A New Approach to Calculus

This paper presents a novel conceptual and computational framework for calculus, built on a geometric re-interpretation of curves as limits of small “angulated” segments rather than purely horizontal or vertical elements. Beginning with the definition of a circle’s circumference through the summation of infinitesimal linear segments, the framework extends to re-definitions of common geometric objects—lines, squares, rectangles, and triangles—using an elemental unit e derived from the limit of segment lengths as they approach zero. The proposed approach embeds these geometric definitions directly into the limit process for derivatives. Using this structure, the derivative is redefined in terms of the “definition” of a function rather than purely symbolic manipulation. This leads to a derivation of the classical power rule for x^m that is consistent with standard calculus but grounded in a physically interpretable, segment-based model. The work provides a bridge between intuitive geometric construction and formal calculus operations, offering a fresh lens through which to view fundamental mathematical principles. The theoretical framework and definitions were independently developed by Omar Admani. The author acknowledges awareness that similar or related ideas may exist within existing mathematical literature but affirms that all derivations and formulations were arrived at independently. Formatting, algebraic formalization, and illustrative examples were assisted by ChatGPT-5 (OpenAI). Earlier drafts and exploratory materials are archived privately within the OSF project to preserve developmental transparency and timestamped authorship records. Also archived on Zenodo for wider access and long-term preservation: https://doi.org/10.5281/zenodo.17500464