Before the philosopher Bertrand Russell began really thinking about the nature of sets, sets were defined informally. Set theory before various axiomatic definitions is now referred to as naïve set theory. Russell’s paradox, discovered by Bertrand Russell in 1901 was also discovered a year earlier by Ernst Zermelo but never published. This paradox shows that […]

### Proof of the determinant of a product of square matrices using Leibniz’s equation

Let us consider the square, dimensional matix. It is possible to re-express the matrix in terms of a vector containing its rows. Where It is then possible to express the product of the matrix , with another dimensional square matrix as the following, by the definition of matrix multiplication. Each row of the matrix can […]

### February Roundup (Maths, Tech, Computing)

Hi everyone, I’m back with a couple of short summaries on topics that I’ve found significant and/or important in February. As before, links to articles that I used on each of the topics will be included at the bottom of the article in case you want to read more into the subjects. Telugu Symbol Crashes […]

### January Roundup (Maths, Tech, Computing)

Welcome to Plancktime’s first ever monthly roundup featuring short summaries on a couple of maths, tech or computing news that I thought were interesting and/or important. Links to articles that I used on each of the topics will be included at the bottom of the article in case you want to read more into the […]

### An Introduction to Calculus of Variations

What is Calculus of Variations? In general, developments in mathematics are motivated by the need for them in applications. Calculus of variations is no exception. In fact, it was first developed in 1969 when Johann Bernoulli asked the greatest mathematical minds of his time to solve the famous ‘brachistochrone problem’. This problem involves a bead […]

### How gravity can seem repulsive (from a certain point of view)

*Insert Star Wars joke here* All forces in the universe are either attractive or repulsive. We know that gravity is a force that is always attractive. Unlike the electromagnetic force, a particle cannot be repelled by gravity *. But what if I told you that from the point of view of a comet or a planet orbiting […]

### The Halting Problem

You’re sat at your desk, watching a progress bar stuck at 99%, wondering if it will ever finish. This is exactly the question that the halting problem asks. Given an input, will a computer program run forever, or eventually halt? For example, the simple program while (true) { print(“This one goes forever.”); } would loop […]

### Topology 5 : Continuous Functions

Continuity is a very important concept in Analysis, and even in General Topology, so it is very important that an intuition for these definitions are had. Say a function were “discontinuous,” like the one below. Figure 5.1 Now, we select an open set in that “separates” this function into two different sections. This will intuitively […]

### The Future of Space Exploration

Space Travel, is the best of it over? Ever since we have seen the stars, mankind has been striving to discover how not only the world works, but how the universe functions as well. One of the most important ways of doing this is via space exploration, which as time has gone by has become […]

### Manifolds 1 : Introduction

Have you ever wondered how we can tell if a topological space looks thin? Something that is like a surface; paper-like in nature. Something like the surface of a sphere (called the -sphere or ) or a blanket. Could such a concept be rigorously defined? We can’t say that it would have to be homeomorphic […]