This is part one of a small series of articles meant to describe the most interesting results of complex analysis in an understandable way. Even if you have not had any course in complex function theory, you should be able to get some valuable understanding of this fantastic field. If you are studying mathematics or […]
Using Newton’s Approximation to find the roots of polynomials is commonly taught. However, this can involve repeated iterations to achieve the required level of accuracy. This post shows how to achieve high accuracy in a single application. The method exploits the amazing symmetry of cubic polynomials around their Inflection Point, (Xip,Yip), to find an optimal […]
Last time we followed Eristarchus’ steps of measuring the circumference of the Earth. Today, we will push forward in our investigation of how big the universe is. Because, again, we are hungry for knowledge. We know what the size of the Earth is. So let us go up there and see the one of the closest […]
Most physics textbooks are similar. They differ (little) for the sequence and for the choice of the mathematical details with which topics are presented. Most authors and teachers do not even ask themselves why they illustrate topics that way. Apparently, there is only one way to introduce something. I believe this is not true. At […]
“Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of F which can neither be proved nor disproved in F.” Gödel’s 1931 paper containing the proof of his first incompleteness theorem is difficult to read. It is 26 pages long, contains 46 […]
Suppose, as we all do on a Friday evening, you are looking at data of all the heights of people attending a university. You can plot a histogram of all the data and it might look something like this: You hear that there will be a new person arriving tomorrow and you want to find […]
These days all your devices are encrypted. Apple boasts about their secure systems, most companies have encrypted servers, even your own laptop probably has some level on encryption going on. However, all of these systems are subject to fault. Of course, this all depends on the type of encryption you’re using. While many agree that […]
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 […]