unitinvariance

Unit invariance is the property that the form of equations and the meaning of quantities do not depend on the particular system of units used to express them. In physics and applied mathematics, a theory or model is unit-invariant if its predictions remain unchanged when base units for length, time, mass, and other dimensions are rescaled or redefined.

In practice, many quantities are dimensionful, so their numerical values change with unit choices. What remains invariant are the dimensionless combinations and the physical relations that are expressed in a way that is independent of units. A common way to achieve or reveal unit invariance is non-dimensionalization: variables are rescaled by characteristic scales to produce dimensionless groups. The Buckingham Pi theorem formalizes how to form such dimensionless quantities that govern system behavior. Natural units, where fundamental constants are set to 1, are another route to unit invariance, removing unit dependencies from equations.

Dimensionless constants, such as the fine-structure constant or the Reynolds number in fluid dynamics, are inherently unit-invariant; changing units does not alter their numerical value. Conversely, dimensionful quantities such as velocity or energy depend on the chosen unit system, even though the underlying physics is unchanged.

Unit invariance is important for model comparison, numerical simulation, and theory development because it clarifies which aspects of a model are physically meaningful rather than artifacts of unit conventions. It also guides experimental reporting, ensuring that results can be compared across different unit systems.

See also: dimensional analysis, dimensional consistency, Buckingham Pi theorem, non-dimensionalization, natural units.