A Comparative Study of the Archimedean Property in Different Ordered Algebraic Structures

Tabsum B

Abstract: The discussion highlights how the Archimedean property shapes the behavior of ordered fields, ordered groups, and normed vector spaces, and it becomes evident that this property draws a clear line between number systems that behave in a familiar, finite way and those that admit infinitesimal or infinitely large elements. The text shows that the real numbers serve as a benchmark for understanding how scaling, order, and magnitude interact, especially since no element in this system can remain indefinitely small or grow without bound when compared with another. This raises another point, the failure of the property in non-Archimedean fields or p-adic constructions introduces a very different mathematical landscape, one with ultra-metric geometries and topologies that behave far from everyday intuition. It is evident that these contrasting structures reveal why the Archimedean condition is not merely a technical requirement but a guiding principle that influences analysis, topology, and algebra in meaningful ways. Taken together, the material suggests that understanding both Archimedean and non-Archimedean frameworks helps clarify how modern mathematics accommodates both classical intuition and more abstract number systems that stretch conventional ideas of size, distance, and order.

Keywords: Archimedean property, ordered fields, normed vector spaces, non-Archimedean fields, p-adic numbers

How to Cite?: Tabsum B, "A Comparative Study of the Archimedean Property in Different Ordered Algebraic Structures", Volume 14 Issue 11, November 2025, International Journal of Science and Research (IJSR), Pages: 1446-1450, https://www.ijsr.net/getabstract.php?paperid=MR251119213828, DOI: https://dx.doi.org/10.21275/MR251119213828