Blogs

1. Two Rupees

   I was driving through the grey haze conjured up by soft rain and bleak sky when my gaze chanced upon a human form crouched in the mud by the road. I looked into my rear-view mirror and discovered that it was a hapless man struggling to rise and stumbling and falling, to rise and fall again, perhaps defeated by the slushy mud, or by fatigue, or by cold. Thrice I saw him struggle to stand and thrice fall, drenched in the rain. The pantomime being enacted had all the makings of a Chaplinesque comedy on screen, except that this was not reel life but real life: so comedy turned to tragedy.

2. On Binet’s Formula

   The formula of Binet for the Fibonacci sequence for the n^th Fibonacci number presents an excellent example to showcase the same mathematical result, arrived at from quite different approaches and perspectives, demonstrating the self-consistency of mathematics.

3. Demystifying Fractional Powers

   It is incredible how a simple question about the square root symbol and an exploration of fractional exponentials can lead us through a spellbinding journey of discovery, and ultimately open the vistas of advanced mathematics. Who would have guessed that the cube root of \(-8\) would have one real and two complex roots? Or that \(z^6 - 1\) would have roots that are multiples of \(\omega\) and \(\omega^5\)? Mathematics can be engrossing and endlessly fascinating as long as we are bold and patient enough to engage with it. The rewards are enormous and often totally unexpected.

4. Gnuplot for the Plotless

   In the recent past, however painful it was, I had adopted a solution using TikZ-PDF, or Matplotlib, or Typst, without giving gnuplot any thought. Now I would venture first with gnuplot. Its syntax is uncluttered and unentangled. It may be easily tested interactively. Changes may be made and checked rapidly, to allow for a faster solution, without compromising on the quality of the output. Yes, gnuplot has come a long way. It is still recognizable. Just as powerful as before, if not more. But it has far more sheen than when I last met it. Power and Beauty have met at last!

5. 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.

6. A Tetrad of Captivating Problems

   This blog is the sandwich filling between two blog-slices: The Exponential and Logarithmic Functions and e Unleashed. It consists of a tetrad of captivating problems that are related to exponents, which assumed centre stage after Euler showcased the number e and explored its facets in the eighteenth century.

7. 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.

8. 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.”

9. 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.

10. 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.