Articles in the Mathematics category

1. A tale of two measures: degrees and radians

   The transition from degrees to radians is often the most traumatic mathematical change that the student has to endure when moving from elementary to intermediate mathematics. The simplicity of 360° seems so much more welcoming than the equivalent of \(2\pi\) radians for the angle of a full circle. \(\pi\) is forbidding, because it is not the convenient fractional fiction \(\frac{22}{7}\), but rather a number which is both transcendental and irrational and therefore, somewhat “untidy”. Surely this tradeoff between simplicity and complexity must have been worth it, or it would not have been so ordained. Here we attempt to fathom the method in the madness.

2. Solving a Mathematics Olympiad problem

   During a casual tour of the Web, my attention was drawn to a problem that was stated so simply that it beckoned an attempt at a solution. It was purported to be from a Mathematical Olympiad, which raised its attractiveness index, as such problems are known to strenuously exercise the grey cells, while still retaining the charm of a sport. Only later did I find out that the problem I had written down had omitted an important constraint that made the problem all the more memorable. This is an account of my escapade into the land of mathematics in search of solution.

3. Eigenvalues and Eigenvectors—Why are they important?

   A university academic friend of mine recently remarked that it was not easy to motivate students to study eigenvalues and eigenvectors, let alone appreciate their importance: the subject itself was abstract, and the applications tended to be domain-specific and somewhat arcane.

4. Varieties of Multiplication

   I want to look at some of the varieties of multiplication that mathematicians have developed over time. It is a survey that will serve as a pinhole through which we can view how a single, simple mathematical idea has been expanded and elaborated into uses far beyond its historical moorings.

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. 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.

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. 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.