The kilogram
Part of my units pages
The SI system’s base unit of mass is the “kilogram”. Whilst a kilogram equals one thousand grams, as we would hope, the definition is backwards: the gram is defined as one thousandth of a kilogram, AKA a millikilogram.
Within the dimension of mass this is merely silly; the real problem arises when we start combining dimensions. The coherence page uses the example of Newton’s second law of motion F = fma; where the constant of proportionality f is equal to 1 in SI, due to the way its base units are defined. This is true and good, but the base unit for m (mass) is the kilogram. If we want to use grams, to avoid the sillyness of having a prefix factor in a base unit, we end up needing a constant of proportionality f = 1/1000. This is why I strategically chose to avoid using mass in most other examples!
This is purely a naming issue, since we can just rename the “kilogram” to something without an ambiguous prefix; for example, it used to be called the “grave”. This would rename grams to milligraves, which explains where the factor of 1/1000 comes from.
Note that there’s an alternative approach, where we treat the gram as the base unit and hence the kilogram’s prefix works properly. Such a system was widely used, and is now know as the “CGS” system. Unfortunately that system uses the “centimetre” as its base unit of length, and hence suffers the same problem, except in the length dimension rather than mass.