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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:
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:
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,
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!
University of Florida
Ph.D., Dartmouth College
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 Bogotá, 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.
Because of the highly visual nature of this course, it is available only in video formats. The viewer can see how the proofs were reasoned as the equations are built on screen and explained by the professor.
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