Dimensional analysis and dimension systems

Physical rules are independent of measuring units. Dimensional analysis explores such invariance, its outcomes and applications. Dimensionless quantities are the invariants under changes of units. Physical rules must ultimately be presented only in dimensionless quantities. The number of dimensionless quantities that can be extracted in a problem is less than that of the original variables, which results in simplification. The dimensionless quantities such extracted may deeply reflect the intrinsic relation among physical variables. The concept of dimension is profound, but its use is simple, so dimensional analysis should be an important section of physics training. Some fundamental concepts and principles involving dimensional analysis and its applications are expounded, including dimension systems and their relation to units systems. Examples are taken from the literature, where there is still some confusion, which I shall attempt to clarify. It is emphasized that, when applying dimensional analysis, we can just use the mass-length-time dimension system.