# Dimensional analysis, dimensionless quantities and ratios

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.

Lets see if we can make some sense of this confusion.

Start from what does a unit mean.

A unit length, for example, could be

1 meter

1 inch

1 foot

1 yard

1 kilometer

1 mile

1 parasangue

etc.

Units need definition and conversion factors from system to system, particularly if one wants to build a bridge or plan a road.

Ratio's of quantities where the units are eliminated are universal, whether you are talking of meters or parasangues ( an ancient persian equivalent of kilometer length)

Take the perimeter of a circle and divide it by its diameter. Whatever units you may have used to inscribe the circle, the ratio is pi, whether a kilometer diameter or an inch diameter.

Given the radius of a circle, whether it is small, in inches, or huge, in kilometers, one can find the perimeter in the appropriate units by multiplying by 2*pi.

This and similar quantities simplify the work not only for geometers, in map making, engineers and architects, but all scientists.

The same is true for units of weight ( don't let me make a long list of them). The ratio allows easy communication and calculations whether for tons or pounds.

etc.

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?

It is usual to use the word "fractional" with these kinds of quantities--as in "The fractional $\mathrm{CO}_2$ emission rate is ..."--to make it clear that the number represents a fraction of the total.

This actually leaves a little ambiguity in the sense of "Fraction of what?". The mass? The number of moles? Etc. So you should usually specify that as well: "The fractional $\mathrm{CO}_2$ emission rate by mass is ...".

The good news is that for on-going work these is usually a convention and once everybody is aware of it the communication hassle settles out.