Number Theory An algorithm from ancient Greece Euclid of Alexandria [Source] The task is straightforward. Given two positive integer numbers x and y, we want to compute their greatest common divisor, i.e., the largest number that divides both x and y without a remainder. For example, suppose that x = 30 and y = 12. It is easy to factor these two numbers and write x = […]
When we look at the operation of integrating a function, we find that it resembles the operation of addition in the sense that it’s kind of an infinite sum of infinitely small numbers. How much are these operations alike? and in particular, can we find a relation between them? This article answers this and gives […]
Biography Interview with Mathematician Ken Ono I recently had the privilege of interviewing Ken Ono, one of the most brilliant Mathematicians in the world. Not only has he made huge… Probability Theory How to Win at Roulette: Intro to Probabilities and Expected Values Why you still shouldn’t play. The Martingale Strategy Mathematics How to Count […]
This is the fourth entry in my series on partial differential equations. In the first three articles, we talked about the one-dimensional heat equation, where it comes from, and how to solve it in a few simple circumstances. Now we’ll cover the second of the three “basic” partial differential equations that are important for physics, […]
Vector Spaces In this article we introduce the reproducing kernel Hilbert space. Photo by Kaleidico on Unsplash These notes give an introduction to the theory of reproducing kernel Hilbert spaces. We take a somewhat algebraic view of some of his results. In later chapters we consider generalizations of this theory to the vector-valued setting. We […]
This proof is accessible to ANYONE — no mathematics knowledge required! (perfect for if you are a bit bored and in isolation, like me right now!) The Königsberg bridge problem shows the beauty of mathematics to transform the impossible to the obvious. It also gives an insight into the mind of the genius Leonhard Euler. […]
If I were to ask you “define Pi for me” you would probably recite It’s the ratio of a circle’s circumference to its diameter And you will definitely be right, the number Pi is strongly tied to circles and trigonometric functions so you might expect that all or at least most of Pi’s appearances in […]
Part 1: Derivation and examples A partial differential equation is an equation that relates a function of more than one variable to its partial derivatives. To introduce PDEs, we’re going to solve a simple problem: modelling the temperature in a thin metal bar as a function of position and time. Along the way, we will […]
In this article, we discuss a proof of the celebrated Turán’s theorem that uses the Probabilistic Method and the famous Cauchy-Schwarz inequality. Turán’s theorem is named after Pál Turán, one of the many brilliant combinatorialists and number theorists that Hungary has produced. The setting of the theorem is quite simple. Suppose that you have N vertices, and the task is to start adding edges in […]
Puzzle Forever Lost Within Reach The Lost Twins Experiment Chess Chess and Mathematics At first glance, these two disciplines may seem unrelated. However, mathematics and chess have a large amount of unexpected connections. Puzzle Heron’s Shortest Path Problem Which path minimizes the distance needed to get water and then transport it to the green village? […]
Ptolemy was an ancient astronomer, geographer, and mathematician who lived from (c. AD 100 — c. 170). He is most famous for proposing the model of the “Ptolemaic system”, where the Earth was considered the center of the universe, and the stars revolve around it. This was the generally accepted view for many centuries, until Copernicus (19 February […]
When two lines are perpendicular (or orthogonal) with each other, it means they form a 90° angle when they intersect. If you were taught about perpendicular lines in Algebra or for the SAT, you were probably forced to memorize something like “the slope of the perpendicular line is the negative reciprocal slope of the original […]
It was a big surprise discovering zero when I was really looking for prime numbers. Back in 1992, I had a simple computer at my disposal and assigned myself the task to find out what was going on with the prime numbers. It took some good detective work, but once that larger answer was known […]
Mathematics has a lot of hidden gems. Today I will try to give us a glimpse to one of the vast domain — modular arithmetics. In the last post, we showed that there are no circles at the origin with a finite number of rational points. We also said that no rational points are lying on a circle […]