08-27