Dimensional Analysis

Units are a necessary part of the specification of a physical quantity. When physical quantities are subjected to mathematical operations, the units must be carried along with the numbers and must undergo the same operations as the numbers. Quantities cannot be added or subtracted directly unless they have not only the same dimensions but also the same units, for example:

In solving problems, one often can be guided by the units to the proper way of combining the given values. Some textbooks refer to this method as the factor-label or unit-factor method; we will call it dimensional analysis. In essence, one goes from a given unit to the desired unit by multiplying or dividing such that unwanted units cancel. For example, consider converting 5.00 inches to centimeters, given the conversion factor 2.54 cm/in. We might try two approaches:

The first try gives nonsensical units, signaling a misuse of the conversion factor. On the second try, the inch units cancel, leaving the desired centimeter units.