The elementary unit ― count and numbers in the International System of Units

Abstract

The International System of Units (SI) defines counting quantities with unit one, symbol 1. Conventionally, the unit symbol 1 is not shown explicitly. This leads to ambiguities in expressing quantities per count and implies an insufficient distinction between count and number. Building on a critical reading of the SI Brochure (9th edition, version 3.01, August 2024) and formal analysis of counting, the Euclidean Unit U_X is recognized as the individual entity: a single item of, e.g., an event, atom, molecule, particle of type X), distinguished from the abstract elementary unit. The symbol ‘1’ is replaced by ‘x’, representing the abstract unit of U_X and count N_X.

A count N_X with unit x is then defined as N_X = N·U_X, where U_X with unit x represents a single elementary entity of type X and N is a dimensionless number. This notation ensures consistency between units of extensive quantities, such as cell count (N_ce = 2·10⁶ x = 2 Mx), and quantities normalized per count. The inverse Avogadro constant N_A^-1 [mol/x] and elementary charge e [C/x] link the count to amount and charge, respectively. Consequently, the units of the fundamental constants are distinct from the units of the elementary quantities amount [mol] and charge [C].

A consistent nomenclature resolves persistent terminological confusion between numbers, units, and entities. The symbol ‘x’ instead of ‘1’ aligns the unit of count with the principles of quantity calculus. Treating x like all symbols of SI units improves dimensional consistency, supports accurate interpretation of physical quantities, and enhances the rigor of communication across the natural sciences.

Cite:

Gnaiger E (2025) The elementary unit — count and numbers in the International System of Units. BEC preprints 2025.4. https://doi.org/10.26124/becprep.2025-0004