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.
About Prof. Bruce H. Edwards
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Published January 1, 2012
by Great Courses Teaching Company.