Algebraic Explanation and Geometric Representation of Calculus Concepts Using Hyperreal Numbers

This study presents a novel algebraic framework for calculus based on hyperreal numbers, integrating infinitesimal and infinite values to unify algebraic, differential, and geometric perspectives. We define a truncation function to reinterpret the two formulas studied by Euler, introducing new variants of the natural constant \ (e\), and reformulate the \ (-\) and \ (-N\) limit definitions, enabling direct substitution of infinitesimal and infinite values. This approach provides an alternative to L'Hôpital's rule for evaluating limits. Derivatives are categorized into \ (00\) and non-\ (00\) forms, with a new operator, derived from Taylor expansions and the binomial theorem, facilitating the derivation of differential formulas and revealing the inverse relationship between multiplication and division rules. Based on whether the increments \ (x\) and \ (y\) equal their differential counterparts \ (dx\) and \ (dy\), we explore four distinct cases for differentiation and integration, distinguishing functions that appear linear at infinitesimal scales from those that remain curved. This framework clarifies differential form invariance and streamlines conversions between parametric equations and composite function derivatives. Geometric illustrations enhance the intuitive understanding of these concepts. By aligning calculus with algebraic principles, this rigorous yet accessible framework holds significant promise for theoretical mathematics and educational applications.