Roger Penrose is not only a mathematical physicist: he’s also a pure mathematician. So it’s not a surprise that Penrose expresses the deep relation between mathematics and the world (or nature) in the following way: “[T]he more deeply we probe the fundamentals of physical behaviour, the more that it is very precisely controlled by mathematics.” What’s more: “[T]he mathematics that we […]
If you went to college, it is a fair bet that you took some sort of calculus class. If you went for engineering or science, you probably took a LOT of calculus classes. Unfortunately, a lot of math teachers like to show you how smart they are and they make simple ideas pretty hard to […]
In this article, we will discuss a beautiful combinatorial result that relates maximum matchings to minimum vertex covers in bipartite graphs, and was proved in 1931 by Dénes Kőnig, and independently by Jenő Egerváry, whose proof also applies to weighted graphs. The result, formally stated below, is now known as Kőnig’s theorem or Kőnig-Egerváry theorem. The Kőnig-Egerváry theorem […]
Analysis “Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability” — Franzén… 06 Jul 2020 — 10 min read Georg Cantor (1845-1918)’s correspondence with mathematician Richard Dedekind (1831-1916) in the years 1873-74 has been narrated in a previous newsletter. Their letters detail the process by which Cantor arrived at the […]
Between 1609 and 1619, Johannes Kepler used data collected by Tycho Brahe to deduce the laws that determine the motion of the planets around the sun: Every planet moves in an elliptical orbit with the Sun at one of the foci. The planet moves in its orbit, a line drawn from the Sun to the […]
Let’s understand the simple yet beautiful reason why mathematicians say 0 point 9 recurring equals 1. (the image above is actually an infographic of transfinite induction. But hey, it looks cool. I’ll do an explainer on transfinite induction another time!) We are familiar with the real number line. But how do mathematicians view it? Mathematicians […]
A vacuum is a space with as little energy in it as possible. However, a vacuum is not entirely empty. It contains quantum fields. Quantum fields are obtained after the quantization of classical fields, which are functions of spacetime coordinates (such as, for example, the electromagnetic field). Mathematically quantum fields are operator-valued functions of space and time. Figure 1: The […]
Cryptography Could We Break RSA Encryption Without A Quantum Computer? We discuss a range of integer factorization algorithms that can run on classical computers and explore their future in the face of Shor’s algorithm. Fermat Pierre de Fermat is Much More Than His Little and Last Theorem A journey through Fermat’s pseudoprimes and Fermat numbers […]
Pi Day is coming up soon! And there are many ways to calculate or estimate our all-time favorite number π which is approximately 3.14159…. Let’s have a look at these methods and let’s discuss three ways to estimate π using Monte-Carlo Simulations! What is Pi? Pi is the famous circle number approximately given by 3.14159… […]
The ancient Romans had a story about the foundation of the ancient city of Carthage. According to Virgil’s epic poem the Aeneid, Dido, the queen of the Phoenician city of Tyre, was forced to flee with her life when her brother usurped the throne and murdered her husband, the king. She eventually arrived in Libya, where […]
Every once in a while, we will see some breaking news about Artificial Intelligence (AI), the discipline of automating how we think. Not very long ago, there was news stating “[there is] no need [for humans] to prove mathematical theorems anymore!”, referring to a 2019 Google paper (Bansal et al., 2019) about using AI to prove mathematical […]
Remember the harmonic series? More often than not, it serves as one’s first encounter with a series wherein individual terms diminish successively yet the series diverges to infinity. The below quote nicely sums the up notoriousness of the harmonic series: Today I said to my calculus students, “I know, you’re looking at this series and […]
If your math teacher were to start a lesson from asking about your associations of the central angle theorem, would you answer by talking about the Thales theorem? If yes, what got back would be, “Yes, but only in one-third.” That is because an arc on which our inscribed angle could be subtended may have […]
When I was a small child I got lost at a store with lots of aisles. I remember being very scared that my mom would never find me. I walked from aisle to aisle on the verge of tears hoping to run into her. Eventually, to my great fortune, I did. However, it occurs to […]