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B Why I think radians should *not* be dimensionless

  1. Dec 3, 2016 #1
    As most know and rest can find out, radians are a dimensionless unit. Actually, it's even worse: angle in general is a dimensionless quantity. That stems from it being defined as a ratio of the arc versus the radius. I now present my opinion of disagreement to this concept.

    In the strict scientific core of things, I believe no opinion is right or wrong. Units of measurement are there to help us. What does really make them dimensionless or not? Common examples like the aforementioned angle definition, or another classic I like: the efficiency of something, may at times seem "right" to define as dimensionless, their definition including ratios of same units and all. But is it all really not just our choice of convenience?

    When doing nuclear physics homework at university, very frequently we assumed c=1 and dimensionless. The results weren't wrong. Obviously some units needed appropriate adjustments afterwards....to return to SI. But it's not like we hadn't already fully understood (after a bit getting used to) what the results said.

    How about the second (s). Imagine a place full of strange sentient robots with processors that run at 1Hz. Their perception of life may be so linked with this unit, the second, that they might just as well not define any unit for time...it's just 1. Everyone understands this and the full laws of physics having it dimensionless.

    If until this point we have an agreement that dimensions are actually subjective, I can proceed to my humble personal view: it sits better with me that angle is not dimensionless. Rads, degrees....makes perfect sense. Notice how, even while saying angle is dimensionless, we have units for it? Isn't it a little strange? Or even suspicious? I can hardly believe, when talking about angle, that one TRULY thinks of it as a ratio. I think of it as..a wideness thing, an openess thing, a cheese pie piece thing. Contrast this to efficiency: J/J? W/W? Sounds messy. 100%? Yeah baby! 5%? Aw come on! Interestingly enough, although efficiency is a "quantity", there are no "units" of it around.

    So what's everybody's view on the subject?
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  3. Dec 3, 2016 #2


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  4. Dec 4, 2016 #3


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    I wouldn't use the word "subjective". I would say "conventional". The dimensionality of a quantity is a matter of convention for a given system of units.

    The BIPM (the organization in charge of defining the SI conventions) had decided that angles are dimensionless. You can certainly choose a different convention, but it isn't SI.

    Personally, when I am doing dimensional analysis I often consider angles to have a dimension. But not when I am writing a paper or communicating results, then I use SI units.
  5. Dec 5, 2016 #4


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    This might be partly because the use of pi in radian angles. If we would express angles as the fraction of a full circle (rather than half a circle), the bare numeric values would be more intuitive.
  6. Dec 15, 2016 #5
    If I am not mistaken, angles do have dimension, particularly in non-Cartesian coordinate systems.

    What makes them dimensionless is the the fact they can't be described by base units; angles are ratios of the base units of length. That is to say, any fundamental quantity can be expressed without them. For example, luminosity is the amount of energy released per time and it does not concern itself with where the energy is being released. However, we may observe the radiant intensity which includes units of solid angle^-1. We use this observation for convenience which can then be used to express the luminosity of the object. Treating them as a dimension can make observations or calculations possible (that is why we give them units), but they are not fundamental.

    c is a constant and can be treated dimensionless however it is not unit-less. It will still have units of m/s in SI convention. Per wikipedia on Gaussian units: "As another example, quantities that are dimensionless (loosely "unitless") in one system may have dimension in another."

    I would imagine these robots would have a hard time grasping special relativity. Instead of seconds, I imagine they would invent some unit such as "per computation" so that they could measure velocities accurately.

    After my introductory quantum class, everything seems a little strange. :-p
  7. Dec 15, 2016 #6
    There is a very good reason for the units assigned to an angle. An angle is most simply defined as the number of revolutions. That is measured by the ratio of a) the arclength along a circle (according to the number of revolutions, which is often not an integer), and b) the arclength, along the same circle, of one revolution.

    By dividing two quantities having length as their dimension, one gets a so-called "dimensionless" quantity of angle. But that is definitely not because dimensions have been ignored when taking account of an angle, but because they have been taken into consideration.

    Perhaps for maximum clarity, we ought to be saying that the units of an angle are length0.
  8. Dec 22, 2016 #7

    Stephen Tashi

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    Are you using the word "angle" to refer to a particular type of physical phenomena? - or are you using it to refer to a property of a particular type of physical phenomena ?

    A big semantic confusion in such discussions is whether "angle" is used to denote a physical phenomena or whether it is used to denote a property.

    Consider the similar example of the term "mass". It's common to read problems in textbooks that say things like "A mass of 2 kg is resting on an inclined plane...". But the property "2 kg mass" can't rest on an inclined plane. A teapot or a circular saw or some other physical object can rest on an inclined plane and have the property that its mass is 2 kg.

    The semantics is further complicated by the fact that different properties of a physical situation can be measured in the same units. (For example, 3 newton meters , could quantify the property of "work" or the property of "torque". Likewise, a property measured by meters/sec isn't necessarily the velocity of something, etc. ) As @zinq indicates, the fact that a unit of measure is dimensionless doesn't imply that all dimensionless units refer to the same physical property.

    If we wish to consider the question of whether an "angle" is dimensionless, we should clarify whether "angle" refers to a physical situation ( e.g. a ladder leaning against a wall, a rotating wheel , etc.) or whether "angle" refers to a particular property of a physical situation (i.e. a "dimension" in the sense of dimensional analysis).

    For example, in the case of a rotating wheel, we might be interested in the property of "the angle a radius sweeps out in in 2 seconds" and that angle might be, say, 780 deg, which we would consider a different physical result than 60 degrees. But in the case of a ladder leaning against the wall, in measuring the property we call "the angle the ladder makes with the wall", we would consider 780 deg the same result as 60 deg.
  9. Dec 22, 2016 #8
    I agree with Stephen Tashi that there are basically two types of angles to consider: a) the angle between two rays in a plane having a common origin, and b) the total angle swept out by a ray in a plane rotating about its origin, perhaps over a specified period of time. (Or a situation similar to b), like how many times a wire has been wrapped about a cylinder.)

    Cases a) and b) are certainly distinct geometrical concepts. But in my opinion they are not philosophically distinct kinds of things. (So I would not call one "referring to a physical situation" and the other, by contrast, a "property".) Possibly good terminology would be a) angle and b) total angle. But it's hard to change long-established ambiguity!

    Either one of these cases a) and b) would nevertheless be measured in terms that describe the number of revolutions that is relevant. (Whether the units be radians, degrees, revolutions, or something else.) In the case a), this number will always be at least 0 but less than 1. In the case b), this number could be any real number (assuming the plane in question has been given an orientation).
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