A Brief History of the Most Remarkable Numbers $$\pi $$ π , $$g$$ g and $$\delta $$ δ in Mathematical Sciences with Applications

Abstract

This paper deals with a brief history of the most remarkable numbers $$\pi , g \text {and} \delta $$ in mathematical sciences with their many examples of applications. Several series, products, continued fractions and integral representations of $$\pi $$ are discussed with examples. The celebrated Newton method of approximation of $$\pi $$ to many decimal places is included. The appearance of $$\pi $$ in many problems, formulas, elliptic integrals and in probability and statistics is presented with examples of applications including the Tchebycheff problem of prime numbers, the Buffon needle problem and the Euler quadratic polynomial. The golden number $$g$$ and its applications to algebra and geometry are briefly discussed. The Feigenbaum universal constant, $$\delta $$ is discovered in 1978 and it is found to occur in many period doubling bifurcation phenomena in the celebrated logistic map and the Lorenz differential equation system with chaotic (or aperiodic) solutions. Included is a numerically computed Lorenz attractor which resembles a butterfly or figure eight. The Lorenz attractor is a strange attractor because it has a non-integer (or fractal) dimension. The major focus of this article is to provide basic pedagogical information through historical approach to mathematics teaching and learning of the fundamental knowledge and skills required for students and teachers at all levels so that they can understand the concepts of mathematics, and mathematics education in science and technology and pursue further research.