Notations do not matter to the essence of mathematics. But poor notations can be misleading. Notations based on exponents, radicals and logarithms definitely are. They are very distinct, even though they are supposed to describe very similar relations between numbers. The triangle of power is a recently proposed alternative. In short, I am convinced!

This is a guest post I wrote on White Group Mathematics on December 30, 2012.

It was when hearing the sounds of hammers that Pythagoras realized the ubiquity of numbers in mathematical harmony. He would go on laying down the mathematical foundations of music, based on octaves, perfect fifths and major thirds. This mathematics of music would then become the favourite playground of all musicians, from Beethoven to Gangnam Style.

No one believed him. Not even fellow mathematicians. They thought he was wrong. They thought he was crazy. Even he ended up doubting himself and went crazy. And yet, he had mathematically proved it all. Georg Cantor had figured out how to manipulate the infinite. Even more remarkable, he showed that there were actually several infinities; and some are bigger than others!

You've probably learned early on that there are three primary colours. But why three? And why these three? Surprisingly, the answer lies in the beautiful mathematics of linear algebra and (high) dimension spaces!

This article retraces the endless pursuit of the infinite that is at the basis of mathematical analysis. From the first approximations of pi to the shape of our limitless universe, from the essential usefulness of differential equations to the troubles with infinite sums, we present the great ideas of mathematical geniuses all along History.

The power of algebra lies in abstraction, and abstraction is basically forgetting. By retracing the History of algebra from its roots to more recent advancements, this article unveils the numerous breakthrough in our understanding of the world, by abusing of the power of forgetting.

I know this sounds crazy. Even stupid. But Adidas did design a cubic ball, called brazuca, for the 2014 World Cup. And, yet, this cubic ball is rounder than any previous ball in football History. How is it possible? This article explains it.

2048 is the Internet sensation of the year. This very addictive game has been downloaded hundred of millions of times. Interestingly, this game raises plenty of intriguing mathematical questions. This article unveils some of them!

In an effort to make mathematics more computable, a consortium of today's greatest mathematicians have laid out new foundations. Amazingly, they all lie upon one single axiom, called univalence. The goal of this axiom is to make formal mathematics more similar to informal mathematics. With univalence, our Arabic numbers aren't just like natural numbers; They are natural numbers. Univalence also has unforeseen and mesmerizing consequences.

In this article, we explore the possibilities allowed by higher inductive types. They enable a much more intuitive formalization of integers and new mind-blowing definitions of the (homotopical) circle and sphere.

In 2013, three dozens of today's brightest minds have just laid out new foundation of mathematics after a year of collective effort. This new paradigm better fits both informal and computationally-checkable mathematics. There is little doubt that it will fundamentally change our perspective on rigorous knowledge, and it could be that, in a few decades, the book they published turns out to be the bedrock of all mathematics, and, by extension, all human knowledge! Have a primer of this upcoming revolution, with this article on type theory, the theory that the book builds upon!

Take a square sheet of paper. Can you glue opposite sides without ever folding the paper? This is a conundrum that many of the greatest modern mathematicians, like Gauss, Riemann, and Mandelbrot, couldn't figure out. While John Nash did answer yes, he couldn't say how. After 160 years of research, Vincent Borrelli and his collaborators have finally provided a revolutionary and breathtaking example of a bending of a square sheet of paper! And it is spectacularly beautiful!

In 1988, Euler's identity was elected most beautiful theorem of mathematics. It has been widely taught worldwide. But have you ever stopped to really sense the meaning of this incredible formula? This article does.

In recent years, at the interface of game theory, control theory and statistical mechanics, a new baby of applied mathematics was given birth. Now named mean-field game theory, this new model represents a new active field of research with a huge range of applications! This is mathematics in the making!

In 1832, Évariste Galois died. He was 20. The night before his death, he wrote a legendary letter to his friend, in which he claims to have found a mathematical treasure! Sadly, this treasure had long been buried in total indifference! It took nearly a century to rediscover it! Since then, Galois' legacy has become some of the finest pure mathematics, which represents a hugely active field of research today with crucial applications to cryptography. Galois' work is now known as Galois theory. In essence, it unveils the hidden symmetries of numbers!

Linear algebra is a one of the most useful pieces of mathematics and the gateway to higher dimensions. Using Barney Stinson's crazy-hot scale, we introduce its key concepts.

Last summer, I got to discover Morellet's artwork on inclined grids. Amazingly, this artwork is a display of the irrationality of $\sqrt{2}$! It's also a strong argument for the existence of this number. In this article, after discussing that, I take readers further by discussing what numbers can be constructed geometrically, algebraically, analytically or set theoretically using the power of mathematics!

Amusingly, the age difference between a 45-year-old man and a 25-year-old woman doesn't seem as big as the age difference between them 20 years earlier, when the woman was a little 5-year-old girl. This remark was the insight the late science popularizer Albert Jacquart liked to give to his readers to explain logarithms. This article pays tribute to the great scientist by introducing age difference as he liked to tell it.