Mathematical Vignettes
Read more vignettes here for free. Now available on Amazon.com. Mathematicians Don't Work with Numbers!Upon hearing that I am a mathematician people often say to me, "You must be good with numbers." There is really no response to this, for how do I explain that numbers are not the subject of modern mathematics? Sometimes mathematicians do indeed work with numbers but in such cases the numbers themselves are never the point. Mark Kac was a mathematician recruited to work on the Manhattan Project, the secret WWII project that developed the atom bomb. Everyone working on the Manhattan Project had to reduce their theoretical results to numbers, for otherwise a machinist would not know what to make nor how big to make it. No general methods would suffice here, numbers were paramount. Kac captured this sense of mathematicians getting their hands dirty, figuratively speaking, by humorously commenting that he was reduced to working with numbers and, even worse, some of them actually had decimal points! The Manhattan Project was an important special case and no one working on it chose to do anything else, but that is not what professional mathematicians otherwise do. The next time (the first time?) you meet a mathematician say something different like, "What branch of mathematics do you work in?" Any mathematician worth his salt will answer by saying, "I work in _____, which has applications to _____." Yours will be a much better opening statement than if you say what comes across as, "Golly gee whiz, you must be good with numbers!" In the meantime, this book is my answer to the question, what do mathematicians work with? It consists of mathematical "vignettes", most being an elementary and brief description of one part of mathematics. Some vignettes use numbers but you will see they are incidental to the general principles involved. 
The First Seven Days
Limits and Continuity: Bridging the Gap between Algebra and Calculus
Now available on Amazon.com. Calculus is often called Infinitesimal Calculus, even when it is developed using limits and there are no infinitesimals. The historical confusion between the two formulations is finally untangled.
These seven chapters can be covered in seven lectures in a formal course or in seven sessions for selfinstruction.
The opening chapter "All the Preliminaries" lays the groundwork for the rest of the book.
The next chapters. "What is the Problem?" and "Developing a New Intuition", make plain the mathematical obstacles that caused a two century delay between the invention of calculus and the first rigorous formulation of the subject.
The succeeding chapters, "Limits: A First Attempt at Rigor" and "Limits: Rigor That Works" show why the more obvious definition does not work and why the modern definition solves that problem.
The final chapters, "Continuity: The Key to Everything" and "Derivatives: Putting It All Together", launch the student into calculus with a solid understanding of the limitbased formulation and how it differs from the infinitesimalbased formulation.
