Exploring Alternate Dimensions: A Physical Inquiry

[RoyalCat]

Well, it's more of a general inquiry than a specific question, but this looked like as good a place as any to bring it up.

Can a quantity have dimensions NOT of the form: $$[L]^{x}[M]^{y}[T]^{z}$$, where x, y and z are real numbers?

This includes two primary cases as far as I can see. One is where x, y or z are complex numbers with a non-zero imaginary component, and the other, is where the dimension is the product of a function.
That doesn't sound too clear, I know, here's an example:

Let K be a physical quantity.
$$[K] = [ln(L)]$$

Would such a size have any physical meaning? Are there any cases where such quantities do come into play?

On a related note, can functions (Such as $$cos(x), ln(x), e^{x}$$) receive values that are not pure numbers, where $$x$$ has dimensions?

It seems like it would be plausible, for instance, if there's a system whose displacement is given by a function of the form:
$$x(t)=e^{kt}$$
$$[k]=[ln(L)][T]^{-1}$$

But are there any examples of such functions with a physical meaning that are not artificially constructed to demonstrate the point I've been trying to make?

Thanks in advance, Anatoli. :)

 

[Pengwuino]

Functions like the arguments of exponentials, logarithms , and any other transcendental equation must be pure numbers. Consider sin(x) - If you look at the taylor series, you'll have x^2, x^3, x^4 terms and if they do indeed have dimensions, you'll be trying to add up quantities of different dimensions which isn't valid. Also, something doesn't have to have units of mass, time, and length. You can add to that list charge, tesla, farad, etc etc.

 

[RoyalCat]

Functions like the arguments of exponentials, logarithms , and any other transcendental equation must be pure numbers. Consider sin(x) - If you look at the taylor series, you'll have x^2, x^3, x^4 terms and if they do indeed have dimensions, you'll be trying to add up quantities of different dimensions which isn't valid. Also, something doesn't have to have units of mass, time, and length. You can add to that list charge, tesla, farad, etc etc.

Ah, yes, yes, my mistake for missing out on the 4 other fundamental SI units.

I see what you're getting at, but what if I have a quantity the dimensions are which are, to use your example of $$sin(x)$$, $$[sin^{-1}(L)]$$?
Would the same logic apply there since you cannot, in fact, derive a quantity with such dimensions since it would be a hodge-podge sum of $$[L]^{n}$$ with $$n$$ running to infinity, and as a result, not have dimensions?

 

[Pengwuino]

sin(x) is dimensionless quantity, thus it's inverse is dimensionless. The logic would follow using the inverse as well.

 

[RoyalCat]

sin(x) is dimensionless quantity, thus it's inverse is dimensionless. The logic would follow using the inverse as well.

Okay then, thank you very much! :)