Related papers: A Visual Proof that $\pi^e < e^\pi$

In connection to the two fascinating constants $e$ and $\pi$, there are many beautiful visual proofs to the inequality $\pi^{e}<e^{\pi}$. The aim of this classroom capsule is to give three visual proofs to the more general inequality…

In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.

In a recent Note (Am. J. Phys. 92:397, 2024; arXiv:2309.10826), Vallejo and Bove provide a physical argument based nominally on the second law of thermodynamics as a way of resolving the mathematical question appearing in the title. A…

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of…

The question of the title is a famous puzzle in the field of recreational mathematics, and can be addressed by several approaches. A compilation of solutions, some of them very ingenious, can be found in [1]. In this contribution we present…

This short note delivers, via elementary calculations, a product representation of pi.

A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.

We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.

This article contains the proof of a theorem on orthogonal-Pin duality that was cited without proof in a previous article in this journal.

We point out that the proof of irrationality of $\pi$ by Niven can be modified to a proof by contraposition. As a warm-up, we also give a proof of irrationality of $\sqrt{2}$ and $\sqrt{3}$ in a similar way.

Continued fractions are used to give an alternate proof of $e^{x/y}$ is irrational.

The number $e$ has rich connections throughout mathematics, and has the honor of being the base of the natural logarithm. However, most students finish secondary school (and even university) without suitably memorable intuition for why…

By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.

This paper presents a new, significantly simpler proof of one of the main results of applied pi-calculus: the theorem that the concepts of observational and labeled equivalence of extended processes in applied pi-calculus coincide.

A vector variational principle is proved.

Our goal in the present paper is to give a new ergodic proof of a well-known Veech's result, build upon our previous works.

Recently Z.W.Sun found over hundred conjectured formulas for 1/pi. Many of them were proved by H.H.Chan, J.Wan andW.Zudilin (see [3], [8] in the paper). Here we show that several other formulas in [6] are simple transformations of known…

In this short paper we show that the inequality of arithmetic and geometric means is reduced to another interesting inequality, and a proof is provided.

A simple way is shown to construct the length $\pi$ from the unit length with 4 digits accuracy.

We describe the formalisation in Coq of a proof that the numbers e and $\pi$ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex…