Quote by twofishquant
I think we are have different definitions of what a "scalar" is. I'm defining it as a quantity that doesn't change when you change coordinate systems. I measure something in my coordinate system. You measure something in your coordinate system. We get the same number. There are some things that you can measure that have that characteristic (electric charge if you vary only space and time coordinates). There are some things that you can measure that *don't* have that characteristic (volume). Classifying things according to how they behave turns out to be useful.

That is my definition also.
Once you have measured something the observation is a scalar. If you perform some experiment and the number 7.43 pops out on your measuring device then no change of coordinate systems can possibly change that number to anything other than 7.43. Therefore, the number measured is a scalar.
It may be that you claim that 7.43 is a length and I disagree, but regardless of how we interpret the number in terms of physical quantities in our favorite coordinate system, we will agree that the number is the same. That makes it a scalar.