Prove It: The Art of Mathematical Argument

Course No. 1431
Professor Bruce H. Edwards, Ph.D.
University of Florida
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Course No. 1431
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Course Overview

Mathematical proof is the gold standard of knowledge. Once a mathematical statement has been proved with a rigorous argument, it counts as true throughout the universe and for all time. Imagine, then, the thrill of being able to prove something in mathematics. The experience is the closest you can get to glimpsing the abstract order behind all things.

Only by doing a proof can you reach the deep insights that mathematics offers—that tell you why something is true, not merely that it is true. Such insights are invaluable for getting a grasp of the key concepts in every branch of mathematics, from algebra to number theory, from geometry to calculus and beyond.

And by advancing from one proof to a related one, you begin to see how mathematics is a magnificent, self-consistent system with unexpected links between different ideas. Moreover, this system forms the foundation of fields such as physics, engineering, and computer science.

But you don’t have to imagine the exhilaration of constructing a proof. You can do it. You can prove it! Consider these proofs that are not only profound and elegant, but easily within reach of anyone with a background in high-school mathematics:

  • The square root of 2: Can the square root of 2 be expressed as a rational number—that is, as a fraction of two integers? The proof discovered by the ancient Greeks had dangerous consequences for one mathematician.
  • Gauss’s formula: What is the sum of the first 100 positive integers? As a child, the great mathematician Carl Friedrich Gauss discovered that the solution has a simple formula, which can be proved in several different ways.
  • Geometric series: Is the repeating decimal 0.99999… less than 1? Or does it equal 1? The proof surprises many people and provides a launching point into the analysis of infinite geometric series in calculus.
  • Countably infinite sets: An eye-opening proof shows that the set of all rational numbers is the same size as the set of all positive integers, even though there are infinitely many rational numbers between two consecutive integers.

Mathematicians marvel at the clarity that comes from completing a proof. It is as if a light has suddenly switched on in a dark room, bringing simplicity and understanding to what was formerly obscure and confusing. And since research mathematicians spend much of their time working on proofs, you can get a feel for what it’s like at their esoteric heights by putting pencil to paper and working out elementary proofs.

Prove It: The Art of Mathematical Argument initiates you into this thrilling discipline in 24 proof- and information-filled lectures suitable for everyone from high school students to the more math-savvy. The course is taught by award-winning Professor Bruce H. Edwards of the University of Florida. The author of many widely used textbooks, Professor Edwards has a knack for making mathematics as exciting to his audience as it obviously is to him.

In the course, Professor Edwards walks you through scores of proofs, from the simple to the subtle. The accompanying guidebook includes additional practice problems that help you gain confidence and mastery of a challenging, satisfying, and all-important mathematical skill.

Techniques and Tips

The modern concept of mathematical proof goes back 23 centuries to the Greek mathematician Euclid, who introduced the method of proving a conjecture by starting from axioms, or propositions regarded as self-evidently true. Once proved by logic, a conjecture is called a theorem. The beauty of Euclid’s system is that the same conjecture can often be proved in markedly different ways.

In Prove It: The Art of Mathematical Argument, Professor Edwards introduces you to the principles of logic to give you the tools to reason through a proof. Then he surveys a wide range of powerful proof techniques, including these:

  • Direct proof: Start with a hypothesis, do some math, then arrive at the conclusion. This is the most straightforward approach to a proof and is based on the simple logical relation, “P implies Q.”
  • Proof by contradiction: Assume that a mathematical proposition is false. If that leads to a contradiction, then it must be true. Using this technique, Euclid devised an elegant proof showing the true nature of the square root of 2.
  • Induction: Logical induction is used to prove that a given statement is true for all positive integers. The first step is to prove a “base case.” This case establishes that the next case is true, and the next, and the next, ad infinitum.
  • Visual proof: Sometimes geometric figures can be used to show that a mathematical conjecture must be true. One such “proof without words” is credited to James A. Garfield, who later became president of the United States.

A teacher with a knack for bringing abstract material down to earth, Professor Edwards has many practical tips to help sharpen your proof-writing skills. For example,

  • First things first: Before you try to prove a conjecture, stop and ask yourself if it makes sense. Do you believe what is being proposed?
  • Try some examples: Plug in numbers. You may see right away that the conjecture is true and that you’ll be able to prove it.
  • Know where you’re going: Keep your goal in mind as you work on a proof. Use scratch paper to jot down ideas. Often, you’ll see the way to a proof.
  • Don’t be daunted: When you’re studying a finished proof, remember that you don’t see the mathematician’s notes. The proof could be easier than it looks.

Tales of Proofs

Throughout the course, Professor Edwards tells stories behind famous proofs. For example, the Four Color theorem says that no more than four colors are needed to color the regions of a map so that no two adjacent areas have the same color. It’s simple to state, but attempts to prove the Four Color theorem were fruitless until 1976, when two mathematicians used a computer and a technique called enumeration of cases to solve the problem. You get a taste for what’s involved by working through several simpler proofs using this technique.

You also hear about celebrated paradoxes in which logic leads to baffling conclusions, such as Bertrand Russell’s paradox that shook the foundations of set theory. It involves a barber who cuts the hair of all the people who do not cut their own hair—in which case, who cuts the barber’s hair?

And often in Prove It: The Art of Mathematical Argument, you’ll come across unproven conjectures—deep problems that are so far unsolved, despite the efforts of generations of mathematicians. It just goes to show that there are unending adventures ahead in the thrilling quest to prove it!

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24 lectures
 |  Average 30 minutes each
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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? x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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? x
  • 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. x
  • 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. x
  • 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. x
  • 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. x
  • 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. x

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Your professor

Bruce H. Edwards

About Your Professor

Bruce H. Edwards, Ph.D.
University of Florida
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 Bogot·, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of...
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Prove It: The Art of Mathematical Argument is rated 4.7 out of 5 by 35.
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
Rated 5 out of 5 by from very full of information Have not had time to finish but am enjoying, as usual, the DVDs. Each teacher has a different perspective and covers something new as in this method of proving math.
Date published: 2017-12-02
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