UM ESTUDO SOBRE O π NAS P-MÉRICAS: SIMETRIAS EM EXPOENTES CONJUGADOS E EXTREMOS GLOBAIS

Abstract

This work provides, initially, a historical context of approximations of $\pi$ until the value of $3,1415\ldots$ is found, which corresponds to the Euclidian metric. Generalizing this metric for a real value $p$, we can obtain the $p$-metric, which, for $p \geq 1$, gives the set of metrics used in this paper. For this set, we show that the value of $\pi$ in the $p$-metric, denoted by $\pi_p$, attains its global minimum at $p = 2$ and global maxima for $p=1$ or $p \rightarrow \infty$. In order to present this result we make use of tools and techniques from integral calculus, as well as two particular functions, the Beta and Gamma functions, in the creation of a function to approximate $\pi_p$, which will be called $\Pi_p$. Beyond the minimality and maximality of $\pi_p$, we also prove a property of $\pi$ in this set of metrics, which is a notion of symmetry in the values of $\pi_p$ and $\pi_q$ when $p$ and $q$ are conjugated exponents, i.e., $\frac{1}{p}+\frac{1}{q} = 1$. In this situation the values of $\pi_p$ and $\pi_q$ coincide. We observe that, when altering the value of $p$, we get new values for $\pi$ and all the geometric notions that accompany this change. At last, we show how to construct the geometric figures developed in this work in GeoGebra.