What justifies the "canceling out" of the same units? I have difficulty understanding the point of dimensionless quantities. Usually, when you have a concept like mass over volume, which is density, you can state any division as:
On every unit of volume, there comes $n$ units of mass.
And that makes perfect sense. But what I can't understand is the following, when you're trying to find out the mass in a volume which has a specific density, you'd do it like this:
$$m_t = m\frac{V_t}{V}$$
$V_t/V$ yields two of the same units and "cancels out". But I cannot wrap my head around that, physical units have to be pointed to by a finger, they cannot be expressed numerically. A unit of volume has to have a physical representation like three unique meters perpendicular to each other.
If you have a ratio such as:
$$5k/7k = 5/7$$
Its unit is 1, dimensionless, but that's because the actual ratio has been reduced to it. If both $k$ are equal to three, that means their unit size is the same and if you were to resolve it down to a unit of 1, you'd just have to multiply by k, giving $15/21$.
How can you do that for physical quantities? $m^3/m^3$ for example. Meter is something that you have to point to with a finger and say that's a meter.
Even more, let's say a car has an $CO_2$ emission ratio of $50$ grams per kilogram. That makes perfect sense, per every kilogram comes 50 grams. But, if we were to restate the 50 grams as 0.050 kilograms, the ratio would be 0.050.
And what does it mean to say, the emission ratio is 0.050? A dimensionless quantity. On every unit of the denominator there comes 0.050... WHAT?
I'm so confused. Can someone elaborate more on why are dimensionless quantities justified, to qualify it in any way. I'd appreciate it very much.
