In contrast to dimensional analysis (http://en.wikipedia.org/wiki/Dimensional_analysis), there is the theoretical basis of dimensional synthesis. It is a mathematical as well as physical process whereby the introduction of a physical unit factor on either side of a physical equation will render the physical units of both sides to agree with each other. This process is particularly useful for finding solutions of the rational equation: ����=(��-��)�� where �� equals one unit of the physical dimensions of �� and ��. If �� and �� have physical dimension of energy expressed as joules then �� is one joule. If �� and �� have physical dimension of mass expressed in kilograms then �� is one kilogram. If �� and �� have physical dimension of electric charge expressed in coulombs then �� is one coulomb. If �� and �� have physical dimension of electric current expressed in amperes then �� is one ampere. If �� and �� have physical dimension of length expressed in meters then �� is one meter. If �� and �� have physical dimension of absolute temperature expressed in kelvins then �� is one kelvin. This process can be applied to any physical dimension requiring certain normalization for finding rational solutions to physical equations.

Nonetheless, this process allows a simpler formulation equivalent to the processes of much more complicated renormalizations of both Abelian and non-Abelian quantum theories of particle and field interactions. For example, when it is applied to Newton’s law of universal gravitation: |��|=����₁��₂/|��|², the numerator ��₁��₂ can be expressed as (��₂-��₁)�� where �� is one unit of mass and |��|=��(��₂-��₁)�� /|��|². If mass �� is a simple function of the radius vector �� with spherical symmetry then ��=4����|��|³/3 where �� is the mass density. By direct substitution, |��|=4����(��₂|��₂|³-��₁|��₁|³)�� /|��|². For self-field interaction, ��₁=��₂=��, ��₁=��₂, and ��₁=��₂, hence |��| vanishes, i.e. |��|=0. For more complicated functions of mass, L’Hôpital’s rule for the limit of indeterminate forms can be applied.