An overwhelmingly large portion of our modeling of the universe — the enterprise which we call physics — is done by posing and solving differential equations. Somewhat strangely, equations which have been developed by certain physicists to describe very specific phenomena regularly turn out to successfully model completely different phenomena, often in entirely different fields […]
As π goes, so goes a circle π is ubiquitous both in mathematics and physics. Its appearance in some contexts are intuitive to grasp. Others require intellectual contortion, a mind-bending that defies all intuition. Wherever π appears, a circle is always hiding, to be discovered irrespective of one’s intuition! Let us get started slicing π, […]
“What on earth are you talking about? There’s no way that’s true!” — My mom This is what my mom said to me when I told her about this little mathematical anomaly. And it is just that, an anomaly. After all, it defies basic logic. How could adding positive numbers equal not only a negative, but […]
Keep drawing integers from a random number generator (or a customized lottery machine) until you get a number that is smaller than the previous pick. If your run ends after three or fewer draws (including the last picked smaller number) then you pay $10. Otherwise, you receive $20. Will you accept this bet? Think about […]
“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” Ramanujan replied, “it is a very interesting number; it is the […]
I’ve been putting this off for too long. In several of my recent articles, I have taken advantage of the following result without much explanation: Derivatives are evaluated at x. This is called the Taylor series expansion of f(x) about x. If ε is a very small number, then Taylor’s Theorem says that the following approximation […]
Mathematics An introduction to Georg Cantor’s transfinite paradise 17 Dec 2018 — 30 min read Georg Cantor (left) and his remarkable 1874 publication “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” in Journal für die Reine und Angewandte Mathematik (1874). An introduction to Georg Cantor and his transfinite paradise This story is also available on […]
“Nature Always Wears The Colors Of The Spirit” — Ralph Waldo Emerson Light, color & eyes — invisible energy waves, the core of beauty, & the portal to souls. Few topics have captivated the greatest minds like our collaborative search to more deeply understand how & what we see. Just in a line of polymaths alone, we […]
Suppose that we have n fair coins, and we decide to toss them, one by one, and count the number X of HEADS that we see. What can we say about this number X? This is the main question we will try to answer here. Photo by Josh Appel on Unsplash To start, we index the coins from 1 to n and define the […]
Let’s prove one of the most important theorems in elementary analysis! The only requirement is an elementary understanding of derivatives. Statement: Above we see an informal statement.The straight line connecting f(a) with f(b) is equal to the gradient line at one the points between a and b. However, in mathematics it is important to give […]
The Traveling Salesman Problem is one of the great classic problems in mathematics. It’s easy to state, but trying to solve it is enormously hard (more on that later). The papers written on it range from straight-up computer science to theoretical combinatorial optimization. This article will describe a crazy twist on it: what if the […]
Henri Poincaré was a very distinguished astronomer, mathematician, physicist, and philosopher. And, because of this variety of areas to which he contributed he is known as the “last universalist”, and treated as a “polymath”. Some take him as the father of relativity theory. According to Adlam, before Einstein, Poincaré “had the correct equations for the Lorentz […]
One can occasionally feel sorry for the Flat Earthers. It is quite possible to live an entire human life without encountering any first-hand evidence of the curvature of the Earth. As such, it is easy to see why some might believe such an outlandish theory in a society where we are increasingly being told to […]
However much it has served, there is no clear consensus today on exactly where Newtonian gravitational acceleration originates, let alone what it is, except for what it does. In deed, the Oxford dictionary of physics defines gravity as “the attractive force by which bodies are drawn towards the centre of any celestial body, such as […]