International System of Quantities

Lore & Background

It was developed jointly by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC). The system includes seven base quantities that cannot be expressed in terms of each other, while all other quantities are derived from them. The dimension of a base quantity is represented by a single uppercase roman sans-serif letter, and derived quantities are expressed as products of powers of these base dimensions. For example, velocity has the dimension LT⁻¹. The ISQ also defines dimensionless quantities, such as plane angle and solid angle, which may use the units radian and steradian.

Reader's Guide

The International System of Quantities provides the conceptual foundation for the International System of Units (SI), ensuring consistency across scientific disciplines. By establishing a set of seven base quantities and rules for deriving others, it enables precise dimensional analysis and communication of physical measurements. The system's treatment of logarithmic quantities, such as sound pressure level (measured in decibels) and information entropy (measured in natural units), extends its applicability beyond classical physics. The ISQ's dimensional notation—using symbols like L, M, T, I, Θ, N, J—allows any physical quantity to be expressed as a product of powers of base dimensions, facilitating error checking and unit conversion. Its recognition by the CGPM underscores its role as the official quantity system behind the SI, making it essential for metrology, engineering, and fundamental science.

Did You Know?

• The ISQ includes seven base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
• The dimension of a derived quantity is expressed as the product of powers of the base dimensions, such as LT⁻¹ for velocity.

The Seven Pillars of Physical Measurement

The International System of Quantities serves as the conceptual backbone for all of modern physics and science. At its core sit seven base quantities—length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity—that are chosen by convention to form a minimal, mutually independent set. No one of these seven can be rewritten using the others, yet every other measurable quantity in the system can be constructed from them. This independence is what gives the framework its structural integrity. Each base quantity carries a symbolic identity: its name appears in italics, while its dimension is rendered as a single upper-case letter set in upright sans-serif type. Crucially, a dimension captures only the kind of quantity, never its numerical magnitude or the particular unit attached to it. The ISQ thus operates at a level of abstraction above the International System of Units; it defines what is being measured and how quantities relate to one another, without prescribing which specific units scientists must employ. This separation of quantity from unit is what allows the system to remain flexible across disciplines while maintaining internal coherence.

Building the Derived Landscape

A derived quantity, within the ISQ framework, is any measurable property whose definition rests exclusively on the seven base quantities. The system accommodates a vast array of such derived quantities, each paired with its own derived unit. Their dimensional signatures follow a precise algebraic pattern: the dimension of any derived quantity is expressed as a product of powers of the seven base dimensions, written in the form L^a M^b T^c I^d Θ^e N^f J^g. The exponents a through g may be positive, negative, or zero, and any base dimension whose exponent is zero simply drops out of the expression. A familiar illustration is velocity, whose dimension reads L T⁻¹, capturing the idea of length traversed per unit of time. At the other extreme sit dimensionless quantities—those in which every exponent vanishes and the dimension symbol reduces to the numeral 1. Historically called dimensionless, these quantities can be understood as ratios of two quantities sharing the same dimension. The ISQ acknowledges two named dimensionless units, radian and steradian, which serve to distinguish plane angles from solid angles even though both are dimensionally one. This layered structure lets the system handle everything from simple kinematic relations to the most abstract ratios in theoretical physics.

Logarithmic Measures and Information

Beyond the familiar algebraic combinations of base dimensions, the ISQ carves out a special category for logarithmic quantities. The most prominent is the level of a quantity, defined as the logarithm of the ratio between a given value and a stated reference value. The definition differs depending on whether the quantity in question is a root-power quantity (a term the standard notes has been deprecated in favor of the older label field quantity) or a power quantity; for ratios of other kinds, no level is defined within the system. Sound pressure level, measured in decibels, stands as a canonical example, and the SI Brochure lists several level units as non-SI units. The ISQ also recognizes logarithmic frequency ratios: the octave, corresponding exactly to a doubling of frequency, and the decade, corresponding to a factor of ten. Perhaps most strikingly, the system extends to information entropy, for which the coherent unit is the natural unit of information, symbolized as nat. These logarithmic and information-theoretic quantities demonstrate that the ISQ is not limited to classical mechanical or thermal measurements but reaches into acoustics, signal processing, and information science, providing a unified dimensional language across highly varied disciplines.

The Standard Behind the Standard

The standard is the product of a joint effort between the International Organization for Standardization and the International Electrotechnical Commission, which together provide information and definitions covering quantities, systems of quantities, units, quantity and unit symbols, and coherent unit systems, with particular emphasis on the ISQ. The name International System of Quantities itself is used by the General Conference on Weights and Measures to describe the system of quantities that underlie the International System of Units. This governance structure—linking ISO, IEC, and the CGPM—ensures that the conceptual framework of measurement remains globally coherent, continuously updated, and firmly anchored in the practical needs of both fundamental research and applied engineering.