Can Angles be Assigned a Dimension? - Comments

[DaTario]

Yes, I was not sure what DaTario meant by a different "form". From the reference to units, I presumed DaTario would regard the appearance of a factor of 2π as being a different form, but, like you, I would consider that the same in form, just different in detail.

Sorry, I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

I have just found a site with a modest exposition of this idea:
http://www.johndcook.com/blog/2013/11/15/dimensional-analysis/

It seems to be in accordance to the " cloudy" reference I have claimed to have read a long time ago.

Best wishes,

DaTario

 

[haruspex]

I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

That is one very good reason for not allowing such.
If the concept of attributing a dimension to angles has any validity, it must be possible to write any correct equation such that it is dimensionally consistent in that regard.

 

[DaTario]

That is one very good reason for not allowing such.
If the concept of attributing a dimension to angles has any validity, it must be possible to write any correct equation such that it is dimensionally consistent in that regard.

Two other questions:
1) Following your axioms, would it be correct to say that solid angles are truly adimensional?

2) the square root of an adimensional quantity has the angle dimension?

Best wishes,
DaTario

 

[haruspex]

Following your axioms, would it be correct to say that solid angles are truly adimensional?

Good question. I found a reason for saying they also have the angle dimension. E.g. if we consider two angle vectors $\vec {d\theta}$ and $\vec{d \phi}$, and a vector radius $\vec r$, the two arc elements are $\vec r\times\vec{d\theta}$ and $\vec r\times\vec{d\phi}$. The vector area element they form is $(\vec r\times\vec{d\theta})\times (\vec r\times\vec{d\phi})$. Counting the angle elements and cross products that has angular dimension.

the square root of an adimensional quantity has the angle dimension?

No, that would be ambiguous. Not unusual for square roots.

 

[DaTario]

Good question. I found a reason for saying they also have the angle dimension. E.g. if we consider two angle vectors $\vec {d\theta}$ and $\vec{d \phi}$, and a vector radius $\vec r$, the two arc elements are $\vec r\times\vec{d\theta}$ and $\vec r\times\vec{d\phi}$. The vector area element they form is $(\vec r\times\vec{d\theta})\times (\vec r\times\vec{d\phi})$. Counting the angle elements and cross products that has angular dimension.

But in the case of solid angles (stereoradians - sr) the operation is:
$$ \frac{s}{r} \frac{s}{r}. $$

So it is area divided by the square of the radius.

 

[haruspex]

But in the case of solid angles (stereoradians - sr) the operation is:
$$ \frac{s}{r} \frac{s}{r}. $$

So it is area divided by the square of the radius.

That's only after reducing it all to scalars. To see how angular dimension fits in, in my scheme, it seems to be necessary to work with vectors wherever appropriate.

 

[Stephen Tashi]

Sorry, I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

Equations are not "invariant" under a change of units. For example F = MA is correct if F is in Newtons, A is in m/sec^2 and M is in kilograms. However if we measure M in grams, F = MA isn't correct.

In general, changing units changes equations by multiplying one or both sides of the equation by constant factors.

So the prohibition against adding unlike units can't be explained by the invariance of equations. We have to explain why a particular type of variation is the only permissible kind.

 

[Baluncore]

In general, changing units changes equations by multiplying one or both sides of the equation by constant factors.

Those constants actually have units, but not dimensions. For example, to convert metres to cm, multiply by 100 cm/metre. The metre units cancel, leaving the new unit cm in it's place. The units in the factor ratio must have the same dimension.

Notice also that conversion factors are always equal to one. For example 100 cm/metre = 1.
I would say that when all you are doing is multiplying something by one, it is invariant.

 

[Stephen Tashi]

I would say that when all you are doing is multiplying something by one, it is invariant.

But the numerical factor resulting from 1000 grams/kg is "1000", not "1". You are introducing a different concept of "1" than the purely arithmetic concept of "1". I agree that if we use a concept of "invariance" that says the equation F = MA in one system of units can be "the same equation" as F = 1000 MA in a different system of units then that definition allows equations that are not arithmetically invariant to be considered "invariant".

However, this extended concept of invariance doesn't explain why unlike dimensions can be multiplied and divided, but not added. Any explanation goes in the other direction - i.e. if we assume that conversion factors involving products and quotients of units of arbitrary dimension will be used, then particular conversion factors can be regarded as a type of "1" (e.g. "1000 g/ kg" is a such "1"). From that, we get a type of "invariance" of equations (different from arithmetical invariance) as a consequence.

 

[diogenesNY]

Submitted for your approval:

Benoit Mandelbrot seems to have had something to say on this issue, or at least in the same neck of the woods. In fact in part he seems to consider this effort a step towards getting Mathematics and Physics to play nice together without adult supervision. I am not sure that he actually accomplishes this, but that is another issue.

In this short and highly readable paper he addresses the Coastline Paradox and suggests a way of dealing with it that is borderline mind-expanding. Highly recommended.

Title: How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
This was originally published in Science in 1967.

Here is a wikipedia article _about_ the paper:

https://en.wikipedia.org/wiki/How_L...ical_Self-Similarity_and_Fractional_Dimension

Here is the article at the Science although you may need a subscription or academic site license to read it:

http://science.sciencemag.org/content/156/3775/636

Here is the article at JSTOR - this should be available at most (site licensed) academic institutions.:

http://www.jstor.org/stable/1721427

Okay, I just found a direct link to a (seemingly weirdly formatted) .pdf of the original article here:

http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf

I hope y'all find this a worthy contribution.

diogenesNY

 

[Baluncore]

However, this extended concept of invariance doesn't explain why unlike dimensions can be multiplied and divided, but not added.

Please give an example of where you might need to add or subtract two differently dimensioned values?
What concept is involved and how is it meaningful to physics?

 

[Stephen Tashi]

Please give an example of where you might need to add or subtract two differently dimensioned values?
What concept is involved and how is it meaningful to physics?

My interest is in finding a correct justification for why conventional dimensional analysis allows multiplication of different dimensions, but not addition of different dimensions. I'm not advocating a revision of dimensional analysis to allow adding different dimensions. I'm advocating that we find a coherent statement of why multiplying different dimensions is allowed and adding different dimensions is not allowed.

The explanations we often hear are just dogma - e.g. "It would be nonsense to add apples and oranges". (Yet it does make sense to multiply apples by oranges ?)

Some methods for justifying a statement S are the following.

1) S is true because it is empirically true. We assert S is observed to be true without offering any proof or explanation of why S is true.

2) S is a theorem. S can be proven from other statements we accept as truths. This includes indirect proofs. e.g. Reasoning that begins "Suppose we did add apples and oranges then it would follow that ..."

3) S is an assumption or definition.

In my (current) opinion, the coherent justification for the principle "You can't add apples and oranges" is 1). It is an empirical fact.

People who are familiar using conversion factors become so familiar with the convenience they offer that they wish to use method 2) and offer words to prove that "You can't add apples and oranges" as a theorem. Dignified treatments of dimensional analysis used method 3), they simply assert "We may multiply apples by oranges but we do not add apples and oranges" as an assumption or make it a consequence of a definition.

Using method 1): Consider how often in physics it is sufficient to know only the product of two dimensional quantities in order to make a prediction. For example, to predict whether we can unscrew a bolt, it is often sufficient to know the available torque we have in ft-lbs. If we need a torque of 20 ft lbs, we can realized this torque by a 10 lbs force acting on a 2 ft lever, or a 20 lbs force acting on a 1 ft lever etc. The measurement of ft lbs is ambiguous as to how many ft and how many lbs are involved in the phenomena. ( In the particular case of "zero ft lbs", we at least know that that there were 0 ft or 0 lbs involved. ) The ambiguity in a measurement like 20 ft lbs often doesn't matter because there are important behaviors in Nature that are completely specified by the product of two dimensioned quantities and don't depend on how the factors in that product are implemented.

Dimensional analysis does not assert that it always makes sense to multiply quantities of different dimensions. It only asserts that one may multiply quantities of different dimensions. The justification is that we find empirically that there are many situations where we can make a useful prediction knowing the value of a product without knowing the value of its factors.

Empirically , we do not find any notable situations where knowing the value of a sum of differently dimensioned quantities (without knowing the values of the summands) allows us to make a useful prediction. For example a measurement of "20 apples+oranges" is ambiguous about whether there were 0 apples and 20 oranges , versus 10 apples and 10 oranges, versus -50 apples and 70 oranges, etc.

If someone can offer an explanation of why Nature operates so that the ambiguity of products is often useful, but the ambiguity of sums is not, then I'd like to hear it. As far as I can see, it is an empirical fact, not a theorem.

 

[Baluncore]

My interest is in finding a correct justification for why conventional dimensional analysis allows multiplication of different dimensions, but not addition of different dimensions.

We can simplify 3x*4y to 12xy.
Now explain why 3x+4y cannot be simplified.
That should satisfy your interest.

 

[haruspex]

Yet it does make sense to multiply apples by oranges ?)

It can. Tommy is allowed to take one apple and one orange from 3 apples and 4 oranges. What is the set of possible choices? 12 apple-orange pairs.
But fruit does not make a good analogy because you can also argue for adding apples and oranges. The discreteness creates a natural unit of measure.
Here is a possibility... Mutiplying makes sense because you can create new units to match. In some scenario, I take K kg and M metres to compute the product KM kgm. If you prefer to work in pound-inches, you know how to convert that without having to know K and M separately. But if we try to invent the concept of mass plus distance, and I tell you (K+M) "kg+m", you cannot convert the single number K+M to an equivalent number of pound+inches.

 

[Stephen Tashi]

We can simplify 3x*4y to 12xy.
Now explain why 3x+4y cannot be simplified.
That should satisfy your interest.

That makes no connection to physics.

 

[Stephen Tashi]

It can. Tommy is allowed to take one apple and one orange from 3 apples and 4 oranges. What is the set of possible choices? 12 apple-orange pairs.
But fruit does not make a good analogy because you can also argue for adding apples and oranges. The discreteness creates a natural unit of measure.

Dimensional analysis does not say that a physical unit can take a values corresponding to any given real number. So I see nothing wrong with your analogy.

Here is a possibility... Mutiplying makes sense because you can create new units to match. In some scenario, I take K kg and M metres to compute the product KM kgm. If you prefer to work in pound-inches, you know how to convert that without having to know K and M separately.

I'd put it this way: Changing the units in a product and "knowing how to convert" the result (using conversion factors in the standard manner" introduces no new ambiguity in the description of the physical situation.

We "know how to convert" because we know to use conversion factors. But the conventions of using conversion factors doesn't explain why those conventions are physically useful. In my opinion, it is just an empirical fact that the ambiguity in knowing the product of dimensions, but not knowing the individual factors often doesn't detract from the usefulness our knowledge in making physical predictions.

But if we try to invent the concept of mass plus distance, and I tell you (K+M) "kg+m", you cannot convert the single number K+M to an equivalent number of pound+inches.

That depends on what you mean by "equivalent". Perhaps you mean we have no obvious rules to convert it to a unique number of different units. For example, the conversion 4 apples+oranges to units of half_apples+oranges in a naive fashion is ambiguous because it might be done by converting 4 apples+oranges as (3 apples + 1 oranges) which is converted to (6 half_apples + 1 oranges) = 7 half_apples+oranges. Or it might be converted from (1 apples + 3 oranges) as ( 2 half_apples + 3 oranges) = 5 half_apples+oranges. So , if there is an empirical difference between the physical situation producing the measurement 7 half_apples+oranges and the situation producing the measurement 5 half_apples+oranges, this is a argument that converting units in such a manner introduces a harmful ambiguity.

 

[Baluncore]

That makes no connection to physics.

Yes it does. Physics requires that a mathematical equation be evaluated. If you cannot evaluate an equation to a single numerical value then you do not need to add the dimensions. The mathematical reason why differing dimensions are not added in physics is the same reason that 3x + 4y cannot be simplified. 3x + 4y = 3x + 4y.

 

[Stephen Tashi]

Yes it does. Physics requires that a mathematical equation be evaluated. If you cannot evaluate an equation to a single numerical value then you do not need to add the dimensions. The mathematical reason why differing dimensions are not added in physics is the same reason that 3x + 4y cannot be simplified. 3x + 4y = 3x + 4y.

"Equations" are mathematical statements that two functions are equal. Functions that can't be "simplified" can still be "evaluated".

 

[Baluncore]

Functions that can't be "simplified" can still be "evaluated".

So how do you evaluate 3x + 4y = ?

 

[Stephen Tashi]

So how do you evaluate 3x + 4y = ?

We don't "evaluate" 3x + 4y unless we are give numerical values for x and y. Likewise we don't evaluate 6xy unless we are given numerical values for x and y.

 

[Baluncore]

We don't "evaluate" 3x + 4y unless we are give numerical values for x and y. Likewise we don't evaluate 6xy unless we are given numerical values for x and y.

If x was the unit metres and y was the unit seconds, what would you do then?

 

[Stephen Tashi]

If x was the unit metres and y was the unit seconds, what would you do then?

Are you trying to formulate an argument that says "We can't add unlike dimensions because we can't add unlike dimensions"?

As I said, I'm not advocating adding unlike dimensions. I'm investigating what justification we can give for declaring that adding 3 meters and 2 seconds to obtain a composite unit of 5 meters+seconds can never be done ( or should never be done) - no matter what interpretation we give to "meters+seconds".

Declaring "we can't add 3 meters to 2 seconds" or saying "adding 3 meters to 2 seconds doesn't make sense" isn't an explanation. If we want to demonstrate that there is no way to do something, then issuing the challenge "Somebody show me how to do it" isn't a demonstration. For example, "Show me some positive integers $a,b,c$ such such that $a^3 + b^3 = c^3$" isn't a proof of Fermat's theorem.

 

[haruspex]

Dimensional analysis does not say that a physical unit can take a values corresponding to any given real number. So I see nothing wrong with your analogy.

I'd put it this way: Changing the units in a product and "knowing how to convert" the result (using conversion factors in the standard manner" introduces no new ambiguity in the description of the physical situation.

We "know how to convert" because we know to use conversion factors. But the conventions of using conversion factors doesn't explain why those conventions are physically useful. In my opinion, it is just an empirical fact that the ambiguity in knowing the product of dimensions, but not knowing the individual factors often doesn't detract from the usefulness our knowledge in making physical predictions.

That depends on what you mean by "equivalent". Perhaps you mean we have no obvious rules to convert it to a unique number of different units. For example, the conversion 4 apples+oranges to units of half_apples+oranges in a naive fashion is ambiguous because it might be done by converting 4 apples+oranges as (3 apples + 1 oranges) which is converted to (6 half_apples + 1 oranges) = 7 half_apples+oranges. Or it might be converted from (1 apples + 3 oranges) as ( 2 half_apples + 3 oranges) = 5 half_apples+oranges. So , if there is an empirical difference between the physical situation producing the measurement 7 half_apples+oranges and the situation producing the measurement 5 half_apples+oranges, this is a argument that converting units in such a manner introduces a harmful ambiguity.

Maybe something along these lines...
We have two systems characterised by vectors of attributes, S=(x₁, x₂, ..) , S'=(x₁', x₂', ...). We can invent measures of attributes, i.e. maps to reals, m₁, m₂,... We believe (and this is the physics) that the attributes have an additive property such that in some sense we can partition an attribute and recover the whole by summing its parts. This being so, we require our measures to be linear in this sense.
Given two measures, m₁ and n₁ for X₁, if one is linear and there is a nonzero constant μ such that n₁(x)=μm₁(x) for all x, then clearly theother is linear. [Can we show that for two linear measures such a ratio exists?]

Anyway, if we want the product of measures of two attributes, we may choose m₁(x₁)m₂(x₂) or n₁(x₁)n₂(x₂). Armed with the knowledge of the measure ratios, μ, ν, we can compute the measure ratio μν for the measure products.

If we try to do the same summing two measures, there is no constant ratio that can convert one sum of linear measures to another.
This is much the same as I wrote before, but highlighting the dependence on an essential additivity in the underlying attributes, and the consequent requirement of linearity in the measures.

 

[Stephen Tashi]

that the attributes have an additive property such that in some sense we can partition an attribute and recover the whole by summing its parts. This being so, we require our measures to be linear in this sense.

I think that's a fundamental idea in dimensional analysis - but it's a concept that's hard to state only using terms from arithmetic. The notion that the whole attribute "can be recovered" from the "sum" of its parts isn't quite the same notion as the concept that the whole numerical value of something is equal to the arithmetic sum of its "parts".

One attempt to express the physics is that "the presence of an attribute x_1" has the same effect as the simultaneous presence of the mutually exclusive "parts" of the attribute. (That uses "has the same effect" instead of the arithmetic relation "equal" and "simultaneous presence" instead of the arithmetic sense of "sum".)

Alternatively, we could state the concept by embedding it in the definition of the "parts" of an attribute. For example "x_a and x_b are each "half" of attribute x_1" shall mean the following are satisfied: 1) The simultaneous presence of x_a and x_b has the same effect (in whatever phenomenon we are studying) as the presence of x_1 alone - and 2) The presence of x_a alone has the same effect as the presence of x_b alone.

Taking that approach, we could argue that is is useful to make an "isomorphism" between the physical language and mathematical language. i.e. "same effect" maps to "equal". "simultaneous presence" maps to "sum", "half of" in the physical sense maps to "half of" in the arithmetic sense.

As I understand what you are doing with a measure ratio, you are assuming the attribute x_1 is a numerical value at the outset.

If we try to do the same summing two measures, there is no constant ratio that can convert one sum of linear measures to another.

I agree that the use of ratios (i.e. conversion factors) for individual attributes doesn't define how to convert a sum of different dimensions measured in one set of units to a unique sum of the same dimensions measured in a different set of units. When people ask why we "can't" add different dimensions together, I think this inability is a good explanation of why we "don't" add different dimensions.

However, the statement that "it is impossible to add different dimensions" is claim that goes beyond what is convenient or inconvenient. Are we saying that no matter what scheme you make up for adding different dimensions, it won't work? What would it mean for a scheme (which may not be based on conversion factors) to work or not to work?

The general situation as I see it:

The product of xy of units x, y with unlike dimensions omits the information about the individual magnitudes of x and y. There are many physical situations where knowing the product xy is useful even though we don't know the individual values of x and y. (e.g. When a result of interest can be expressed as a function of one variable p = xy instead of function of two variables (x,y).) There are also physical situations where the knowledge of xy is not as useful as knowing the individual values of x and y. For example, if x is the diameter of your coffee cup in cm and y is the length of your cat's tail in cm then the product xy doesn't summarize the situation as well as knowing the individual values of x and y.

By contrast, empirically, there are few situations in physics where knowing the sum x + y of units of different dimensions is useful.

Can we go even further to say "Any physical situation where the result is expressed as a function of the sum of units of different dimensions can be rewritten as a different function of the same variables that does not involve the sum of units of different dimensions"?

 

[DaTario]

In general, changing units changes equations by multiplying one or both sides of the equation by constant factors.

So the prohibition against adding unlike units can't be explained by the invariance of equations. We have to explain why a particular type of variation is the only permissible kind.

Yes, I agree. Admissible invariances seems to be those which involves solely a scale factor.

 

[Baluncore]

By contrast, empirically, there are few situations in physics where knowing the sum x + y of units of different dimensions is useful.

Can you please give examples of those few situations in physics.

 

[haruspex]

@Stephen Tashi , @Baluncore :
I am pleased to have engendered such an interesting debate, but it does seem to have wandered a long way from my original article. How would you feel about having it moved to a different thread?
Meanwhile, you might be interested in http://nvlpubs.nist.gov/nistpubs/jres/65B/jresv65Bn4p227_A1b.pdf. Seems like much of my endeavour is a rediscovery of Page's work. See p 231 and Appendices 2 and 3. He did miss the trick of giving i angular dimension, and did not discuss Planck's constants. The rest of his article might be relevant to your own discussion.

 

[Stephen Tashi]

How would you feel about having it moved to a different thread?

If this thread is getting too long to be readable for new contributors then the discussion should be restarted in another thread. What would the topic of the new thread be?

It interests me to discuss the general foundations of dimensional analysis. I'm not particularly interested in debating the narrower issue of whether angles should be assigned a dimension until the big picture is clear. However, perhaps others would be interested in a new thread restricted to the topic of angles.

The discussion about whether angles should or shouldn't be assigned a dimension just meanders around because the reasons underlie dimensional analysis haven't been established. Some people just repeat the mantra "Angles are dimensionless" because it is a convention of standard dimensional analysis. More liberal participants say that angles can be assigned a dimension because such-and-such an aspect works out to be pleasing and others counter by saying that a different aspect of the situation isn't pleasing. Until the significance of the various aspects is clear, the importance of satisfying them is just a matter of individual tastes.

Discussions about whether angles may be assigned dimensions can become to similar to discussions about other "notorious" topics such as "Is 0.9999... = 1" or "Are dy and dx numbers?", "Can we divide by infinity?". Such discussions tempt us to ignore the arbitrary nature of definitions. We tend to assume that words and notations like "angles", "0.9999...", "dx", "infinity" are particular things that exist independently of anyone's definition of them. Then we use our intuitive concept of these pre-existing things in debating what their properties "are" instead of what we may define those properties to be.

 

[Baluncore]

How would you feel about having it moved to a different thread?

Sorry about being off topic, I plead a little bit guilty. Go ahead and split or fork the thread.
We need more discussion about those interesting bundles of {numbers, units and dimensions}.

 

[haruspex]

What would the topic of the new thread be?

It's your topic. Maybe "why does dimensional analysis work?"

 

[Aufbauwerk 2045]

I think this is an important topic. What is more basic to physics than how we measure things?

In SI units the angle is not a base quantity. We state the angle in radians, which is a ratio of two lengths, namely the length of the subtended arc to the length of the radius, giving a dimensional ratio of L/L = 1. So we say the angle is dimensionless.

Some people have argued for making the angle a base quantity. See for example https://arxiv.org/ftp/arxiv/papers/1604/1604.02373.pdf.

It may be more convenient to keep angles dimensionless. Consider two similar triangles. Perhaps they are both 30-60-90 triangles but the hypotenuse of triangle #1 is twice the length of that of triangle #2. The corresponding angles are equal, but the corresponding sides are not. I suppose it's fair to ask why this is more convenient.

I think this question is related, at least subjectively, to time. The ancients came up with 360 degrees because it corresponds to a 360-day year in some ancient calendar. You can associate a point moving around on the circumference of a circle with the passage of time. We don't care how long that circumference is. We just want to know how many units of time have passed. We can associate units of time with degrees around a circle.

For example, consider our standard analog clock. It may be a wristwatch or Big Ben. In either case, we know that when the little hand is at a certain angle from straight up, it means 20 minutes past the hour.

In physics, in general, the study of periodic motion is an enormously important topic. Therefore, we want our system to be convenient for the mathematics of periodic motion.

I plan to read the above paper and think about this some more. It would also be a good time to review Bridgman's Dimensional Analysis.

 

[Stephen Tashi]

It's your topic. Maybe "why does dimensional analysis work?"

My question is "What do we mean when we say an particular convention of dimensional analysis 'works'?". For example, what criteria are we using to say that assigning angles a dimension "works" or "doesn't work" ?

 

[haruspex]

My question is "What do we mean when we say an particular convention of dimensional analysis 'works'?". For example, what criteria are we using to say that assigning angles a dimension "works" or "doesn't work" ?

My criterion is quite simple: can a dimension be assigned in a way which is consistent and has the power to detect blunders? I believe I have shown that the answer is yes.
But from your posts in the thread, I thought you were focused on the more fundamental question of why DA works at all. E.g., your observation regarding products versus sums.

 

[scottdave]

Interesting concept. One thing I noticed was: in cases where you multiply (or divide) by 2pi, the article suggests that pi has the dimension of Angle. Since there are 2pi radians in a full circle (or full cycle of oscillation), then shouldn't it be 2pi containing the dimension, rather than just pi?
I do like the fact that this way puts a direction in as a dimension, when quantities are vectors.

 

[scottdave]

Interesting concept. I'm not sure if I am ready to adopt it, but it gives some things to think about. You have certainly put a lot of thought into how to handle various situations. In your section 3.6 discussing Planck's Constant. You show hbar to have dimension of ML2T^−1Θ, but I think you should have ML2T^−1Θ^-1, to reflect π in the denominator, and to cancel out the ω.