Blogs

1. 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.

2. Zygon—The First-Year Specialist

   “Zygon and I started high school together, many decades ago,” recounted Seven. “He was restless spirit, always seeking, never finding. It was as if he had never defined a goal to start with but calibrated the result of his quest against an inner feeling that functioned as his barometer of success. Sometimes, he was satisfied for a while, and he kept at what he was doing. The moment either boredom, or dissatisfaction, or satiety, or some other feeling crept in, he would start his restless seeking again.”

3. Differential Equations

   A differential equation (DE) connects a function with its derivative(s). In calculus, the function 𝑦⁡(𝑥) is known, and its derivative, 𝑦′⁡(𝑥), needs to be found. In differential equations, the derivative, 𝑦′⁡(𝑥), is known, and the function, 𝑦⁡(𝑥), needs to be identified. As with indefinite integration in calculus, the solution will throw up one or more arbitrary constants. The values of these constants must be determined by the initial conditions provided in the problem in order to obtain a unique solution.

4. Expressions, Equations, and Formulae

   My dear friend, Solus “Sol” Simkin, casually asked me one summer day if I would write a blog demystifying the meanings and uses of four mathematical terms: expression, equation, formula, and differential equation. I thought that this was spoken in jest, and let his request lie in a dusty corner of my mind, as a memento to his humour.

5. 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.

6. Same Action: Four Castes

   This blog was originally penned on 24 October 2008. It is one among many dozen blogs that have not been previously released publicly on the Web. I will be gradually refreshing and releasing all those unpublished blogs on this website. They are generally short and tightly focused pieces of writing, readable in a few minutes. Where possible, I have retained the original contextual immediacy and topical relevance. I have used the IAST transliteration scheme for the Sanskrit terms; consult the link for the correct pronunciation. The upshot of this blog is that caste is spiritual in nature, and that all individuals may exhibit all four castes at different times—according to attitude to action—regardless of birth or occupation.

7. 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.

8. From Calculus to Analysis: Limits

   At high school you were taught how to integrate and differentiate. You were exposed to all sorts of tricks and special techniques—such as the chain rule for differentiation, and integration by parts for integration. If you revelled in mastering and applying such techniques, you might find that what succeeds high school calculus, is a horse of an entirely different colour, called analysis, at university.

9. How Are Numbers Built?

   Decimals and continued fractions are two different ways to represent numbers succinctly, with complementary strengths.

10. Doron’s Mathematical Amnesia

    Poor Doron has suffered a strange cognitive deficit. His language and speech skills have survived intact, but he has developed ‘selective mathematical amnesia’. He has forgotten much of high school mathematics—which for a person with a Master’s degree in engineering—is a rather tragic state of affairs. In fact, his knowledge of mathematics now resembles Emmental cheese—full of holes.

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