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haruspex submitted a new PF Insights post
Can Angles be Assigned a Dimension?
Continue reading the Original PF Insights Post.
Can Angles be Assigned a Dimension?
Continue reading the Original PF Insights Post.
That is not a claim, it is part of the definition of Θ.robphy said:What is the justification of the claim: "The cross product operator also has dimension Θ"?
I'm not especially attached to the part relating to i. It is independent of the rest and probably needs more thought. There might be a way around the 1+i problem similar to how I resolved s = rθ, i.e. one would have to agree to treating complex algebra in a slightly different way.robphy said:If i carries units, is there any meaning to (say) 1+i?
That's why I assigned i the dimension Θ, to make iθ dimensionless.robphy said:Note: in exp (x), the x must be dimensionless
That sounds very much as though it is not a new idea, which is at once heartening and disappointing. Thanks for the reference.robphy said:You might be interested in this old article from the American Journal of Physics.
http://scitation.aip.org/content/aapt/journal/ajp/65/7/10.1119/1.18616
"Angles—Let’s treat them squarely" by K. R. Brownstein
By definition, real numbers are dimensionless, so I do not understand what you mean by saying it is a true dimension. Indeed, the fact that angles have units but not dimension is somewhat awkward, as I mentioned in the article.A. Neumaier said:Angles have the dimension of 1. That this is a true dimension
Not if you redefine trig functions as taking arguments of dimension Θ, as I did.A. Neumaier said:for s=sinθ to make sense, the dimension of θ must be 1.
I don't understand your point. They are normally considered dimensionless anyway; I'm looking for a way to give them dimension.Ygggdrasil said:Angles can be defined as dimensionless quantities if one thinks of them as fractions of a circle (multiplied by the constant 2π).
haruspex said:I don't understand your point. They are normally considered dimensionless anyway; I'm looking for a way to give them dimension.
Would thinking of all masses as fractions of 1kg make mass dimensionless?
Fair question.Ygggdrasil said:I guess I don't get the point of trying to give angles a dimension. Angles are defined as a ratio (arc length : circumference) which is a dimensionless quantity and seems fundamentally different than something like mass.
the concept of larger and smaller for like quantities (if there exists a finite B such that A+B=C then C > A)
A ratio can have no dimension since it must be a ratio of two things of the same dimension. But at an angle is not a ratio. You can say it is a certain fraction of a complete circle, but whether that has dimension depends on whether you consider the complete circle as having a dimension. You are not used to thinking of it that way, but that does not mean it cannot be done.Baluncore said:Angles are ratios, parts of a circle.
Not if the dimension has the unusual property that it becomes dimensionless when raised to some finite power. The ϑ2=1 axiom means that a polynomial function of an angle is fine if all the terms are even powers (dimensionless result) or all odd powers (result of dimension ϑ).Baluncore said:Polynomials and dimensions are incompatible.
Which circle are you referring to?haruspex said:A ratio can have no dimension since it must be a ratio of two things of the same dimension. But at an angle is not a ratio. You can say it is a certain fraction of a complete circle, but whether that has dimension depends on whether you consider the complete circle as having a dimension. You are not used to thinking of it that way, but that does not mean it cannot be done.
Baluncore said:Polynomials and dimensions are incompatible. Transcendental functions that are approximated by polynomials must have dimensionless inputs and outputs.
You could also ask "The unit circle with center (0,0)? The unit circle with center (15,12)?"robphy said:Which circle are you referring to?
The unit circle? Or maybe the circle of radius 7?
You have to use a circle with its center at the vertex of the angle, so the measurement process isn't really independent of which circle is used unless we think of "a circle" as a portable measuring instrument, just as we think of a meter stick as portable measuring instrument.One feature of the angle measure (defined as the ratio of circular-arc-length to radius) is that it is independent of the circle used to make that measurement.
In this general discussion, one needs to distinguish an "angle" from an "angle measure".
Stephen Tashi said:You could also ask "The unit circle with center (0,0)? The unit circle with center (15,12)?"You have to use a circle with its center at the vertex of the angle, so the measurement process isn't really independent of which circle is used unless we think of "a circle" as a portable measuring instrument, just as we think of a meter stick as portable measuring instrument.
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I agree. It's the distinction between "a dimension" (e.g. length) and "a unit of measure" (e.g. meters).
robphy said:Before somehow specifying an angle-measure, one could talk about all sorts of properties of angles at this stage. Then, when introducing an angle-measure, it probably should be explicitly defined---maybe operationally.
Let me see if I understand you viewpoint.But all of this "angle-measure" discussion is distinct from the "angle" discussion in the previous paragraph.
haruspex said:Not if you redefine trig functions as taking arguments of dimension Θ, as I did.
Stephen Tashi said:Let me see if I understand you viewpoint.
In the PDF linked in the Insight and post #10, the author, A. Sonin, makes a distinction among:
1) A physical object or phenomena (e.g. a stick)
2) A "dimension", which is a property of a physical object or phenomena (e.g. length)
3) A "unit of measure", which is a way to quantify a dimension (e.g. meters)
The author is careful to point out that a "dimension" is not a physical phenomena. It is a property of a physical phenomena.
You describe "an angle" in mathematical terms, but since you say an "angle" can have various properties, I think you mean an "angle" to denote a physical phenomena, which is alternative 1)
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Yes, I think that is why I have never been satisfied with the view that angles are utterly dimensionless.Stephen Tashi said:It's interesting to consider the distinction between a mathematical definition of a function and a physical definition of a function. To define ##\sin(\theta)## mathematically (i.e. a mapping from real numbers to real numbers) one would have to unambiguously answer questions like "What is ##\sin(0.35)##?" without any discussion of "units of measure" - e.g. 0.35 deg vs 0.35 radians. From a mathematical point of view, ##\sin(\theta\ deg)## and ##\sin(\theta\ radians)## are different functions, even though we use the ambiguous notation ##\sin(\theta)## to denote both of them. Only the family of trig functions where ##\theta## is measured in radians satisfy mathematical laws like ##D sin(\theta) = cos(\theta)##.
To give a physical law in the form of a function we may do it by assuming certain units of measure. Then it is assumed that changing the units of measure appropriately produces a new mathematical function which states the same physical law. So a physical definition of a function defines a set of different mathematical functions that are regarded as physically equivalent.
The physical definition of ##\sin(\theta)## defines a set of different, but physically equivalent mathematical functions.
Whatever circle Baluncore had in mind.robphy said:Which circle are you referring to?
I haven't forgotten this. I want to take a look at the Brownstein article first.robphy said:At this stage, my question of the consistency of "1+i" in post 2 stands out as still unresolved, despite your reply in post 5.
Raising a dimensioned entity to a power is fine, because we can still express the dimensions of the result. For other functions, such as exp, log and trig functions, it is more problematic. If you ever find you have an equation of the form ##e^x##, where ##x## has dimension, you can be pretty sure you have erred.
If you ever find you have an equation of the form ##e^x## where ##x## has dimension, you can be pretty sure you have erred.
Drakkith said:I was not aware of this fact. Very interesting.
That is because some interval has been fixed upon, making λ purely a number. If you want to vary the interval you can make λ a rate:Stephen Tashi said:Is it also an error to have a term ##e^c## where c is a constant with dimensions?
The Poisson distribution has density ##f(k) = \frac{ \lambda^k e^{-\lambda}}{k!}## where ##\lambda## is "The average number of events in the interval". So I assume ##\lambda## has a dimension since "the interval" might mean 1 second or 1 hour etc.. How are the dimensions going to work out in that formula? .
As I thought I showed, you can think of them as fractions of a standard angle, but that does not make them ratios.atyy said:angles are ratios
The + in 1+i is a different beast from that in 1+1. The 1 and the i retain their separate identities. That we write the sum of a real and an imaginary that way is mere convenience. We could instead have the notation <x,y> to represent complex numbers. Addition would be just like vectors, but a unique rule for multiplication.robphy said:If i carries units, is there any meaning to (say) 1+i?
But what would we mean by "fixed upon"? If we are doing a physics problem, does "fixing upon" an interval of 1 meter give different numerical results than fixing upon an interval of 1 kilometer ? If someone determines an equation with a given ##\lambda## applies when the units of length are meters, shouldn't we be able to to deduce what equation applies when the units of length are kilometers by the usual conversion of units ?haruspex said:That is because some interval has been fixed upon
Doesn't "varying" the interval require having some reference length stated in particular units to vary it from? If ##\lambda## is dimensionless, then ##\lambda t## presumably has a dimension of time [T].If you want to vary the interval you can make λ a rate:
##f(k, t) = \frac{ (\lambda t)^k e^{-\lambda t}}{k!}##
atyy said:I agree with Baluncore - angles are ratios, so they do not have a dimension.
In your Poisson example, yes. λ was specified as the average number of events in some unstated but fixed interval. It was not the rate of events, so was indeed dimensionless. If you change the interval (but keep the same process) then λ will change.Stephen Tashi said:does "fixing upon" an interval of 1 meter give different numerical results than fixing upon an interval of 1 kilometer ?
If you allow for different intervals then, as I posted, you must change the definition of λ to be a rate. So λt is dimensionless.Stephen Tashi said:Doesn't "varying" the interval require having some reference length stated in particular units to vary it from? If λ is dimensionless, then λt presumably has a dimension of time [T]
But if I am stating an equation that describes a physical situation, I can't get away with giving an equation that applies to an unstated interval.haruspex said:In your Poisson example, yes. λ was specified as the average number of events in some unstated but fixed interval.
I'm sorry, I am not grasping your point.Stephen Tashi said:But if I am stating an equation that describes a physical situation, I can't get away with giving an equation that applies to an unstated interval.
Suppose the equation that fits my experimental data is ##f(k) = \frac{ (2.3)^k e^{-2.3}}{k!} ## and an experimenter attempts to duplicate my results. He uses an interval of 10 seconds to define ##\lambda##. In order to compare his results to mine, he needs to know what interval I used. He asks me and I tell him "My interval was 5 seconds long". The version of my equation that he can check against his data is ##f(k) = \frac{(4.6)^k e^{-4.6}}{k!} ##.
Are we to say that this conversion of equations takes place by some method other than by converting units using conversion factors ?
One may object: "You should have reported your equation in dimensionless form". That would side-step the need to convert units. However, reporting results in dimensionless form isn't a requirement in science.
Lets try this: Suppose there is a random variable X , measured in meters, that has its density defined on interval ## [0, \ln (2) ] ## by ##f(x) = C( 2 - e^{x}) ## where ##C## is the normalizing constant ##\int_{ 0}^{ln (2)} {(2 - e^ {x})} dx##.haruspex said:I'm sorry, I am not grasping your point.
Preferably it belongs to physics. Otherwise, I am afraid that your next question will be: "To which category of mathematics does it belong?"Demystifier said:I have a related question for everybody. Does the dimensional analysis belongs to mathematics? Or should it be considered as a part of physics? Can the notion of dimension (like meter or second) make sense without referring to a physical measurement?