Kit. wrote:It could mean, for example, that there is a theory that postulates that the peak performance of an ideal lawnmower is directly proportional to its mass.

Such a theory would involve a constant of proportionality, which would have dimensions of area per unit time per unit mass.

mikrit wrote:What I find weird about dimensional analysis is that one is allowed to multiply apples and oranges to get appleoranges, or divide them to get apples per orange, but never ever add them. Why is multiplication more flexible than addition? And am I allowed to raise an apple to the orangeth power?

One idea that might help is that units are fundamentally multiplicative. When you say "2m" you really are multiplying the dimensionless number 2 by the unit "metre". 4m is twice 2m, i.e. 2x2xm. 4m

^{2} is 2m x 2m = 2 x 2 x m x m. We don't normally write the times sign in there, but it's in there implicitly. Similarly, you sometimes see plots in which some quantity is divided by a unit to give a dimensionless number to plot on the axis: for example you might see an axis labelled "height/m".

If you start adding quantities that have different units, you have something like 2m+2kg; we tend to say that this simply isn't meaningful, but it's more the case that it's not particularly useful because those two terms will remain forever separate. It could still be useful: you might have 2m of licorice and 2kg of chocolate: if there are 10 of you you each have (2m+2kg)/10=20cm+200g. But note how careful you have to be to keep the units as a multiplicative factor in their respective terms.

On the other hand, if the

units have the same

dimensions, you can work with them: if I have 2m+6in, I can note that in=0.0254m, so I have 2m+6x0.0254m=2.1524m. Note how the normal rules of multiplication apply.

Also, just because things have the same units or dimensions doesn't mean that adding them up is meaningful. Having 2kg of chocolate and 2kg of sand is different to having 4kg of gold; if you want to know whether you're over your baggage allowance it's ok to add, but if you want to know what it's worth at today's prices, you need to factor in the price per unit mass before you add.

In terms of raising to a power, my hunch is that it's unlikely that you could ever have a meaningful formula in which the exponent had dimensions, but I'll need to think about it further!

[EDIT: s/your/you're/]