Infinitesimals inside the Familiar Field of Complex Numbers

TL;DR

This paper demonstrates that the complex numbers contain non-zero infinitesimals due to non-Archimedean subfields, challenging traditional views and suggesting that infinitesimals are inherently present in standard mathematical structures.

Contribution

The paper reveals that the complex field includes non-zero infinitesimals by identifying non-Archimedean subfields within it, based on Steinitz's theorem, which is a novel insight.

Findings

Complex numbers contain non-zero infinitesimals.

Presence of non-Archimedean subfields in .

Infinitesimals simplify analysis and formal language.

Abstract

We show that the field of complex numbers $\mathbb C$ contains non-zero infinitesimals by observing that $\mathbb C$ contains non-Archimedean subfields. Our observation is based on an old theorem in algebra due to E. Steinitz, discussed in the article in detail. The presence of infinitesimals in $\mathbb C$ was surprise to the author and might be surprise to the readers as well, since $\mathbb C$ is commonly defined in terms of the field of reals $\mathbb R$, which is Archimedean. An additional intrigue arises from the fact that $\mathbb R$ was historically introduced in 19-th century (by Dedekind, Cauchy and others) exactly to make infinitesimals in Leibniz-Newton infinitesimal calculus redundant. It seems that mathematics will never get rid of infinitesimals completely - they are all around us whether we like it or not. In the last section of the article we explain how our result fits to analysis, both standard and non-standard. With examples from history of calculus as well of first-class recent achievements in analysis we try to convince the reader that presence of infinitesimals in analysis simplifies its formal language and improves its efficiency. The motivations of our research is related to an attempt to simplify the properties of particular algebras of generalized functions of Colombeau's type, shortly discussed in the text.