Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Redressing grievances with the treatment of dimensionless quantities in SI



David W. Flater


The International System of Units (SI) and its associated methods adhere to a design principle in which all kinds of quantities are described in terms of a standard set of dimensions. It is a corollary of that principle that quantities having no extent in any of those dimensions ("dimensionless quantities") will not be distinguishable as different kinds through dimensional analysis alone, and the formality of quantity calculus will be of no help in preventing mistakes where different kinds of dimensionless quantities are confused.

This corollary has led to controversies involving abuse of the unit hertz, elimination of radians and steradians as being equivalent to the number 1, and the definition of the mole. In addition, official rejection of the use of so-called counting units in conjunction with accepted units has been repeatedly criticized and is often ignored.

The proposed resolution is to extend quantity calculus to discern different kinds of dimensionless quantities as well as it currently discerns different kinds of dimensional quantities. The extended system provides a formal interpretation of commonly-used workarounds in terms that are compatible with SI. Should the extended quantity calculus become accepted by consensus in the future, a guide to its use would expand the set of references provided by the Joint Committee for Guides in Metrology (JCGM), which currently lacks a definitive reference on quantity calculus.



SI, International System of Quantities, counted quantities, one as a unit, dimensionless quantities


Flater, D. (2017), Redressing grievances with the treatment of dimensionless quantities in SI, Measurement (Accessed June 25, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created October 3, 2017, Updated May 27, 2020