It’s been really fun arguing with Keiths recently about dimensionless units. I can’t get enough of the guy lately. I’ve certainly learned a lot in the process.
One thing that I learned along the way, but something that I’ve suspected for a very long time, is that there is such a thing as dimensionless units. I only got confirmation of that suspicion while reading the wiki entry quoted below.
I came up with a guess that when we state angular measurement in radians, that:
Radian = 1
Thus when I say something like
Sin ( π radians) = Sin ( π )
From this wiki entry, my intuition was confirmed because it listed radians as a dimensionless unit:
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is thus a bare number, and is therefore also known as a quantity of dimension one. Dimensionless quantities are widely used in many fields, such as mathematics, physics, engineering, and economics. Numerous well-known quantities, such as π, e, and φ, are dimensionless. By contrast, examples of quantities with dimensions are length, time, and speed, which are measured in dimensional units, such as meter, second and meter/second.
Even though a dimensionless quantity has no physical dimension associated with it, it can still have dimensionless units. To show the quantity being measured (for example mass fraction or mole fraction), it is sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol). The quantity may also be given as a ratio of two different units that have the same dimension (for instance, light years over meters). This may be the case when calculating slopes in graphs, or when making unit conversions. Such notation does not indicate the presence of physical dimensions, and is purely a notational convention. Other common dimensionless units are % (= 0.01), ‰ (= 0.001), ppm (= 10−6), ppb (= 10−9), ppt (= 10−12), angle units (degrees, radians, grad), dalton and mole. Units of number such as the dozen and the gross are also dimensionless.
Curiously the listing of mole surprised me since it is an SI base unit. So I suppose even SI base units like moles can be dimensionless in the formal sense even though they are base units for describing physical systems. I really don’t know how it can be formally argued that moles are actually dimensionless units.
Now even with dimensionless units, we still do dimensional analysis to ensure we’re using the right units. If I had an angle alpha of 45 degrees, I could express it in degrees or radians
α = 45 degrees = 45 degrees x π radians / 180 degrees = 0.25 π radians
The dimensional analysis can be applied to dimensionless units, even though the units are dimensionless. The moral of the story is that just because dimensional analysis can be applied in order to ensure the right numbers for the corresponding units emerge in a calculation, it doesn’t necessarily mean the units involved were ever for physical dimensions in the first place. I suppose one could argue they are sort of conceptual dimensions, but it appears that in formal terms, dimensionless units are not a measure of actual physical dimensions.
Finally, one thing RDFish and I would agree with, that beating puppies simply for enjoying the sense of power is morally reprehensible.
All reasonable people with a normal moral sense agree that puppy torture is evil.
That’s a great insight, imho.