A Novel Mathematical Approach to Pattern-Based Numerical Analysis Involving π and e

Abstract

Mathematics advances through the discovery of patterns, constants, and unifying principles that connect diverse areas of study. Fundamental constants such as π and e play a central role in linking geometry, algebra, calculus, and number theory. This research article presents a novel approach to numerical analysis based on structured pattern recognition and logical formulation, with particular emphasis on the natural emergence of the constant e in growth, limits, and continuous processes. The study develops generalized mathematical ideas that simplify complex calculations and reveal deeper relationships among numbers. The proposed approach is useful for students, educators, and researchers by offering efficient methods that reduce computational effort while preserving mathematical rigor. The findings contribute to number theory and applied mathematics by demonstrating how systematic reasoning leads to elegant results involving both π and e.