dimensionful

Dimensionful describes physical quantities that carry a physical dimension and must be specified with a unit. These are contrasted with dimensionless quantities, which are pure numbers without units, such as mathematical constants or ratios.

In physics, dimensions are built from base quantities such as length (L), mass (M), and time (T), with other bases like electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J) used in some systems. Derived quantities combine these base dimensions, and their units reflect those combinations. For example, energy has dimensions M L^2 T^-2, velocity has dimensions L T^-1, and force has dimensions M L T^-2.

Numerical values of dimensionful quantities depend on the chosen unit system. Changing units alters the numbers but not the underlying quantity; for instance, 2 meters equals 200 centimeters or about 6.56 feet. Dimensional analysis uses these dimensions to check the consistency of equations, ensuring that both sides have the same dimensional structure.

In physics, there is a distinction between dimensionful and dimensionless constants. Dimensionful constants, such as the speed of light c, Planck constant ħ, and Newton’s constant G, carry units. Some constants are dimensionless, like the fine-structure constant α. Unit choices can change the numerical values of dimensionful constants, while dimensionless constants remain invariant under unit redefinitions.

In field theory and related areas, fields can have mass dimensions that depend on the spacetime dimensionality, influencing how terms in equations scale. Dimensionful quantities are thus central to measurement, calculation, and the formulation of physical laws.