Blogs

1. e Unleashed

   This blog follows on from the previous blog The Exponential and Logarithmic Functions. We begin with a brief review of the life of Euler both as a human being and as a mathematician. We look at the complex exponentials, the hyperbolic functions, the catenary, and the linear and logarithmic spirals. We conclude with the recognition that the complex exponentials may be viewed as vectors undergoing linear transformations when they are differentiated or integrated. There is a third blog A Tetrad of Captivating Problems. It is meant to be read in conjunction with these two blogs on e.

2. The Exponential and Logarithmic Functions

   The number \(e\) is associated with logarithms, exponential growth, exponential decay, compound interest, the differential and integral calculus, the circular and hyperbolic functions, probability, queueing and reliability theories, the Fourier transform, and many other areas of mathematics. This linkage, across sub-disciplines, was not known initially, but only recognized gradually as ‘things fell into place’ later on.

3. The Pi of Archimedes

   “This blog—and its companion, The Wonder That Is Pi—began life in 2004, as part of a series of lectures I delivered to some very bright first-year engineering students at an Australian university.

   The number π (pronounced ‘pie’) has been recognized from time immemorial because its physical significance can be grasped easily: it is the ratio of the circumference of a circle to its diameter. But who would have thought that such an innocent ratio would exercise such endless fascination because of the complexities it enfolds?

   Not surprisingly, some high school students I met recently wanted to know more about π and how it got its unusual value of ²²⁄₇. Accordingly, I have substantially recast and refreshed my original presentation to better accord with the form and substance of a blog. The online references have also been updated to keep up with a rapidly changing Web.

   My original intention was to write a single blog on π. But because I did not want it to become yet another overly long slog, I have decided to divide the material into two parts.”

4. The Wonder That Is Pi

   This is a sequel to the blog “The Pi of Archimedes”. We look at π as a number rather than the ratio of two lengths, and try to unravel how and why it is ubiquitous in mathematics.