IEFInternational Education Forum3083-49022981-8605Bio-Byword Scientific Publishing Pty. Ltd.10.26689/IEF.v4i9.15725ArticleRe-Understanding the Essence of Calculus Operations and Analysis of Its Number Field Foundationshttps://artdesignp.com/journal/IEF/4/9/10.26689/IEF.v4i9.15725YangJinhe2026492026-10-20This study summarizes the series of confusions that arise when studying calculus theory within the traditional limit-based frameworks. The study analyzes several questionable aspects of constructing calculus systems based on limits from eight dimensions: rules of the division operation, objective laws of partition, differential concepts, logical deduction, semantic connotations, learning and practice, the transition from quantitative to qualitative change, and numerical field operations. It argues that the limit is merely an outward manifestation of calculus rather than its essence, introduces the concepts of explicit and implicit real fields, explains the internal operation mechanism of differentiation and integration, and contends that calculus is an inherent natural property of real numbers, and is a mathematical discovery rather than a human invention, in order to construct a cognitive system of calculus that can be connected with elementary mathematical operation rules, and provide a new perspective for understanding the underlying logic of calculus.Calculus, Limit, Implicit real field, Differential, Integral

[1] Department of Mathematics, East China Normal University, 2021, Mathematical Analysis (Volume 1), 5th edition. Higher Education Press, Beijing. [2] Chen JX, Yu CH, Jin L, 2019, Mathematical Analysis (Volume 1), 3rd edition. Higher Education Press, Beijing. [3] Kline M, 2002, Mathematical Thoughts of Ancient and Modern Times (Volume 2), translated by Zhu Xuexian, Shen Youcheng, Ye Qixiao. Shanghai Science and Technology Press, Shanghai. [4] Fichtenholz GM, 1954, A Course of Differential and Integral Calculus (Volume 1), translated by the Higher Mathematics Teaching and Research Office of Peking University. People’s Education Press, Beijing. [5] Xu LZ, 2007, General Principles of Calculus. Dalian University of Technology Press, Dalian. [6] Robinson A, 1980, Nonstandard Analysis, translated by Lu Chuanwu and Ma Jifang. Huazhong Institute of Technology Press, Wuhan.