The hunt for the beautiful nature of the fastest thing in the Universe — What is light? This question has been repeatedly asked throughout history and has been a great mystery of science and philosophy since the dawn of the human race. In this article, we’ll go through the history and the pursuit of the true nature of light from the ancient Greeks to…

There is always a prime between n and 2n — In 1845 Joseph Bertrand conjectured that for every integer n > 1, there is at least one prime p such that n < p < 2n. Introduction This conjecture is known as Bertrand’s postulate and is so famous that there’s even a rhyme attached to it: Chebyshev said it and I’ll…

The essence of homology — What does a 3-dimensional sphere look like? After all, we humans are able to see in three dimensions so we should actually see something. But a 3-dimensional sphere is the surface of a 4-dimensional ball just like a 2-dimensional sphere (the good old sphere as you know it) is the…

And the cotangent function — It has been known for a long time that the infinite series 1 - 1/3 + 1/5 - 1/7 + ⋅⋅⋅ = π/4. We are also going to see (as a side-effect of our calculations) why this is true but the main question in this article will be that of…

An indescribably beautiful proof — In this article, we will carefully try to understand why π is an irrational number. We will do this by going through what is in my opinion one of the most beautiful and creative proofs in all of mathematics. In my last article, we proved that the star of calculus…

And a thorough guide to proving it — When I first heard of this fact, I was studying mathematics at University and I was immediately intrigued by it. Let me explain why. Some mathematical facts, usually called theorems, are obvious. For example, the fact that the number 2 is the only even prime number is obvious if you…

An elusive and mythical value — It all started in 1650 when Italian mathematician Pietro Mengoli asked a question that no one (including himself) was able to solve. Pietro had earlier shown that the infinite sum 1 + 1/2 + 1/3 + ⋅⋅⋅ diverges to infinity which is a fancy way of saying that no matter…

Can energy be lost or created in our Universe? — This is a story of a woman who against all odds became one of the best mathematicians of her time and developed many of the modern theories that we use in abstract algebra today. She proved a theorem so powerful that it explains the ultimate fate of our Universe and…

Prime numbers and generalizations of the twin prime conjecture — Prime numbers are the whole number equivalent of atoms in molecules in the sense that every whole number greater than one can be written as a unique product of prime numbers just like molecules are built up uniquely by atoms (of course, the analogy is not strong enough to draw…