A calculator will happily multiply the wrong quantities. Dimensional analysis catches a useful class of mistakes before they become convincing numbers.
An equation needs compatible dimensions
Distance has the dimension of length. Speed has length divided by time. Multiplying a constant speed by a duration gives a distance:
The units follow the same pattern: metres per second multiplied by seconds gives metres. Adding a distance to a time, on the other hand, cannot produce a meaningful physical sum.
Check a circuit equation
Electrical power can be written as
The last two forms use Ohm's law for a resistor. Each gives watts. A stray expression such as does not have the dimensions of power, even if its numerical result happens to look plausible. **
Dimensions cannot prove an equation**
Both and have dimensions of length. Units alone cannot choose between them. They also do not establish whether your assumptions—constant velocity, an ideal resistor, negligible friction—are appropriate.
Think of dimensional analysis as a fast filter: passing it is necessary for many physical equations, but it is not sufficient to establish correctness.
## A habit worth keeping
Write units beside inputs. Convert prefixes explicitly: one millimetre is metres, but one square millimetre is square metres. Check the final dimensions, then ask whether the magnitude makes physical sense.