Prove It: The Art of Mathematical Argument

Rated 4 out of 5 by from Kinda Good I bought this a few weeks ago and I like it. I liked how you slowly did these problems. The only problem is that
Date published: 2020-08-27
Rated 5 out of 5 by from Learned More Than Just Proofs Professor Edwards is a teacher I wish I could have studied math under. I see why the Great Courses uses him in their advertisements. He explains each step of a problem and does not rush through solutions. This was an excellent course that provided understanding of theorems as well as the proof. The proofs contain much algebra and require a style of thinking that is not natural to me, but I found them fascinating. Truth is I learned much more than just the proofs, especially about Infinite Series, Fibonacci Numbers and e. I will watch this course again and highly recommend it.
Date published: 2020-01-30
Rated 5 out of 5 by from Good intro into the actual mathematician's craft. Easy to follow explanations, wide range of areas, and professor Edward's style is very enjoyful.
Date published: 2020-01-15
Rated 5 out of 5 by from Outstanding Work! Essential to the ones that want to know how the world works.
Date published: 2019-10-02
Rated 5 out of 5 by from Instructor is excellent. This course was amazing. Easy to understand and want to learn more.
Date published: 2019-05-20
Rated 5 out of 5 by from I was a previous college professor who retired several years ago. This series is a wonderful review for me. I will never stop learning!
Date published: 2019-04-06
Rated 4 out of 5 by from Good Review of Proof Approaches; High School Level This is a well-done overview of the variety of ways mathematicians have devised to approach proving mathematical statements. Those who like learning about math for its own sake (this includes me) may find the course inherently interesting. Those studying math at a high school or early college level may gain a helpful extension of their basic understanding. Professor Edwards clearly loves his subject, as his unfailing smile and good humor demonstrate. He generally speaks in short, unadorned sentences, as if any syntactical complexity might impede our understanding of his mathematical points. This approach is also manifest in his frequent use of such comic book exclamations as "Ouch!" and "Wow!". (He is also the first professor I have ever seen actually employ, in real life, the stereotypical lecturer's gesture of touching the fingertips of each hand to each other; very endearing!) A few relatively minor complaints: I very much wish that more time was spent helping us to achieve an intuitive understanding of the abstract math. Venn diagrams, for example, would have been a great help in the early lectures on logic. (They were introduced briefly later.) And while the essential number 'e' is the subject of the entire final lecture, the natural logarithms (logarithms of base 'e') are discussed earlier with no attempt to explain 'e' or why it should be an especially important logarithmic base. There is also frequent and unnecessary repetition of simple points, and sometimes inadequate (at least for me) explanation of some of the more complex ideas. It is also worth noting that the level of the presentation ranges from junior high school to second year college. At times I was bored; at others, working hard to keep up. And I could certainly have done without TGC's annoying sound effects - irritating beeps to emphasize important points. So - I do recommend this course for intermediate-level math aficionados, and for motivated high schoolers. It is unlikely, I think, that you will appreciate each lecture equally, but overall I found it a worthwhile investment of my time.
Date published: 2018-11-04
Rated 5 out of 5 by from Big-time ^ Truly Great Before I bought this course I had had one “very readable book,” which happens to be in this course’s bibliography, and a webpage on four proof techniques. My 30-year-old virtual memories had so eroded, and Dover’s_Real Analysis_is so poorly written that I couldn’t get from one paragraph to the next. Thus began a growling curiosity for the elegance of theoretical math. The first time through the course a few years back, I recall the content as clearer than my Cambridge book, and more thorough and expansive than the webpage, but something essential eluded me. I still couldn’t reason properly in the course’s terms and moved on to other courses, including Prof Grabiner’s Math, Philosophy, and the Real World [#1440], which is paced very slowly but explains the atmosphere of a lot of basic reasoning. (Yes, I agree Prove It is a good step toward Prof Benjamin’s Discrete Mathematics [#1457], which is a barn burner, an artwork to be proud of; and I too would like to see a course on vector calculus and analysis.) The second time through Prove It: The Art of Mathematical Argument (an unfortunate title because most mathematicians “reason” mathematically, and only argue about sports or pizza), with the guidebook, and determined to master again the entirety, it didn’t just survive a second viewing (an ultimate commendation), I was able to see so much deeper into the subject. As a complete physics innumerate, I’m fresh off Physics in Your Life [#1260] by the rapid-fire Prof Wolfson and was at first shocked by the abundance of apparently dead air and the grade-school pace of Lecture One in Prove It. As the metronome clicked at about 30 bpm, I noticed that the very sharply edited new setting and updated production does not necessarily improve the content over older courses with outdated graphics (I’ve learned just as much from the oldest courses as from the most recent). The beauty abundant in this course, though, whether the setting, or Professor Edwards’s colorful neckties (he dresses exceedingly well), is appreciated. Regarding my second time through, a burst of fire here, a sudden movement there, by Lecture Two I was reminded how powerful and serious this man is, touched with the most pleasant humor. He’s excellent with body movement, addressing or gesturing audience stage right while onscreen graphics are moving that way. Indeed, he employs his whole body to transmit these ideas. He is a grandmaster of a teacher. The camera angles are so fine it’s like you’re in his classroom. At age 65 during production, he comes out with the most amazing asides from across his epochal career. By Lecture Five, I began to appreciate the repetition and slow tempo; because by then, from his masterpiece of organization, the whole orchestra was singing. By Lecture Six I was often hitting pause and rewinding, and viewing each subsequent lecture multiple times. (I too find set theory dreadfully onerous, but essential.) By the end of the 24-lecture course—and it’s all Professor Edwards, not multiple professors as mistakenly posted below—I was still challenged and will be going over and over this course every two years for the rest of my life. If you’re into math, I have found gold here and wouldn’t limit in the slightest the set of lifelong learners to whom I would recommend this truly wonderful offering. Thank you, Dr. Edwards, and all behind this exquisite product. This is big-time.
Date published: 2018-10-04
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Prove It: The Art of Mathematical Argument
Course Trailer
What Are Proofs, and How Do I Do Them?
1: What Are Proofs, and How Do I Do Them?

Start by proving that two odd numbers multiplied together always give an odd number. Next, look ahead at some of the intriguing proofs you will encounter in the course. Then explore the characteristics of a proof and tips for improving your skill at proving mathematical theorems.

32 min
The Root of Proof-A Brief Look at Geometry
2: The Root of Proof-A Brief Look at Geometry

The model for modern mathematical thinking was forged 2,300 years ago in Euclid's Elements. Prove three of Euclid's theorems and investigate his famous fifth postulate dealing with parallel lines. Also, learn how proofs are important in Professor Edwards's own research.

31 min
The Building Blocks-Introduction to Logic
3: The Building Blocks-Introduction to Logic

Logic is the foundation of mathematical proofs. In the first of three lectures on logic, study the connectors "and" and "or." When used in combination in mathematical statements, these simple terms can create interesting complexity. See how truth tables are very useful for determining when such statements are true or false.

30 min
More Blocks-Negations and Implications
4: More Blocks-Negations and Implications

Continue your study of logic by looking at negations of statements and the logical operation called implication, which is used in most mathematical theorems. Professor Edwards opens the lecture with a fascinating example of the implication of a false hypothesis that appears to pose a logical puzzle.

30 min
Existence and Uniqueness-Quantifiers
5: Existence and Uniqueness-Quantifiers

In the final lecture on logic, explore the quantifiers "for all" and "there exists," learning how these operations are negated. Quantifiers play a large role in calculus-for example, when defining the concept of a sequence, which you study in greater detail in upcoming lectures.

30 min
The Simplest Road-Direct Proofs
6: The Simplest Road-Direct Proofs

Begin a series of lectures on different proof techniques by looking at direct proofs, which make straightforward use of a hypothesis to arrive at a conclusion. Try several examples, including proofs involving division and inequalities. Then learn tricks that mathematicians use to make proofs easier than they look.

31 min
Let's Go Backward-Proofs by Contradiction
7: Let's Go Backward-Proofs by Contradiction

Probe the power of one of the most popular techniques for proving theorems-proof by contradiction. Begin by constructing a truth table for the contrapositive. Then work up to Euclid's famous proof that answers the question: Can the square root of 2 be expressed as a fraction?

31 min
Let's Go Both Ways-If-and-Only-If Proofs
8: Let's Go Both Ways-If-and-Only-If Proofs

Start with the simple case of an isosceles triangle, defined as having two equal sides or two equal angles. Discover that equal sides and equal angles apply to all isosceles triangles and are an example of an "if-and-only-if" theorem, which occurs throughout mathematics.

31 min
The Language of Mathematics-Set Theory
9: The Language of Mathematics-Set Theory

Explore elementary set theory, learning the concepts and notation that allow manipulation of sets, their unions, their intersections, and their complements. Then try your hand at proving that two sets are equal, which involves showing that each is a subset of the other.

30 min
Bigger and Bigger Sets-Infinite Sets
10: Bigger and Bigger Sets-Infinite Sets

Tackle infinite sets, which pose fascinating paradoxes. For example, the set of integers is a subset of the set of rational numbers, and yet there is a one-to-one correspondence between them. Explore other properties of infinite sets, proving that the real numbers between 0 and 1 are uncountable.

32 min
Mathematical Induction
11: Mathematical Induction

In the first of three lectures on mathematical induction, try out this powerful tool for proving theorems about the positive integers. See how an inductive proof is like knocking over a row of dominos: Once the base case pushes over a second case, then by induction all cases fall.

31 min
Deeper and Deeper-More Induction
12: Deeper and Deeper-More Induction

What does the decimal 0.99999... forever equal? Is it less than 1? Or does it equal 1? Apply mathematical induction to geometric series to find the solution. Also use induction to find the formulas for other series, including factorials, which are denoted by an integer followed by the "!" sign.

31 min
Strong Induction and the Fibonacci Numbers
13: Strong Induction and the Fibonacci Numbers

Use a technique called strong induction to prove an elementary theorem about prime numbers. Next, apply strong induction to the famous Fibonacci sequence, verifying the Binet formula, which can specify any number in the sequence. Test the formula by finding the 21-digit-long 100th Fibonacci number.

30 min
I Exist Therefore I Am-Existence Proofs
14: I Exist Therefore I Am-Existence Proofs

Analyze existence proofs, which show that a mathematical object must exist, even if the actual object remains unknown. Close with an elegant and subtle argument proving that there exists an irrational number raised to an irrational power, and the result is a rational number.

31 min
I Am One of a Kind-Uniqueness Proofs
15: I Am One of a Kind-Uniqueness Proofs

How do you prove that a given mathematical result is unique? Assume that more than one solution exists and then see if there is a contradiction. Use this technique to prove several theorems, including the important division algorithm from arithmetic.

31 min
Let Me Count the Ways-Enumeration Proofs
16: Let Me Count the Ways-Enumeration Proofs

The famous Four Color theorem, dealing with the minimum number of colors needed to distinguish adjacent regions on a map with different colors, was finally proved by a brute force technique called enumeration of cases. Learn how this approach works and why mathematicians dislike it-although they often rely on it.

31 min
Not True! Counterexamples and Paradoxes
17: Not True! Counterexamples and Paradoxes

You've studied proofs. How about disproofs? How do you show that a conjecture is false? Experience the fun of finding counterexamples. Then explore some famous paradoxes in mathematics, including Bertrand Russell's barber paradox, which shook the foundations of set theory.

30 min
When 1 = 2-False Proofs
18: When 1 = 2-False Proofs

Strengthen your appreciation for good proofs by looking at bad proofs, including common errors that students make such as dividing by 0 and circular reasoning. Then survey the history of attempts to prove some renowned conjectures from geometry and number theory.

31 min
A Picture Says It All-Visual Proofs
19: A Picture Says It All-Visual Proofs

Before he became the 20th U. S. president, James A. Garfield published an original proof of the Pythagorean theorem that relied on a visual argument. See how pictures play an important role in understanding why a particular mathematical statement may be true. But is a visual proof really a proof?

32 min
The Queen of Mathematics-Number Theory
20: The Queen of Mathematics-Number Theory

The great mathematician Carl Friedrich Gauss once said that if mathematics is the queen of the sciences, then number theory is the queen of mathematics. Embark on the study of this fascinating discipline by proving theorems about prime numbers.

29 min
Primal Studies-More Number Theory
21: Primal Studies-More Number Theory

Dig deeper into prime numbers and number theory by proving a conjecture that asserts that there are arbitrarily large gaps between successive prime numbers. Then turn to some celebrated conjectures in number theory, which are easy to state but which have withstood all attempts to prove them.

31 min
Fun with Triangular and Square Numbers
22: Fun with Triangular and Square Numbers

Use different proof techniques to explore square and triangular numbers. Square numbers are numbers such as 1, 4, 9, and 16 that are the squares of integers. Triangular numbers represent the total dots needed to form an equilateral triangle, such as 1, 3, 6, and 10.

30 min
Perfect Numbers and Mersenne Primes
23: Perfect Numbers and Mersenne Primes

Investigate the intriguing link between perfect numbers and Mersenne primes. A number is perfect if it equals the sum of all its divisors, excluding itself. Mersenne primes are prime numbers that are one less than a power of 2. Oddly, the known examples of both classes of numbers are 47.

31 min
Let's Wrap It Up-The Number e
24: Let's Wrap It Up-The Number e

Prove some properties of the celebrated number e, the base of the natural logarithm, which plays a crucial role in precalculus and calculus. Close with a challenging proof testing whether e is rational or irrational-just as you did with the square root of 2 in Lecture 7.

32 min
Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.


Dartmouth College


University of Florida

About Bruce H. Edwards

Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.

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