Coincidences, Chaos, and All That Math Jazz

I normally don't have time to become engrossed in new books but, yesterday, one of my students, J.B., presented me with this 265 page paperback whose title is the same as that of this post. It has already received recognition from the BookLovers Review.
The authors, Edward B. Burger and Michael Starbird are esteemed mathematicians who have each been recognized as distinguished teachers. After reading part of their book, I can understand why! The ability to communicate difficult concepts for all to understand is rare indeed.

I started reading it last night, worked my way through the mesmerizing explanation of public key cryptography which explains in the simplest terms how sensitive information can be transmitted in encoded form, then decoded. While the encoding method is completely revealed to the public, there's a key factor (figuratively and literally in the mathematical sense) in the decoding process that no person and no computer could determine in real time except for the person holding the key! If that sounds too cryptic (sorry, I couldn't resist that), read the chapter 'Secrets Held, Secrets Revealed.' The secret lies in the realm of Number Theory (congruences and residues) but the ideas are explained using a simple numerical example that all can verify on their calculators. Considering that a large security firm purchased an encryption company for $400,000,000 to own the rights to this method, I guess I can now tell my students there is another reason to study math beyond the stories of Bill Gates and Paul Allen!

The chapter on chaos (Understanding Uncertainty - Chaos Reigns) is brilliant and again brought to a level everyone can understand. The following quote about the source of chaos is profound and embodies the essence and spirit of this exceptional work:

"Even mathematics, complete and precise, is subject to the perils of tiny variations in initial conditions, which, when multiplied and magnified by the tyrant of repeated application, end by leading us far astray..."

I've only scratched the surface of this book. I want to personally thank my student for this special gift to me. She took the time to find the perfect way (for a math nerd like me) to express her gratitude for my writing a college recommendation that, she perceived, helped her gain admission into a prestigious university. Trust me, this exceptional young lady needed no help from me!