A question on constants and dimensionless equations

[help1please]

Homework Statement

There is not really a problem, just a question which i am about to ask. The known data is

$$\alpha$$ the fine structure constant
$$\hbar$$ Plancks constant
$$c$$ the speed of light
$$\mu$$ permeability

Homework Equations

$$2\alpha \hbar = \pi^2 \mu_0 c$$

The Attempt at a Solution

A paper I downloaded a while back, but can't actually link to states this equation

$$2\alpha \hbar = \pi^2 \mu_0 c$$

in a derivation. It seemed a little odd, are the dimensions right? I noticed all the data in this equation are actually made up of constants. Is it wise to say that when you deal with dimensionless objects, they are always constants (for a separate question).
Thank you!

 

[rude man]

Homework Statement

Is it wise to say that when you deal with dimensionless objects, they are always constants (for a separate question).
Thank you!

No. For example, angles are dimensionless.