Given some two dimensional surface in R³, a sensible question to ask when getting acquainted with the beast is what the curvature is at any point (this would be sensible if you are approaching it from a differential geometer’s perspective; from a algebraist’s perspective you might as well ask what the colour 5 smells like). […]
Let’s start off easy. We know the shape of the earth. It is more or less spherical. Next, our galaxy, the Milky Way. Somewhat surprisingly to Edwin Hubble (1889-1953) who discovered it, it is barred spiral shaped, i.e. the shape of a disk with spiral arms. What about the observable universe? Is it spherical? It certainly seems to be, as it is expanding outwards and light […]
This is the second part of my piece on Raymond Louis Wilder (1896–1982) and his philosophical, historical and anthropological account of mathematics. I suggest that the reader refer back to the introduction to my ‘Raymond L. Wilder’s Anthropology of Mathematics: Platonism and Applied Mathematics’for Wilder’s biographical details. Introduction R.L. Wilder called for mathematics to be analysed […]
Numerical Analysis Deriving the Quadratic Equation From the Roots Up Solving Quadratics using gradients at the roots Numerical Analysis Sextic Polynomials — A Practical Approach At a young age a builder taught me how to mark out a right angle using the 3,4,5 triangle rule with a tape measure. That math practicality has stuck with […]
Fermat’s theorem on sums of two squares had famously been proven in just one single sentence. Can prime numbers be written as the sum of two squares? For 13 this is possible (2²+3²), but for example, 3 cannot be written in this way. Pierre de Fermat came up with a theorem on this topic, for which Don Zagier, an […]
In my last article on special relativity, I explained how and why our intuitive understanding of space and time needed to be modified to account for developments in electromagnetic theory that took place in the 19th century, in particular the results of the Michelson-Morley experiment and the non-covariance of the electromagnetic wave equation under the […]
Particle Physics The Discovery of Spontaneous Radioactivity Sweety, let me see what you got inside. Geometry Horrocks’ Measurements of How Far Away The Sun Is Understanding the sun-to-planets absolute distances was a long-run investigation proceeding with new scientific-tools and new laws. As a police case requires time and symmetry of action that concatenates all threads […]
The equation shown above is one of the most famous equations in physics which changed the world forever. It is the mathematical expression of the law of universal gravitation. Here, F is the gravitational force of attraction between two bodies, M₁ and M₂ are the masses of those two bodies, r is the distance between them, and G is the gravitational constant. There are two […]
Suppose that x and y are non-negative real numbers, not necessarily distinct. The famous arithmetic-geometric mean inequality says that: With equality if and only if x=y. This generalizes to the case of n non-negative numbers: Again with equality if and only if all of the numbers are equal. The quantity on the left-hand side of this inequality is the average, also called […]
Mathematics How do you like them apples? 31 Jul 2019 — 10 min read Photo: © 1997 Miramax Pictures The purpose of this article is to narrate you through solutions to the two math problems solved by the fictional character Will in the 1997 Academy Award-winning movie Good Will Hunting. The narration is based largely on the excellent […]
Radicals Radically Utilitarian Developing an Intuition for Radicals Economics Is Economics a Science? A Debate on the Scientific or Ideological Nature of Economics Mathematics Proving Arrow’s Impossibility Theorem Let’s prove the result behind why group decisions, from democracy to take-away choice, are always flawed. Game Theory The Nash Equilibrium, explained A strategy profile is a […]
The previous article discussed the trailing digits of the powers of two. Now, this article examines the leading digits of the powers of two and the interesting law they follow. But first, here’s a question: Taking any power of 2, what are the chances that the leading digit (the leftmost digit) is 1? How about 2? 3? […]
By definition, a scientific theory according to Cambridge English dictionary, is a formal statement of the rules on which a subject of study is based or of ideas that are suggested to explain a fact or event or, more generally an opinion or an explanation. Formulating a scientific theory requires an imaginative power to abstract concepts, […]
Larmor had an intense, almost mystical devotion to the principle of least action…To [him] it was the ultimate natural principle — the mainspring of the Universe.~Arthur Eddington, quoted in Zangwill (2013)¹. Two beads with masses M₁ and M₂ slide without friction on a ring of radius R. The masses are connected over the minor arc A […]