Algorithm Engineers & Algorithms Why engineers should know their algorithms Linear Algebra A Guide to Inverting Matrices for Analysts One of the oldest and most thoroughly studied problems in both linear algebra and numerical analysis is finding solutions for Ax = f. In a first linear algebra class, one might learn about how to identify […]
The knowledge of which geometry aims is the knowledge of the eternal. — Plato Inphysics, most of the time we need to apply integration. But there are some integrations which can’t be solved using normally known methods. For them we have some special tricks such as Feynman’s Integral Trick, the Fourier Transform, Laplace’s Transform, the Residue […]
The following essay is about what the theoretical physicist and mathematician Roger Penrose has said about “seeing” mathematical truths. Penrose’s overall Platonic position is also discussed. Indeed the question as to whether Penrose actually has an overall Platonic position is also asked. But before all that, let’s place Penrose’s position in some kind of context. Does Penrose Go Beyond […]
Prime Numbers Mersenne Primes Number Mysteries and the Hunt for Large Primes Prime Numbers The Goldbach Conjecture An Equivalent Formulation of an Unsolved Mystery Number Theory The Euler Totient Function A Probabilistic Approach Field Theory A Finite Number Of “Numbers” What exactly is arithmetic on a field of numbers which are merely elements in some […]
Logic Formal Logic vs. Material Logic? In a purely logical argument, even if the premises aren’t in any way (semantically) connected to the conclusion, the argument may still be both valid and sound. Professor Edwin D. Mares displays what he sees as a problem with purely formal logic when he offers us the following example Logic Lewis […]
Group theory is the language of many of the mathematical disciplines. An indispensable tool in understanding the underlying nature of nature. A theory that holds the secrets of the fundamental particles and forces of the Universe itself! We use it to understand shapes in higher dimensions, the proof of insolvability of higher degree polynomials, and the […]
I have always been fascinated with “origins”: the origin of life on Earth, the origin of human beings, the origin of the Universe, and of course to fundamental particles. I reserve a sort of special place to a mathematical entity that some people regard as fundamental: prime numbers. A prime number is a whole number greater than 1 whose only factors are 1 and itself. […]
Graph Theory From GPS navigation to network-layer link-state routing, Dijkstra’s Algorithm powers some of the most taken-for-granted modern services… 09 Nov 2019 — 23 min read From GPS navigation to network-layer link-state routing, Dijkstra’s Algorithm powers some of the most taken-for-granted modern services. Utilizing some basic data structures, let’s get an understanding of what it […]
Proof Theory A Simple Proof of the Quadratic Formula Using Symmetry You Don’t Need to Remember the Formula 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 […]
Entropy is a bit of a buzzword in modern science. Usually it is used as a synonym for “disorder,” but it is so much more interesting than that. The concept itself has a long and interesting history. In order to fully understand what entropy is, we really need to see where it came from. The very first […]
“The infinite! No other question has ever moved so profoundly the spirit of man.” — David Hilbert Hilbert’s hotel paradox is a beautiful demonstration of the countably infinite. A coach full of the natural numbers, stretching backwards forever, arrives at a hotel with endless rooms and a maths-wizz receptionist. After allocating every guest a room the […]
In 1956 a few writings of Wittgenstein that he didn’t publish in his lifetime were revealed to the public. These writings were gathered in the book Remarks on the Foundations of Mathematics (1956). There, we can see that Wittgenstein had some discontentment with the way philosophers, logicians, and mathematicians were thinking about paradoxes, and he even registered […]
The Fibonacci Sequence is one of the cornerstones of the math world. Fibonacci initially came up with the sequence in order to model the population of rabbits. In reality, rabbits do not breed this way, but Fibonacci still struck gold. This sequence has so many beautiful mathematical features it has its very own journal dedicated to it […]
The history of mathematics has profound roots. We can seek first traces of using algebra in works of Babylonian mathematicians tracing back to almost 4000 years ago. With firm confidence, we even can say that they had enough knowledge for solving classic quadratic equations. Babylonian clay tablet with an approximation of square root of 2 from Wikimedia Since then, […]