General question about dimensional analysis

[V0ODO0CH1LD]

I am actually going to post the problem that sparked the question I am about to ask, but I don't need help with the answer and that's why I didn't start this thread in the homework and coursework section. Actually, I just realized I read the problem statement in the wrong way, which makes my question pointless with regards to the problem. Anyway, I am still wondering about it independently of weather it helps with the problem or not. So here it it:

The problem stated that an acceleration has magnitude $\mu(r+\frac{a^3}{r^2})$, where $a$ is the initial displacement and $r$ is the distance from the origin. What the problem asked was to check what were the dimensions of $\mu$. When you read it right it's an easy problem, right?

But in the way I had originally read it, it stated: an acceleration has magnitude $\mu(\frac{r + a^3}{r^2})$. Which makes the thing inside the parenthesis look like $\mu(\frac{1}{r}+\frac{a^3}{r^2})$.

Well, $\frac{a^3}{r^2}$ would just be something like $\frac{(am)^3}{(rm)^2}$, where $m$ is some unit of distance and that would simplify to $\frac{a^3}{r^2}m$. Which is fine.

But what would $\frac{1}{rm}$ mean? Like, a dimensionless something per unit of distance? How should I think about $\frac{1}{r}m^{-1}$? Or does it not even exist on the account that I read the problem wrong? At first I was like: well; I have something like 1/r inverse meters, so that must mean I have r regular meters.. But that makes no sense. Is there a correct way to view this?

 

[mathman]

I believe you are overthinking. I assume μ(r+a³/r²) is correct. You state that r and a are distances, so μ has dimension 1/distance x distance/time², so that μ has dimension 1/time².

 

[V0ODO0CH1LD]

I believe you are overthinking.².

Sorry, I tend to do that..

I understood the problem after reading the problem statement correctly, but it still got me wondering what (meter)^-1 means. Not any other unit over meters, just inverse meters. What does it mean if I say 3m^-1? Is it undefined?

 

[Nytik]

Depends on the context. The wavenumber of a wave has units of m^-1; that is, the number of wavelengths per unit distance.

 

[mikeph]

The meaning is left for you to decide/discover, but whatever you find must reconcile with the fact that there are three of them per metre.

It might be three full wavelengths of EM radiation per metre, and you'd be specifying light in the UHF frequency band, or it might refer to the linear density of turtles as they follow a one-dimensional path towards the sea.

Whatever it is, there are three of them in a metre.