Prefix Precedence

Part of my units pages

The SI definition explicitly states that prefixing a unit with a multiplier, like “kilometre”, gives us a new, “derived” unit. This is important for resolving otherwise ambiguous quantities like “3cm³”: according to SI, this is 3(cm)³ = 0.000003m³, whereas interpreting the prefix as a simple number would give 3c(m³) = 0.03m³. Similar problems arise when dividing, e.g. “per kilometre”.

Whilst the SI method is well-defined, it still places a mental burden on the user. What’s especially annoying is that the SI rules for units are opposite to the usual rules of multiplication and exponents, e.g. algebraically we would say that abⁿ = a(bⁿ)

This also allows derived units (e.g. cubic centimetres) to creep into our calculations. My preferred solution to any potential ambiguity is to use explicit parentheses, hence:

“3cm³” is well-defined by SI, but invites ambiguous interpretation. Better to be safe and add explicit parentheses.
“3(cm)³” is unambiguous. I would still try to avoid this, since it’s using derived units (cubic centimetres). We could instead use 0.000003m³, or 3×10⁻⁶m³, or 3μ(m³). The parentheses in the latter are a bit ugly, but aesthetics are less important than specificity, and it’s arguable which of these forms looks nicer.
“3c(m³)” is also unambiguous (although a different quantity to the above). In this case there aren’t many zeros, so we could just write 0.03m³.

Another way to reduce ambiguity and ugliness is to give explicit names to our units, like calling the cubic metre a kuub. The quantities above would then be called 3 microkuub and 3 centikuub, respectively.