Infinitesimals inside the Familiar Field of Complex Numbers

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.

Theorems & Definitions (28)

• Theorem 4.1: Steinitz
• proof
• Corollary 4.2: Uncountable Cardinality
• proof
• Definition 4.3: Algebraic Definition of $\mathbb C$
• Corollary 4.4: Two Fields
• proof
• Theorem 4.5: Erdös, Gillman, Henriksen
• proof
• Theorem 5.1: Main Result