Two veteran math educators demonstrate how some "magnificent mistakes" had profound consequences for our understanding of mathematics' key concepts.
In the nineteenth century, English mathematician William Shanks spent fifteen years calculating the value of pi, setting a record for the number of decimal places. Later, his calculation was reproduced using large wooden numerals to decorate the cupola of a hall in the Palais de la Découverte in Paris. However, in 1946, with the aid of a mechanical desk calculator that ran for seventy hours, it was discovered that there was a mistake in the 528th decimal place. Today, supercomputers have determined the value of pi to trillions of decimal places.
This is just one of the amusing and intriguing stories about mistakes in mathematics in this layperson's guide to mathematical principles. In another example, the authors show that when we "prove" that every triangle is isosceles, we are violating a concept not even known to Euclid - that of "betweenness." And if we disregard the time-honored Pythagorean theorem, this is a misuse of the concept of infinity. Even using correct procedures can sometimes lead to absurd - but enlightening - results.
Requiring no more than high-school-level math competency, this playful excursion through the nuances of math will give you a better grasp of this fundamental, all-important science.

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, New York. Previously, he had the same positions at the City College of the City University of New York for forty years. He has published over fifty-five books in the area of mathematics and mathematics education, including The Fabulous Fibonacci Numbers (with Ingmar Lehmann).

Ingmar Lehmann is retired from the mathematics faculty at Humboldt University in Berlin. For many years he led the Berlin Mathematics Student Society for gifted secondary-school students, with which he is still closely engaged today. He is the coauthor with Alfred S. Posamentier of The Secrets of Triangles, The Glorious Golden Ratio, and three other books.