Math Education Concepts

One of My Favorite Quotes

A portrait of Pierre de Fermat, French lawyer and mathematician. (Photo credit: Wikipedia)

“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.”Pierre de Fermat

One of Fermat’s habits was to write proofs in the margins of the books he read.  He was most known for writing in his copy of Arithmetica.  One day he came across the book and worked through the many problems published therein.  The author of Arithmetica, Diophantus of Alexandria, shared his proofs and solutions in his text.  The story is told that at some point, about 1637, Fermat decided to expand the Pythagorean Theorem to similar equations with exponents greater than 2.  He concluded that there were no whole number solutions to the equation xn + yn = zn for values of n greater than 2.  However, in the margin of Book II of Arithmetica all he wrote was “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” (Singh, 62)  To his successors’ chagrin, Fermat did not write the proof to this, then, conjecture.  At least none that anyone have found and published.  Fermat died in 1665.  In 1670, Fermat’s son, Clément-Samuel, published Fermat’s discoveries, theorems, notes, and commentaries in Diophantus’ Arithmetica Containing Observations by P. de Fermat.  In the years to follow, mathematicians would unsuccessfully attempt to prove what was known as Fermat’s Last Theorem.  It was given this name because it was the only theorem that Fermat did not provide a proof for.

The quote rings out for me because it leaves an element of wonder, mystique, and brilliance, all at once.  Did Fermat have a proof?  Did he write out the proof?  Did he have it figured out in his mind?  Was this his last puzzle for the world?

WHAT IS YOUR FAVORITE QUOTE? WHY?

REFERENCE: Singh, Simon. New York: Anchor Books, 1997

FERMAT’S ENIGMA

I recently read Fermat’s Enigma, written by Simon Singh.  It tells the story of the quest to prove Fermat’s Last Theorem, the last of Fermat’s conjectures to be proven.  Singh covers the story from the inspiration of the conjecture (Pythagoras’ time), through the time Fermat wrote the conjecture in the margins of Arithmetica, through the proof of the conjecture by Andrew Wiles.  I read this book because I wanted to know more about Fermat’s Last Theorem (FLT), but as I read along, I found myself critiquing the book.

My initial impression of the book, prior to reading it, was that it would be filled with a non-mathematicians’ attempt to explain mathematical concepts.  Then, I re-read the blurb about Singh in the beginning of the book and was reminded that he is a Physicist and, most likely, had a mathematical background.  I’m not sure how a non-mathematician would feel about reading Fermat’s Enigma, but I don’t think they’ll get the same enjoyment as a mathematician would.

One theme throughout Fermat’s Enigma is Singh’s peep into the world of mathematics and mathematicians.  The book tells the story of one man’s enjoyment of puzzles and riddles, hundreds of attempts to prove FLT, dozens of brilliant minds collaborating, and one man’s childhood dream come to fruition.  In telling the story, Singh covers a wide range of mathematics history.  This was an unexpected treat for me.  In addition to learning about the FLT story, I also got a brief account of a segment of mathematics history.

While covering the FLT story, Singh, along with John Lynch, a television editor, spent months talking with Andrew Wiles, conducting interviews with other mathematicians, and researching the history of FLT to give as accurate an account as they could about Fermat’s Last Theorem.  The intent of Lynch, with the help of Singh, was to create a television documentary of the story.  I haven’t seen the documentary, but I am satisfied with having read the book.

As I read the “Epilogue”, I felt sad.  Maybe I wasn’t ready to stop reading Fermat’s Enigma, maybe I was empathizing with Wiles on his feelings of finally proving Fermat’s Last Theorem.  He said “There’s no other problem that will mean the same to me… Having solved this problem there’s certainly a sense of loss.” (p. 285)  I felt that my brief, less intense, obsession with reading this book was finally concluding.  I was so engrossed in it that I had to set my alarm so I wouldn’t miss appointments.  When I’d finally finished the book, I felt that I needed something else to capture my attention the way this story did.  At the same time, I felt a sense of relief.  It was finished.

Fermat’s Enigma is fun and easy to read.  It explores some common mathematical concepts as well as a fair share of mathematics history.  Some other benefits of the book are the proofs and examples given in the “Appendixes,” as well as the list of dozens of references in the “Suggestions for Further Reading” section.  As a math educator, I think this book is a must read.  It is an informative and resourceful account of a momentous event in recent mathematics history.  However, if a reader wants a concise, less dramatic, overview of the story of Fermat’s Last Theorem, this may not be the best choice.