unitless

Unitless quantities, or dimensionless quantities, are those that do not carry physical units. In a consistent system of measurement, a dimensionless quantity has zero dimensions and can be represented by a pure number. They often arise from ratios of like quantities, from normalizing a quantity by a reference scale, or from fundamental constants. Because they do not depend on the choice of units, dimensionless numbers facilitate comparison across different systems and often simplify mathematical models.

Common examples include the Reynolds number (Re = ρ v L / μ), which characterizes fluid flow, the Mach number (M = v / a) for speed relative to sound, the Strouhal number (St = f L / v) for oscillating flows, and aspect ratios such as width to height. Material properties can be dimensionless as well, for instance Poisson’s ratio. Angles are treated as dimensionless in the SI system, with radians often regarded as a derived, but dimensionless, unit. Fundamental constants such as the mathematical constants pi and e, and the fine-structure constant α, are dimensionless.

Dimensionless quantities are central to methods such as nondimensionalization, where variables are scaled to remove units from equations, and to the Buckingham Pi theorem, which identifies dimensionless groups that govern physical relations. In science and engineering, many statistical descriptors—probability values, p-values, correlation coefficients, and z-scores—are also dimensionless, reflecting pure numbers rather than physical units.