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Have different units of measurement for areas and volumes ever been seriously proposed to the BIPM?

Something like

$(\text{cm})^2$ instead of $\text{cm}^2$

or

$(\text{mm})^3$ instead of $\text{mm}^3$

I know that in a convention it is enough to be all in agreement (and ink is, of course, very precious) but I feel a certain daily annoyance in seeing the prefixes of multiples and submultiples lose their original meaning.

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  • $\begingroup$ Don't get me started on notation like $\sin ^2 x$... $\endgroup$
    – Marius Ladegård Meyer
    Commented Nov 23 at 18:45
  • $\begingroup$ Totally agree with you! $\endgroup$
    – Marco
    Commented Nov 23 at 19:16
  • $\begingroup$ I think the problem would be the kg. Of course, it is kind of a mess to have a k in a base unit to begin with. But your approach would make that mess even worse $\endgroup$
    – Dale
    Commented Nov 23 at 19:50
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    $\begingroup$ Just think of prefixes like c and m as being a special kind of multiplication that has a higher precedence than exponentiating does. $\endgroup$
    – Ghoster
    Commented Nov 23 at 21:54
  • $\begingroup$ What you’re proposing isn’t different units; it’s different notation for the same units, namely a square centimeter and a cubic millimeter. $\endgroup$
    – Ghoster
    Commented Nov 24 at 7:09

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I can kind of see where your question is coming from. But just off the top of my head, consider the following issue that can arise, as one example probably out of many similar ones:

The SI base units for pressure are $\mathrm{kg\times m^{-1}\times s^{-2}}$. Now we also know that they're directly related to the units of force in the physically meaningful sense of "pressure is force per unit area". Now, since the SI base units for force are: $\mathrm{kg\times m\times s^{-2}}$ if we adopt your suggestion, it will be perfectly legitimate, perhaps even obligatory to define the base units for pressure as: $\mathrm{kg\times m\times(m^{-2})\times s^{-2}}$, because we're dividing by area, which if I understand your suggestion will now be an independent unit.

So, the nice property of length units partially canceling will no longer be manifest in the definition of pressure, resulting in a rather longer and to my mind more cumbersome definition. While one may argue that this better preserves the physical meaning of each "piece" of the defined unit (length vs. area), I still feel that this would overall be rather confusing.

There are probably examples where this suggested convention can lead to even weirder situations, for example think of something like $\mathrm{m^2/(m^2)}$, where units that previously had completely "disappeared" from the definition, will now oddly remain present, if you see what I mean. The pressure example was just the quickest one for me to come up with.

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  • $\begingroup$ Thank you for your answer. I have to tell you that I was stressing out another fact: with my question I wanted to talk about the (obvious) fact that 1 cm^2 is not 0.01 m^2 and the conventions about notation related to it. $\endgroup$
    – Marco
    Commented Nov 23 at 19:14
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    $\begingroup$ You're welcome. Note, OTOH it is correct that $1^2 \mathrm{cm}^2 = (0.01)^2 \mathrm{m}^2$. That just means we need to be consistent with our powers, right? It is just that when using numbers, we often evaluate all the operations so that no powers are left in the notation, which can lead to confusions such as the one you are rightly pointing to. But I think that's a bit of a separate issue too... $\endgroup$
    – Amit
    Commented Nov 23 at 19:22

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