B What is the difference between "Unitless" and "Dimensionless"?

[jaumzaum]

Hello! I'm having trouble to understand the difference between unitless and dimensionless. Many people on the internet says different things, so I need someone to really confirm me this, because I'm turning crazy kkkk.

So, let's start with temperature. I say T =400 K.

1) What is 400? Is it the unit and is that what the "unit" in unitless stands for?
2) What is Kelvin? I know it is the unit of measure, but is that what the unit in unitless stands for?
3) Also, temperature is a scalar quantity because it is not a vector. Is t classification of a vector/scalar quantity a third classification, or is "scalar" the same as dimensionless or unitless?

 

[hutchphd]

I don't know what a "unitless" quantity is. Please describe what you mean by example or otherwise...
A "dimensionless" number describes the ratio of amount of similar things: atoms, apples, pounds, speed, Mohicans. Such a number will not change if you change your units of measurement.

 

[etotheipi]

1) What is 400? Is it the unit and is that what the "unit" in unitless stands for?
2) What is Kelvin? I know it is the unit of measure, but is that what the unit in unitless stands for?

I think you're getting confused about general dimensional analysis. Our very own @Orodruin has a nice discussion of dimensions and units in his textbook. The gist of it is this,

For each independent physical dimension $\mathrm{X}$ we choose a base unit, $u_X$. For instance, in the SI, the dimension 'length, $L$' is assigned a base unit of metres, $u_L = \text{m}$. A physical quantity $p$ that has dimensions of $X$ is a product of a measured value $\lambda_p$ (a dimensionless number) and the corresponding base unit, i.e. $p = \lambda_p u_X$. Using the same example as before, you might say $\mathscr{l} = 3\text{m}$. But you can also have derived quantities whose dimensions are products of powers of the base dimensions, in which case the dimensions $[p]$ and derived unit $u_p$ respectively of this quantity are $$[p] = \prod_i {X_i}^{\alpha_i} \,, \quad \quad u_p = \prod_i {u_{X_i}}^{\alpha_i}$$For instance, velocity has dimensions $[v] = \text{L}^1\text{T}^{-1}$, whilst its derived unit in the SI is $u_v = \text{m}^{1}\text{s}^{-1}$. Another important thing is that for any physical dimension there are infinitely many possible choices of base units. It's important that the physical quantity is independent of whatever choice of base units:$$p = \lambda_{p} u_p = \lambda_{p}' u_p'$$That's basically just saying something like $\mathscr{l} = 1\text{m} = 3.28 \text{ft}$.

So when you write $T = 400\text{K}$, all it means is that the "$400$" part is the measured value of temperature in Kelvins, and is a dimensionless number, whilst the $\text{K}$ part is the unit. The product of the two gives you a dimensional, unitful quantity $T$. Temperatures are actually a bit funny when it comes to units, but that's a whole other story

On the other hand, a dimensionless quantity is one for which the exponents $\alpha_i$ in the products shown above are all zero. Dimensionless quantities do not carry units, and they're scale invariant.

Also, temperature is a scalar quantity because it is not a vector. Is t classification of a vector/scalar quantity a third classification, or is "scalar" the same as dimensionless or unitless?

A scalar could be dimensionless or dimensional. It could be "the number of molecules of oxygen within a $5\mathrm{cm}$ radius of a point", which doesn't have dimensions, or "the mass of a particle", which does have dimensions.

The definition of a scalar is just a coordinate independent number,$$\phi(x) = \phi'(x')$$

 

[Nugatory]

can you show us an example in which there seems to be a distinction between “dimensionless” and “unitless”?

 

[kuruman]

A plain 10 is dimensionless and (I suppose) unitless in the sense that you can attach any dimension and, once you do that, you can attach any unit. Usually, the next step, if there is one, is to specify what kind of beast this "10" is, e.g. length, force, mass, etc. After that one specifies the units, e.g. if length, meters, light years, rods, etc. Most of the time the two specifications are done simultaneously and the dimensions are inferred from the units, e.g. $\mathrm{9.8~\frac{m/s}{s}}.$

Note that you can infer the dimensions from the units but not the units from the dimensions. Furthermore, a dimensionless quantity is not necessarily unitless. For example, $\mathrm {10~\frac{inches}{mile}}$ is dimensionless but not unitless.

In summary, I think writing down 400 without anything else signals that the number is unitless and dimensionless. Why put something next to it if there is nothing to put? I would love to hear from you what the "many people on the internet" have to say on this.

 

[A.T.]

For example, 10 inches/mile is dimensionless but not unitless.

But you can simplify this to the unitless number 1/6336. So for dimensionless numbers unitless vs. not unitless is just a matter of notation.

 

[vanhees71]

To me the expression "unitless" doesn't make sense at all. Any physical quantity has a unit. Though it can be "dimensionless" like an angle, which is the ratio of two lengths if measured in radians. Of course you can also decide to introduce a new unit like the degree (right angle $90^{\circ}$).

In principle there's even the possibility to introduce a system of units where all quantities are dimensionless (Planck units).

The new SI (valid since 2019) defines its units almost completely by this idea, i.e., it defines numbers to the fundamental constants. Only the Gravitational constant cannot be measured accurately enough, such that for the time unit one still uses a material-dependent though very accurate quantity, the hyperfine transition in Cs atoms in use for that purpose since the mid 1960ies.

 

[kuruman]

But you can simplify this to the unitless number 1/6336. So for dimensionless numbers unitless vs. not unitless is just a matter of notation.

Sure, but this is an example of useful dimensionless numbers. A ratio of $\mathrm{10~\frac{feet}{mile}}$ conveys more practically useful information than $\frac{1}{528}$ to a farmer digging a sloped irrigation ditch. The same idea of practicality applies to quantities with units. How useful is it to a motorist driving from New York to Boston to know that the distance is 2.0×10⁴⁰ Planck lengths with an expected duration (depending on driving conditions) of 2.3×10⁴⁷ Planck times?