Radian describes as the length of the arc divided by the radius of the arc, θ = s / r (Wikipedia) its very cleary that θ is not dimensional because its the ratio between two numbers of the same unit( arc length to radius length), but then why θ has a unit(radian)? in digital communication, we have SNR(signal power to noise power) the SNR is unitless, why θ is different?
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Just because a quantity is dimensionless, doesn't mean it's not convenient to give it a "unit". It would be weird to call a subtended angle "$0.26$" when it's $0.26$ radians, if only because we're used to every quantity telling you what it's $0.26$ of. You certainly wouldn't want to confuse radians with the steradians of solid angle.
Angles aren't alone in this. Toxicity is often measured in LD50, the amount needed to kill 50% of specimens. But it's measured relative to body mass, e.g. as 3 mg/kg as opposed to just saying $3\times 10^{-6}$.
J.G.
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2I wish I could upvote this answer more than once, because it was great. +1 – user729424 Feb 01 '20 at 17:50
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1(+1). An example I always found interesting was that torque / the moment of a rotating force has the same dimension as energy but is usually expressed with different units (Newton-metres vs. Joules) since they are conceptually different. – Jam Feb 01 '20 at 17:58
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6@Jam Ah, that's the other issue: convenient units to characterize dimensionful quantities. My favourite example: you'd rather describe the power collected in sunlight per unit area by a solar panel in Watts per square metre, not kilograms per cubic second. – J.G. Feb 01 '20 at 18:05
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if the θ wants not to be confused with the degree, why is θ not defined as the length of the arc measured by radius, arc(θ)=x radius, so 180 degrees = pi radius. ? – Computer_guy11 Feb 01 '20 at 18:26
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@Computer_guy11 Because it's still useful as a concept of 'angle' despite the pitfall of needing a unit. Sometimes we want to refer to angles without referring to arcs. – Jam Feb 01 '20 at 18:40
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1@Jam : Which may be a good reason to consider the radian as a unit, because the "mystery" behind torque's unit is conceptually clearer if one writes it as $\mathrm{J/rad}$. When you multiply a torque by the angle it turns through, you get work done (energy used, which should be measured in joules), just as when you multiply a force by the distance it pushes over, you likewise get work done - and sure enough, "newtons" are the same as $\mathrm{J/m}$, but length is dimensional. – The_Sympathizer Feb 02 '20 at 02:07
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But, in fact, .. SNR is measured in decibel (or equivalent), just to specify that the figure given is not in Neper or else.
So it is for the radian, just to clear that you are speaking of degrees, etc.
G Cab
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I think the passage you are reading is slightly misleading. It talks as if a thing is inherently dimensionless, when actually it is determined with respect to a given physical quantity.
I'd like to make the point that "dimensionless" does not necessarily mean "can't be given in units."
Let's consider wikipedia's definition of "dimensionless":
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned.
It also goes on to talk about commensurable quantities.
It is true that length is a physical dimension and that the ratio between two lengths in the same unit would be considered a "dimensionless" quantity in terms of length.
But then again, radians can be considered a sort of measure of angle, commensurable with other measures of angles. That's why one could convert between the units "radians" and "degrees". Comparing units is, essentially, assigning the first unit to $1$ on the real number line, and a second number on the number line which is the measure of the second unit on this real line. I don't know if "angle" is considered a physical dimension or not (it seems like it to me, but maybe technically it is not.)
So, being a real number, it can be compared with other measures of angles using different units. Now, a ratio of two angles given in radians would again be "dimensionless" in terms of a measure of angles, but conceivably it too could be a useful unit of something other than an angle.
rschwieb
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tl;dr– Units are always optional. You can omit them if you want, or invent further unit qualifications if you care to. Radians aren't inherently unitless, and recognizing their units can be useful.
Units could be removed from everything.
We don't need to use units. Like, at all, ever.
For example, we could rewrite all of physics, economics, etc. without units. In fact, most mathematical software works this way, stripping off units and just using dimensionless quantities for everything.
It'd be annoying, though. For example, someone could weigh $`` 5 ",$ and be $`` 7 "$ tall, $`` 9 "$ old, and have $`` 11 "$ in the bank; the absence of units would make it easier to confuse numbers, plus it'd force us to, e.g., measure all lengths in meters (or else otherwise qualify different length systems).
Units can be added to most anything.
You're asking if radians are unitless because they're arc-length over radial-length, right? And presumably arc-length and radial-length both have units of length?
Bah! That's just one way of doing it!
To draw an analogy, put yourself in the shoes of an engineer working with a power generator. Power generation efficiency is often described in terms of how much electricity you get out vs. how much heat you put in. And like with your radial logic, that's power-over-power, making it unitless, right? And some engineers write 'em that way, saying, e.g., $\eta = 0.4 .$
But instead of using units of $`` \mathrm{W} "$ for power of all types, we can further qualify thermal energy vs. electrical energy. We can say, for example, a power generator takes $`` 100 \, {\mathrm{W}}_{\text{T}} "$ of thermal power to produce $`` 40 \, {\mathrm{W}}_{\text{E}} "$ of electrical power. And now, $\eta = 0.4 \frac{{\mathrm{W}}_{\text{E}}}{{\mathrm{W}}_{\text{T}}} .$
See? We just took a "dimensionless" quantity and made it dimension-having. Because doing so can be useful in avoiding errors and effectively communicating.
Likewise, why must radians be "dimensionless"? We can say that an arc-length is measured in units of ${\mathrm{m}}_{\text{arc}}$ while radial-length is measured in units of ${\mathrm{m}}_{\text{radial}} .$ And while that's not a standard notation, and I'm not suggesting that anyone start using it, my point's that the choice about whether or not to use such notation is 100% arbitrary language convention. So, it follows that radians being "unitless" is a consequence of an arbitrary convention, making it similarly arbitrary.
Ideally, unit logic should be kept simple.
As noted above, you can just invent units whenever doing so is useful. But, when doing so, it's best to consider how to make unit logic useful.
We usually try to make them obey the normal rules of algebra, e.g. $\frac{\mathrm{unit}}{\mathrm{unit}} = 1 .$ And we try to keep conversions clean, e.g. $\frac{1\,\mathrm{m}}{1\,\mathrm{cm}}=100 .$
Though not all units are completely well-behaved. For example, $\frac{90 \sideset{^\circ}{}{\mathrm{C}}}{45 \sideset{^\circ}{}{\mathrm{C}}} \neq 2 ,$ since degreed units of temperature aren't based at zero. And with angles, $45{}^{\circ} = 405{}^{\circ} .$
If you come up with your own units, there's no strict requirement to keep them simple, but in practice it's pretty rare to use a unit logic more complex than, say, common temperature scales.
Typically, I find sub-typing existing units to be useful in communication. For example, most folks who understand the engineering scenario above will get what $`` {\mathrm{W}}_{\text{thermal}} "$ means, even if they haven't seen it before. This is analogous to type-inheritance, where $`` {\mathrm{W}}_{\text{thermal}} "$ is a sub-type of $`` \mathrm{W} "$ (more in the next section).
Tangential: Connection to type theory.
As programmers know, computers often store integers as 32-bit int's or 64-bit long's. They're conceptually the same thing, just the larger variants have a larger range of values while taking more space and potentially being slower in calculations.
There're also "unsigned" variants that can't store negative values.
One nice feature of unsigned integers is that, if you have one, you can assume that it's not negative – because it can't be.
And in C#, they're looking at adding non-nullable types, which we can ignore the potential for being null because they can't be.
The point here is that types can be invented to avoid errors, just like with non-negative integers and non-null references. These features aren't actually necessary, just they offer type-checking protection.
And that's one of the nice things we get from units, which are largely a typing logic. For example, when an engineer writes down $`` 10 \, {\mathrm{W}}_{\text{T}} " ,$ that "thermal"-qualifier helps avoid mistaking the figure for being a different value related to, e.g., electricity. And even if we omitted that qualifier, that "Watt"-qualifier helps to avoid mistaking the figure for some other sort of quantity.
In short, further qualifying units is like using stronger typing in programming.
