Prime Numbers Primes and The Powers of Two Relating the powers of two with primes and perfect numbers Number Theory The Fibonacci Sequence and The Powers of Two On two of the most popular and well-known patterns by humans and nature Analysis Why is 1 + 2 + 4 + 8 + … = -1? […]
So far, Earth and Moon’s sizes are not puzzles anymore for us. They are quite well known since Erathosteneles derived the first and Aristarchus the second. But it is not like no mysteries out there left. There are dozens still. Where the next, natural step leads to derive the distance between the two of them. Aristarchus and Hipparchus […]
The secretary problem is a famous riddle on the topic of decision making. It is about finding the best strategy when choosing between a sequence of alternatives of which you do not know the best one. The statement of the secretary problem goes as follows: You are the HR manager of a company and need to hire […]
To satisfy your curiosity and save you from the mathematical jargon, the simple explanation is just: x = 1 + 2 + 4 + 8 + … x = 1+ (2 + 4 + 8 + …)x = 1+ 2(1 + 2+ 4 + 8…)x = 1+ 2x x = -1 This is almost the exact […]
There are many counter-intuitive, bizzare theorems in mathematics. One of my favourites has to be the Banach-Tarski Paradox. A result due to Stefen Banach and Alfred Tarski in 1924, the paradox states that given a ball, there is a way you can cut the ball into a finite number of pieces and rearrange these pieces to form two balls that […]
Probability Theory Why Probability Theory is Hard It’s not because you’re stupid or weren’t concentrating in school Probability Theory Better-Than-Random Decisions in Uninformed Settings Two envelopes with different amounts of money in them. Choose the better one with a higher chance than fifty-fifty! Probability Theory Transforming Scores Into Probability How to show what subjective scores […]
Why hasn’t intuition received as much attention as logic for the twenty five centuries since Aristotle? Intuition is elusive, hard to define and quantify, and sometimes deceiving. As a matter of fact, there are even different kinds of intuition. Intuition is the beacon shining over the darkness according to Aristotle; however, it also appears like […]
Relativity Deriving Einstein’s Gravity Equations From Thermodynamics Is Gravity Just an Average of the Behavior of Unknown “Atoms” of Spacetime? Quantum Electrodynamics How Richard Feynman Reinvented Quantum Theory The Space-Time Approach to Quantum Mechanics Quantum Physics Quantum Tunneling, Bubble Universes and the End of the World The Catastrophic Consequences of Living in a Universe with […]
Physics can be easily considered the most fundamental of the sciences: it can describe almost any phenomena Nature gives us. Gravity. Yes. Why does my coffee cool? Yes. Do atoms exist? Yes. But internally, within the vast field there has been one lingering problem: two of the best theories physics has to offer are incompatible […]
Dynamic Programming When computer science and dynamic programming meets math This is the code heavy addendum to the more mathsy article I wrote on the Erdos-Szekeres Theorem. However, this is also self-contained, and is focused on finding monotone sequences rather than proving things about their size. Our task is to write a program to find the longest […]
Everybody knows the angle of a straight line is 180°(π.) Yet, there would be no point in writing an article about it if that would be all. There must be something more, and, indeed, there is. Because every angle involves trigonometry within. They are like twins. And supplementary ones show quite interesting properties. Proving the […]
Let’s discover the ingenious, yet simple, solution to Euler’s Log-Sine integral First we give a label to the value of our integral, I. Our approach will be to prove that I satisfies the equation I = 2I. This of course implies that I = 0 after subtracting I from each side. To do this, we will need to find a […]
In the recent “ask me anything” streaming, Noam Chomsky pointed to an interesting problem. Darwin and Wallace were very puzzled and debated the fact that all humans have arithmetical capacity. [That] it’s just a part of our nature to understand that there are infinitely many natural numbers; that when you add them it works this way and […]
Algorithm Engineers & Algorithms Why engineers should know their algorithms Proof Theory How Mathematical Proofs can Help Unlock the Secrets of the Brain Computational neuroscience, broadly defined, is the mathematical and physical modeling of neural processes at a chosen scale, from molecular and cellular to systems, for the purpose of understanding how the brain represents […]