Articles in the Mathematics category

1. The Fundamental Theorem of Algebra

   It once fell to my lot to teach Mathematics to first-year students at an Australian university. It is often said that you truly learn only when you start to teach. And so it was in my case.

   This blog is written for high school and first year students of mathematics whose primary interest is in the application of what they learn. The Fundamental Theorem of Algebra is usually glossed over with a passing reference, and without even a plausibility argument. Because of the theorem’s importance, I strike a balance here between being too rigorous and too lax, so that there is sufficient tension to sustain student interest, and sufficient detail to ensure that the proof is persuasive.

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. 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. 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.

6. 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.

7. 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.

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. The Two Most Important Numbers: Zero and One

    The unique properties of the numbers zero and one make them mathematically interesting and indispensable. In this slow-paced stroll though the ideas streaming out of these two numbers, we uncover well-known as well as relatively obscure facts about them. It is hoped that in the process we may discover how they cement together disparate areas of Mathematics.