# Fundamental question about dimensional analysis

Let me admit beforehand that this is quite possibly a very stupid question. I was also uncertain of where to post this question, as it doesn't fit cleanly into either physics or math stackexchange.

In dimensional analysis, it does not make sense to, for instance, add together two numbers with different units together. Nor does it make sense to exponentiate two numbers with different units (or for that matter, with units at all) together; these expressions make no sense:

$$(5 m)^{7 s}$$

$$(14 A)^{3 A}$$

Now my question is plainly this: why do they not make sense? Why does only multiplying together numbers with units make sense, and not, for instance, exponentiating them together? I understand that raising a number with a unit to the power of another number with a unit is quite unintuitive - however, that's not really a good reason, is it?

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Another way to look at it

$$e^x = \sum_n \frac{x^{n}}{n!} = 1 + x +\frac{x^2}{2} + \dots$$

which comes down to adding quantities with different dimension, which you have already accepted makes no sense. This is why you can't exponentiate values with units.

And we can do a similar thing with most transcendental functions.

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Just a trivial addition: a general power $x^y$ may be written as $\exp(y \ln x)$ so it has the same problem if $y$ fails to be dimensionless. ... In a similar way, the exponents should always be Grassmann-even (not "fermionic"), and so on. –  Luboš Motl Mar 28 '11 at 6:44
Thank you for a very intuitive explanation! –  Jubilee Mar 28 '11 at 13:27

One further point to note, is that strictly one is just saying that the exponent is dimensionless, not that it does not contain expressions with dimension. So for example we could have some expression like $X=a^{(E/E_0)}$ where the exponent for a is a ratio of energies.

There are several restrictions on the space (sometimes viewed as a vector space) of dimensional quantities: for example units are raised to rational, but not irrational values. This allows a theorem: The Buckingham $\Pi$ Theorem to form.

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Because of the way an exponential is defined. By an expression like $a^b$ we mean to say that the quantity $a$ is multiplied $b$ times with itself. Thus an expression like $(5m)^{7s}$ would mean $5m$ multiplied "7 seconds" times with itself, which is meaningless.