The process of unit conversion, often known as dimensional analysis, is a very important part of many real-world calculations. In mathematics, one learns to manipulate numbers, but in the real world numbers almost always have some sort of unit attached, and this needs to be managed correctly in order to arrive at a correct answer; without a unit, the answer is just a meaninless number.
Dimensional analysis allows one to convert between measurement systems easily. It also allows more obscure conversions, such as calculating how many atoms or molecules are present by using the weight of the substance. As far as real life examples, this sort of calculation is very important to medical personnel, who must use it to figure out dosages and test results.
Unit conversions can be critical in science as well as mathematics. Without knowing these conversions the mathematician would be severely limited in his/her scope of understanding. Even if math is not a chosen field of study, the average citizen can also benefit from knowing these conversions.
Units give meaning to numbers. Without a correlating unit a number is a useless piece of information. Nurses who administer medications use this knowledge very frequently. A qualified nurse that intends to practice safely and effectively must be familiar with this body of knowledge. This information is critical in proper dosage calculations. It is also extremely important in molecular studies performing calculations such as molecular and atomic structure.