Expressing a Physical Quantity

In considering how to express the result of a physical experiment undertaken with a view to measurement, two cases essentially different in character present themselves. In the first the result which we wish to express is a concrete physical quantity and in the second it is merely the ratio of two physical quantities of the same kind, and is accordingly a number. It will be easier to fix our ideas on this point if we consider a particular example of each of these cases, instead of discussing the question in general terms. Consider, therefore, the difference in the expression of the result of two experiments, one to measure a quantity of heat and the second to measure a specific heat - the measurements of a mass and a specific gravity might be contrasted in a perfectly similar manner - in the former the numerical value will be different for every different method employed to express quantities of heat; while in the latter the result, being a pure number, will be the same whatever plan of measuring quantities of heat may have been adopted in the course of the experiment, provided only that we have adhered throughout to the same plan, when once adopted. In the latter case, therefore, the number obtained is a complete expression of the result, while in the former the numerical value alone conveys no definite information. We can form no estimate of the magnitude of the quantity unless we know also the unit which has been employed. The complete expression, therefore, of a physical quantity as distinguished from a mere ratio consists of two parts: (1) the unit quantity employed, and (2) the numerical part expressing the number of times, whole or fractional, which the unit quantity is contained in the quantity measured. The unit is a concrete quantity of the same kind as that in the expression of which it is used.

If we represent a quantity by a symbol, that must likewise consist of two parts, one representing the numerical part and the other representing the concrete unit. A general form for the complete expression of a quantity may therefore be taken to be q [Q], where q represents the numerical part and [Q] the concrete unit. For instance, in representing a certain length we may say it is 5 [feet], when the numerical part of the expression is 5 and the unit 1 [foot]. The number q is called the numerical measure of the quantity for the unit [Q].