# Dimensionless units, RDFish is a Genius

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:

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:

http://en.wikipedia.org/wiki/Dimensionless_quantity

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.[1] 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.

RDFish is an Idiot

That’s a great insight, imho.

## 13 thoughts on “Dimensionless units, RDFish is a Genius”

1. […] beating puppies simply for enjoying the sense of power is morally reprehensible.

Now, which bearded gent could you possibly have in mind when you say that?

2. 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.

Sal is referring to his attempt to defend this statement:

NO! J/K is dimensionless! It only indicates the method used to count the energy microstates.

The discussion starts here, and of course it goes very badly for Sal.

3. Sal:

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.

A mole is dimensionless for the same reason that a dozen is dimensionless. A mole is 6.022 x 10^23 atoms, and ‘a dozen atoms’ is twelve atoms.

4. Running off and starting a new thread to try and points score is shitty, IMHO. Although you may feel there no longer is a downside to your reputation, Sal, I assure you there is a bountiful upside.

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

That’s all very entertaining. However, it completely misses the point.

If you measure an angle in degrees, that’s dimensionless too. That, by default we assume radians is merely a useful convention. It works well in mathematics, because there is a nice infinite series for the sin function when used with radians.

π is dimensionless because it is a ratio. It doesn’t matter whether I measure in centimeters or in inches, I get the same ratio.

I could introduce FunnyPi as the ratio of the diameter of a circle measured in inches, and the circumference measured in centimeters. FunnyPi would still be dimensionless, because it is still a ratio. But it would not be independent of the units on which it is based. So FunnyPi would be a not-very-useful ratio.

You are confusing dimensionless with independence of the units used.

Shannon entropy is perhaps a bit confusing itself. As a measure of information (i.e. abstract information), it is dimensionless and independent of units. But if we are discussing the entropy of an actual communication system, then the units of physical encoding are part of the protocol for that communication system, and Shannon entropy won’t be independent of units of physical encoding.

6. Rich,

Running off and starting a new thread to try and points score is shitty, IMHO.

It is, but that’s Sal’s MO. He wants to put as much distance between himself and the evidence as possible.

7. The discussion starts here, and of course it goes very badly for Sal.

Thanks for highlighting the discussion. But the reality is Keiths got taken to the woodshed several times. Why?

He thinks the following relation is false, he accused me of making it up, of being sloppy, etc. He proudly condemned it as something he refuses to accept:

1 nat = 1.381 x 10^-23 J/K

But then I showed him an entropy conversion calculator that utilize exactly that relation.

2LOT and ID entropy calculations (editorial corrections welcome)

But he seems to be enjoying the continued humiliation. On top of this, when he sort of concedes the relation as valid (he flip flops on its veracity):

1 nat = 1.381 x 10^-23 J/K

he insists the left hand side of the equality is dimensionless while the right hand side is not. He keeps pleading I admit some sort of mistake. He’s failed to demonstrate my error, and in the process put forward several erroneous conclusions which I shot down such as his “correct” equation:

1 nat = (1.381x 10^-23 nat/(J/K))(J/K)

I pointed out his “correct” equation leads to an absurdity.

(1.381x 10^-23 nat/(J/K)) (J/K) = 1.381x 10^-23 nat

but
1 nat ≠ 1.381x 10^-23 nat

Hence his corrected equation disagreed by a factor of 7.24 x 10^22. He finally, in the face of overwhelming evidence, had to say this to me for the first time in all eternity:

You’re right.

Whoa!

8. Poor Sal, trying to spin his way out of another huge embarrassment.

Sal, people can read, you know. Anyone who has followed that link knows that you’re bluffing.

9. Keiths gave this howler:

a dimensionless quantity can be converted to a dimensioned one, and vice-versa.

My copper ingot example shows exactly how it is done.

So what is his copper ingot example? And how is this miracle done? 🙂

I want to be able to convert back and forth between the total mass of the ingots, which is a dimensioned quantity, and the number of ingots, which is dimensionless.

You can only relate the number of ingots to their mass, you can’t convert ingots to their mass. That is a nonsensical statement.

But that doesn’t stop Keiths from trying:

How do I do it? Easy. I set up a conversion factor of 1/(10 kg). Multiplying the total mass by the conversion factor will give me the number of ingots

Keiths uses a conversion factor that is dimensioned, i.e. 1/ 10kg, that is not a dimensionless conversion factor. FAIL!

There is an important rule of conversion factors, namely, they should be dimensionless:

Conversion factors are dimensionless numbers which inter-relate, or convert, different units of the same entity. Thus ‘100 centimetres/metre’ and ’60 second/minute’ are familiar conversion factors – their dimensions are respectively [L]/[L] = [ ] and [T]/[T] = [ ] where ‘T’ is the entity time.

He then concocted this example and attributed the idea to me, which is totally false. But any way, here are the relevant points:

2LOT and ID entropy calculations (editorial corrections welcome)

200 ingots = 2000 kg

200 ingots/2000 = kg

The left hand side is dimensionless.

So he says the LHS is dimensionless. Ok, kg are a measure of mass, therefore they a not dimensionless. But this is a meaningless equation. Think I’m being unfair, he did say, his conversion factor was 1/10kg to be applied to total mass and gave the number ingots. That would actually mean the factor is ingot/10kg.

Thus using his formula:

2000 kg = 2000 kg (ingot/10kg) = 200 ingot

thus
200 ingots = 2000kg

Keiths fails to realize this is a meaningless equation because the LHS if dimensionless and the RHS is dimensioned.

Any physically meaningful equation (and any inequality and inequation) must have the same dimensions on the left and right sides.

OUCH! 🙂

10. Well, well, well, look at this Wiki entry:

nat ≡ kB = 1.380 650 5(23)×10^−23 J/K

http://en.wikipedia.org/wiki/Conversion_of_units

Does that look familiar to Keiths?

I mean, I confess I rounded it off a bit, but it looks like something I’ve been sayin’ all along

1 nat = 1.381 x 10^-23 J/K

but nevertheless Kieth said of this:

So the answer is no, I do not agree with your sloppy equation.

My sloppy equation? Keiths doesn’t agree with an accepted equation in physics. There it is in the conversion tables of Wikipedia, and he insists its my sloppy equation. Willful ignorant blindness. That means Keiths does not agree with this:

1 nat = 1.381 x 10^-23 J/K

despite numerous evidences.

This is an accepted meaningful physical equation:

1 nat = 1.381 x 10^-23 J/K

For it to meaningful, the LHS and RHS cannot be of different dimension, and since the LHS is dimensionless, by inference the RHS must be dimensionless. As it says here:

Any physically meaningful equation (and any inequality and inequation) must have the same dimensions on the left and right sides.

End result, Keiths remains in a state of willful ignorance. In his own words he has said he doesn’t accept this:

1 nat = 1.381 x 10^-23 J/K

So much for your willingness to acknowledge even the most simple truth. Saving face to you is more important than saying what is true. You would have gladly had me go around and declare the above equation as false and sloppy so you could save face. You kept demanding a retraction Keiths. No dice. Wallow in your state of denial and willful ignorance, Keiths.

11. Sal, in order for this conversion to make any sense you have to use a system of units in which kB can be normalised to 1 by effectively measuring temperature in units of energy (whereas thermodynamic entropy is effectively measured in nats). If you measure length, mass, time, electric charge and temperature in Planck units, the fundamental proportionality constants (c, G, h/, ke, kB) are set to 1, simplifying equations. But this is only a convention. It doesn’t mean that the constants are really dimensionless. The Planck units of temperature and energy have these values in SI terms:

Tp = 1.417e+32 K

Ep = 1.956e+9 J

kB = Ep/Tp = 1 (according to Planck’s “nondimentionalising” convention) = 1.38e-23 J/K (in SI units)