Articles in the Mathematics category

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.