Prove It: The Art of Mathematical Argument

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.

ALMA MATER

Dartmouth College

INSTITUTION

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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