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Nondimensionalization is the transformation of an equation to an equivalent equation, eliminating some or all the variables that have units (e.g., seconds, meters, kilograms, etc.), i.e., making each value and the entire equation dimensionless. This can be done to simplify the equation and/or to make apparent the modeled phenomenon's intrinsic characteristics. In place of the variables with units can be variables with ratios.
Such a ratio can be between values in the original equation that had identical or convertible units, but the equation might need to be manipulated so as to provide two such values. If such a manipulation is not available, a quantity might be chosen to form such a ratio, e.g., one that has a natural significance, but other times, a seemingly-arbitrary quantity is chosen and the "nondimensionalization" seems little more than merely adopting new units. For example, expressing a speed as a ratio with c might make the physics more evident, whereas expressing it as its ratio to "1000 meters per second" seems like little more than changing to a more convenient unit.