What is the proper place of units?

It is common to associate an algebraic name such as $x$ with, say, a length.  There are two common conventions for how to do this:

In the numbers convention, common among mathematicians and often used informally by others, especially during calculations, one says e.g. :

amath \qquad\qquad\qquad x= endamath the length of the rod in feet.

and for given rod a mathematician might write e.g.

amath \qquad\qquad\qquad x=6 endamath.

The numbers convention keeps units on the sidelines.  Units are treated as a way of turning quantities into numbers.  Algebraic relations are thought of as relations between numbers.


In the quantities convention, common among physicists, engineers, and other technical folk, especially in formal settings, one says e.g. :

amath \qquad\qquad\qquad x= endamath the length of the rod.

For the same rod as before, one writes, e.g. :

amath\qquad\qquad\qquad x=6 endamath ft.

With the physical convention, the same quantity can be written in different units,

amath\qquad\qquad\qquad x = 6 \text( ft = 2 \text( yd  =1 \text( fathom =182.88 \text( cm. endamath

The quantities convention lets units march on the playing field in company with numbers.  Algebraic relations are thought of as relations between quantities.

The numbers convention is something of a straitjacket.  For historical reasons, it persists in college textbooks for algebra, calculus, linear algebra, and multivariable calculus.

As it is usually practised, however, the quantities convention also suffers from unfortunate constraints.  How this is so, and what to do about it, are matters for a later post.