# Can Angles be Assigned a Dimension?

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### 1. Some Background on Dimensional Analysis

… if you are not already familiar with it.

#### 1.1 Dimensions

Dimensional Analysis is a way of analysing physics equations that considers only the qualitative dimensions – mass, length, time, charge .. – of the quantities involved, not the values that they take in the given problem.  This is not to be confused with Euclidean dimensions.

The fundamental rule is that you can only add, subtract or equate terms that have exactly the same constituent dimensions, and each to the same degree.  An acceleration cannot be a force, or be added to a force, because the latter includes a mass dimension while the former does not.  An area cannot be compared to a distance because it is length squared.

For a detailed discussion see e.g.  http://web.mit.edu/2.25/www/pdf/DA_unified.pdf, or any text on the Buckingham Pi theorem.   (There are also Wikipedia and Khan academy links, but these unfortunately confuse DA with managing units as variables, which is a separate topic.)

#### 1.2 Notation

The standard notation in DA is that if x is a variable in an equation then [x] extracts its dimensionality.  The dimensions themselves are labelled M for mass, L for length, T for time, Q for charge, θ for temperature …

Thus, if F is a force then ##[F]=MLT^{-2}##.  The well-known equation F=ma would be analysed as ##MLT^{-2} = (M)(LT^{-2})##, which is clearly true.

#### 1.3 Uses

##### 1.3.1 Predicting the form of a relationship

DA can be a useful shortcut to establishing the general form of how one quantity depends on others.

Example: We presume that the pressure difference, ##\Delta P##, between the inside and outside of a bubble depends on the radius, ##r## and the surface tension ##S##.

## [ \Delta P]=ML^{-1}T^{-2}##

##[r]=L##

##[S]=MT^{-2}##

The only way to combine the pressure difference and surface tension to obtain a length is

## (MT^{-2})/(ML^{-1}T^{-2})=L##

Hence we can say

##r=kS/\Delta P##

for some constant k.

##### 1.3.2 Error checking

Many algebraic errors can be caught by checking dimensional consistency.

### 2. Angles

Angles have never been considered to have dimension.  Consider, for example, the formula for arc length ##s##:

##s = r\theta##

Since ##s## and ##r## each have dimension ##L##, the angle cannot have a dimension, or so it seems.

#### 2.1 Units Matter

Dimensionless combinations, like ##force/(mass \times acceleration)##, generally have the useful feature that they are invariant to the units used.  As long as the units used for the variables are consistent, the same number results whether you use SI or Imperial, or Babylonian.

This is not true of angles; many problems posted on the Homework forums are resolved when the student remembers to plug in the number of radians as argument to the sine function instead of degrees.

#### 2.2 Distinct Entities with Same Dimensionality

It is somewhat unsatisfactory that, when reduced to mere dimensions, some pairs of quite different entities appear to be the same.

Torque and energy are both force times distance.  In terms of vectors, the first is the vector product, the second the scalar.

Angular momentum and action are both ##ML^2T^{-1}##.  Again, a vector and a scalar.

### 3 Axioms for an “Imaginary” Dimension

Consider assigning angles the dimension ##\Theta##, with some unusual properties:

• ##\Theta^2=1##
• The cross product operator itself has dimension ##\Theta##
• ##i##, the square root of -1, has dimension ##\Theta##

#### 3.1 Vectors and Angles

The following table illustrates the use of the dimensionality of angles with cross and dot products.

Entity Sample Equation Dimension
Arc length element  ##\vec{ds}=\vec r\times\vec{d\theta}## ## L=(L)(\Theta)(\Theta)##
Torque  ##\vec\tau=\vec r\times\vec F## ##ML^2T^{-2}\Theta=(L)(\Theta)(MLT^{-2})##
Work   ##E=\vec r.\vec F##  ##ML^2T^{-2}=(L)(MLT^{-2})##
Angular momentum  ##\vec L=\vec r\times\vec p##  ##ML^2T^{-1}\Theta=(L)(\Theta)(MLT^{-1})##
Gyroscopic precession  ##\vec \tau=\vec \Omega_p\times\vec L##  ##ML^2T^{-2}\Theta=(\Theta T^{-1})(\Theta)(ML^2T^{-1}\Theta)##
Velocity  ##\vec v=\vec r\times\vec{\omega}##  ##LT^{-1}=(L)(\Theta)(\Theta T^{-1})##

E.g. to resolve the ##s=r\theta## case for arc length, we can argue that this should really be expressed as the integral of the magnitude of a vector:

##s=\int|\vec{dr}|=\int |\vec r\times\vec{d\theta}|##.  The ##\Theta## dimension of the angle is neutralised by that of the cross-product operator.

#### 3.2 Functions of Angles

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.

For trig functions, it would be reasonable to require the argument to be an angle, but ##e^{i\theta}## would appear to create a difficulty for assigning angles a dimension.

This can be resolved by giving ##i## the ##\Theta## dimension also.  Based on the power series expansions, we see that the odd trig functions (those for which f(-x)=-f(x)) necessarily return the ##\Theta## dimension but the even functions, such as cosine, return a dimensionless value.

Thus, ##e^{i\theta}=\cos(\theta)+i\sin(\theta)## is entirely dimensionally consistent.

#### 3.3 Areas and Volumes

Areas are naturally generated as cross products of vectors, imbuing them with the dimension ##L^2\Theta##.  Since the vector is normal to the surface, this is analogous to rotations as vectors.

Since volumes arise from the triple scalar product, it would seem that these should also have the ##\Theta## dimension.  This is more surprising.

The solid angle element subtended at the origin by a surface element ##\vec {dS}## at position ##\vec r## is given by ##d\Omega=\frac{\vec r.\vec{dS}}{|r|^3}##.  Or, if we wish to make this a vector in the direction ##\vec r##, ##\vec{d\Omega}=\vec r\frac{\vec r.\vec{dS}}{|r|^4}##.

Alternatively, in polar coordinates, ##d\Omega=\sin(\theta).d\theta d\phi##

Whichever way, the dimension is again ##\Theta##, which feels consistent with the result for volumes.

#### 3.4 Complex Arithmetic

If ##i## is to be given dimension, what are we to make of ##1+i##?

Despite appearances, there is no difficulty.  ##1+i## is a convenient notation, but it is not addition in the same sense as in ##1+1##.  The 1 and the ##i## retain their separate existences.  We might just as easily, if less conveniently, have chosen to write complex numbers as an ordered pair, like <x, y>.  The real and imaginary parts never get crunched together in the same way as in normal addition, so they can have different dimensions without creating any inconsistencies.

#### 3.5 Frequency and Angular Frequency

In wave expressions, frequency, ##f##, is the number of cycles per unit time, while angular frequency, ##\omega##, is radians per unit time.  ##\omega=2\pi f##.

Clearly ##\omega## should have dimension ##\Theta T^{-1}##.  The dimension for ##f## depends on whether the factor ##\pi## is to be taken as an angle or as a dimensionless number performing a conversion of units.  Taking ##f## as having dimension ##T^{-1}## appears to be best.

#### 3.6 Planck’s Constants

Consider the equations

##E=hf##

##E=\hbar \omega##

Since ##h## has dimension of action, ##ML^2T^{-1}##, it has no angular dimension.  That is fine for the first equation since frequency is just ##T^{-1}##.

In the second equation, ##[\omega]=\Theta T^{-1}##.  ##\hbar## is defined as ##\frac h{2\pi}##.  Since ##\pi## is an angle here, that has dimension ##ML^2T^{-1}\Theta##, cancelling the ##\Theta## from ##\omega## and achieving dimensional consistency.

But note that this gives ##\hbar## the units of angular momentum, not action. An implication is that ##\hbar## should perhaps be considered a vector, and we should write the photon energy as

##E=\vec{\hbar}.\vec{ \omega}##

though how one is to justify that ##\vec {\hbar}## is necessarily in the direction of the velocity is unclear.

Likewise for momentum

##\vec p = k\vec{\hbar}##

where k is the wavenumber.  Note that this provides momentum as the vector it should be, rather than just defining its magnitude.

The Heisenberg Uncertainty relations, such as

##\frac 12\hbar\leqslant \Delta \vec p.\Delta \vec x##

would appear to violate both the dimensionality and the notion of making ##\hbar## a vector.  But if we follow the steps in the proof of the uncertainty relation we come to this penultimate statement:

##\frac 14\hbar^2\leqslant (\Delta \vec p.\Delta \vec x)^2##

At this point, giving ##\hbar## an angular component of dimensionality creates no problem.   Neither is there an issue with thinking of it as a vector.  These problems only appear when we overlook the ambiguities that so commonly arise when taking square roots.

### 4. Postscript

Subsequent to penning the original article, I have become aware of numerous prior attempts, dating back as far as 1936.  An excellent summary is by Quincey and Brown at https://arxiv.org/ftp/arxiv/papers/1604/1604.02373.pdf.

But their list misses a key one:

C. H . Page, J. Research National Bureau of Standards 65 B (Math. and Math. Phys.) No. 4, 227-235; (1961). http://nvlpubs.nist.gov/nistpubs/jres/65B/jresv65Bn4p227_A1b.pdf

It appears that most of my work above is a rediscovery of Page’s:

Result Page in “Page”
Cross product has angular dimension, dot product does not  231
##\Theta^2=1## Appendix 2
sine has angular dimension, cosine does not Appendix 2
solid angles have angular dimension (i.e. not squared) Appendix 2
whole cycles, as a unit, are dimensionless Appendix 3

Curiously, Page drew attention to the difficulty posed by ##e^{i\theta}=\cos(\theta)+i\sin(\theta)##, but overlooked the remedy of assigning angular dimension to ##i##.

He did not consider Planck’s constants.

Tags:
105 replies
1. robphy says:

What is the justification of the claim: "The cross product operator also has dimension Θ"?

2. Arnold Neumaier says:

Angles have the dimension of 1. That this is a true dimension can be seen from the fact that one measures angles in different units , namely either degrees or radians, and they convert into each other just like units for other dimensional quantities. Your ##\Theta## doesn't make sense unless it equals ##1## since for ##s=\sin\theta## to make sense, the dimension of ##\theta## must be ##1##.

3. haruspex says:

What is the justification of the claim: "The cross product operator also has dimension Θ"?

That is not a claim, it is part of the definition of Θ.

If i carries units, is there any meaning to (say) 1+i?

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.

Note: in exp (x), the x must be dimensionless

That's why I assigned i the dimension Θ, to make iθ dimensionless.

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

That sounds very much as though it is not a new idea, which is at once heartening and disappointing.  Thanks for the reference.

Angles have the dimension of 1. That this is a true dimension

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.

for s=sinθ to make sense, the dimension of θ must be 1.

Not if you redefine trig functions as taking arguments of dimension Θ, as I did.

4. Ygggdrasil says:

Angles can be defined as dimensionless quantities if one thinks of them as fractions of a circle (multiplied by the constant 2π).

5. haruspex says:

Angles can be defined as dimensionless quantities if one thinks of them as fractions of a circle (multiplied by the constant 2π).

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?

6. Ygggdrasil says:

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?

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.

7. haruspex says:

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.

Fair question.

I have always found it a bit unsatisfactory that some quite different pairs of physical entity are dimensionally indistinguishable.  I mentioned some in the post.  There seemed to be something distinct about rotational entities, such as angular momentum, that was not captured by DA.

As regards utility, as I showed in the table, it can be added to normal DA for an extra bit (literally) of information.

8. Stephen Tashi says:

It would be nice to have a list of what axioms a "dimension" must satisfy.  Until that is available, arguments about what is or isn't a dimension are going to be personal opinions.

A convenient online reference for dimensional analysis is http://web.mit.edu/2.25/www/pdf/DA_unified.pdf.   However, I don't know whether the axioms stated in that work are standard – or appropriate.

For sake of discussion, consider what p10-11 of that book says about dimensions.  It says that there must be an operation of comparison and an operation of addition defined on the dimension.    All sorts of familiar ambiguities and confusions arise with angles because their "wrap-around" feature – e.g. is  360 degrees "equal" to zero degrees?  Does the sum ##pi## radians + ##pi## radians have a unique answer ?  – i.e. are ##2pi## radians and zero radians physically distinct ?    If we assume there is an "addition" is defined for dimensions, are we assuming (as we do in mathematics) that a sum is unique ?

Page 11 postulates that  physical dimension must admit an ordering:

the concept of larger and smaller for like quantities (if there exists a finite B such that A+B=C then C > A)

To me, this is troublesome assumption for any dimension (including mass, length, time) where we wish to give a negative quantity a physical interpretation.    For example, is " – 10 meters"  a physical quantity ?  If we consider "-10 meters" a finite physical quantity  then the above assumption lets us conclude things like  15 meters + (-10 meters) = 5 meters,  so  15 meters < 5 meters.

The topics of "changing units" and "changing coordinates" are usually treated as different subjects.  Are they really different concepts ?  Sometime as  "change of units" can involve more than mutiplying by a factor (e.g. converting degrees centigrade to degrees Kelvin).     In a  coordinate system for a "thing" it is permissible for the same thing to have several sets of different coordinates.   Does saying something is a "dimension" imply that there exists a coordinate system for measuring it where each distinct description in coordinates represent a different physical situation ?

9. Baluncore says:

Angles are ratios, parts of a circle. Neither angles nor ratios have a dimension.

Polynomials and dimensions are incompatible. Transcendental functions that are approximated by polynomials must have dimensionless inputs and outputs.

10. haruspex says:

Angles are ratios, parts of a circle.

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.

Polynomials and dimensions are incompatible.

Not if the dimension has the unusual property that it becomes dimensionless when raised to some finite power.  The ϑ[SUP]2[/SUP]=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).

11. robphy says:

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.

Which circle are you referring to?

The unit circle? Or maybe the circle of radius 7?

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".

Certainly, you can try to make definitions… but they have to lead to a consistent system.

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.

12. Stephen Tashi says:

Polynomials and dimensions are incompatible. Transcendental functions that are approximated by polynomials must have dimensionless inputs and outputs.

I'm curious why you say that polynomials are incompatible with dimensions.  Coefficients of different powers of x in a polynomial can be assigned different dimensions, so that each power of x is converted to the same dimension.

If we have an equation that describes a dimensioned physical quantity as a power series, aren't we assigning different dimensions to each coefficient in the power series ?

13. Stephen Tashi says:

Which circle are you referring to?

The unit circle? Or maybe the circle of radius 7?

You could also ask "The unit circle with center (0,0)?  The unit circle with center (15,12)?"

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.

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.

If we have an object that moves along a path, to measure the property of the path called its "total length" with a meter stick, we have to move the meter stick to various locations on the path.  If we are dealing with an object moving in a circular path and want to measure a property of the path called the "total angle swept out", we may need a measuring instrument that can produce results greater than 360 deg.  Such a measuring instrument could involve a circle, but it would have to have the added feature of keeping track of arc lengths greater than ##2pi##.

In this general discussion, one needs to distinguish an "angle" from an "angle measure".

I agree.  It's the distinction between "a dimension" (e.g. length) and "a unit of measure" (e.g. meters).

14. robphy says:

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.

[snip]

Yes, but I didn't think I had to make further clarification on this. Shall we bring up issues of parallel transport on a non-Euclidean space as well?

I would hope that when one says "arc-length divided by radius" that the rest of this is assumed.

[snip]

I agree.  It's the distinction between "a dimension" (e.g. length) and "a unit of measure" (e.g. meters).

My distinction is this… If two lines (or two segments) meet at a point, then one can talk about the angle [or an angle] at the location where the two lines meet, labeled by the vertex (call it) C or that vertex with a two points, one on each segment–like ACB.  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.

Given two lines (or line segments) meeting at a point, one could define an angle-measure the usual way (essentially with a circular protractor, appropriately calibrated in the likely possibility that protractors have different radii), or maybe in a different way (e.g. https://mathnow.wordpress.com/2009/11/06/a-rational-parameterization-of-the-unit-circle/ ) although it might not give your angle-measure the desired properties of additivity, or maybe using a hyperbolic-protractor (as one might use in special relativity).

Issues of "units of angle-measure" come into play here.

But all of this "angle-measure" discussion is distinct from the "angle" discussion in the previous paragraph.

15. Stephen Tashi says:

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.

But all of this "angle-measure" discussion is distinct from the "angle" discussion in the previous paragraph.

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)

When you say "angle measure", I'm not sure whether you mean alternative 3) or alternative 2).  But does alternative 3) (units of measure)  make any sense without the existence of alternative 2) (dimension) ?

As I mentioned previously,  I haven't yet seen a precise statement of what mathematical or physical properties a "dimension" must have.   I don't know whether other thread participants agree with those listed by A. Sonin.

In regards to "dimensionless ratios",  a dimensionless ratio can associated with a property of a physical object.  Different dimensionless ratios can be associated with different properties (e.g. height of a person/ length of that persons right leg,  weight of a person now / weight of that person at birth).     "Dimensionless ratios" can obviously be quantified.    So it is rather confusing to consider the question of whether a "dimensionless ratio" is (or isn't) a associated with a "dimension".

16. Stephen Tashi says:

Not if you redefine trig functions as taking arguments of dimension Θ, as I did.

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.

17. robphy says:

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)

[snip]

I think the bottom line here is: clearly define terms, especially when one is trying to change definitions.

18. haruspex says:

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.

Yes, I think that is why I have never been satisfied with the view that angles are utterly dimensionless.

Which circle are you referring to?

Whatever circle Baluncore had in mind.

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.

I haven't forgotten this.  I want to take a look at the Brownstein article first.