As Cantor showed us, there are uncountably many transcendental numbers. Despite being overwhelmingly more numerous than algebraic numbers, it is not so easy to find them in Nature: we know, for example, that pi is transcendental, but proving this is a major result. We will discuss these facts, then go on to discuss the Kontsevich-Zagier theory that most transcendental belong to a countable class, the periods of integrals of algebraic functions with rational coefficients. Finally, we consider Yoshinaga’s number: a specific number that is not a period.
The famous Gauss-Lucas theorem describes a geometric connection between the roots of a polynomial and the roots of its derivative. In particular, the theorem states that the critical points of a polynomial p all lie within the convex hull of the roots of p. Building on the works of Pemantle-Rivin and Kabluchko, I will describe a probabilistic version of the Gauss-Lucas theorem, which holds for a large class of random polynomials.