Equations and Units

Abstract

We must distinguish between dimensional equations that are valid in all self consistent systems of units and equations that give the numerical values only in a specified system of units. A dimensional equation, for instance, is

$$ {\text{u = Ri}}{\text{.}} $$

(1)

This equation holds regardless of what the units are. In contrast, the equation

$$ u=\underline{23}i\ , $$

(2)

where u represents a voltage and i a current, holds only in the system of units for which the numerical value 23 has been specified. Strictly speaking, this equation is incorrect in the form written because 23 is a number of zero dimension; and volts as given by the left-hand side, can never be amperes, as given by the right-hand side. To draw attention to this fact, 23 is underlined to indicate that it is not just a number, but also has a dimension. It is poor practice to omit dimensional symbols, because a dimensional check is always the first step in tracing computational errors. The equations in this book, therefore, refer to dimensional magnitudes almost without exception, and they hold for every coherent system of units.