I believe that degree is also dimensionless. It is also defined as the ratio of two quantities that have units of length.
Also degree and radian are related by a factor. So if radian is dimensionless then degree also is.

Please provide the definition of a degree that is the ratio of two quantities that have units of length.

The only definition of the degree that I am aware of is that the angle swept by a full circle is 360°. That is very analogous to saying that the distance covered between the equator and the pole is 40 million meters.

 

I am not at computer right now. However, I agree with the formulas for definition s of radian and degree by David here. The degree is also a ratio of two quantities of length unit.
http://math.stackexchange.com/questions/1097581/why-are-angles-in-degrees-dimensionless

 

I would argue that many of the claims on that page are somewhat self-fulfilling -- for instance, the claim that degrees and radians are related by a dimensionless constant is "true" only because they have chosen to ignore the units on that constant. It is also a weak argument to say that because one measure of an angle is dimensionless that all measures of an angle are therefore dimensionless. As noted earlier in this thread, there are many measures of physical quantities that are normalized and, hence, dimensionless. You could normalize most measures very easily. For instance, you could normalize the measure of length to the length of the Planck length. That does not make other measures of length dimensionless. Radian measure is merely an angular measure that is normalized -- specifically, it is normalized to the angle subtended by an arc whose arclength is equal to the radius of the arc -- nothing more.

 

I would argue that many of the claims on that page are somewhat self-fulfilling -- for instance, the claim that degrees and radians are related by a dimensionless constant is "true" only because they have chosen to ignore the units on that constant.

I am not sure why the constant is not dimensionless. Circumference of a circle divided into 360 equal parts. Let's call the circumference c, and the length of each part is x.
c = 360*x. Both c and x have unit of length so the constant 360 should be dimensionless here.

It is also a weak argument to say that because one measure of an angle is dimensionless that all measures of an angle are therefore dimensionless. As noted earlier in this thread, there are many measures of physical quantities that are normalized and, hence, dimensionless. You could normalize most measures very easily. For instance, you could normalize the measure of length to the length of the Planck length. That does not make other measures of length dimensionless. Radian measure is merely an angular measure that is normalized -- specifically, it is normalized to the angle subtended by an arc whose arclength is equal to the radius of the arc -- nothing more.

I think we should make a distinction between a measure and a normalized measure.
For example, length can be measured in meter, inch, foot, yard,... or a normalized Planck length as you said. But normalized length is a different type. It doesn't have unit, just a pure number.
For that reason, it seems nonsense to say that normalized length has unit and also to conclude that Planck length is dimensional.

 

I am not sure why the constant is not dimensionless. Circumference of a circle divided into 360 equal parts. Let's call the circumference c, and the length of each part is x.
c = 360*x. Both c and x have unit of length so the constant 360 should be dimensionless here.

Let's see where this claim leads.

Divide the length of a stick into twelve equal parts. Let's call the length x and the length of each part y.

x = 12 y

Both x and y have the unit of length so the constant 12 should be dimensionless here.

Okay, fine. So what?

I think we should make a distinction between a measure and a normalized measure.

Fine. No problem there.

For example, length can be measured in meter, inch, foot, yard,... or a normalized Planck length as you said. But normalized length is a different type. It doesn't have unit, just a pure number.

You've just made my point.

We can measure a distance in meters, inches, feet, yards, or a dimensionless number that is normalized to a chosen reference length, such as the Planck length. All of these are measures of length. All but the last one have dimensions of length, but the last one is just a pure number precisely because it is normalized.

The same is true with angular measure. We can measure it in degrees, grads, or a host of other dimensioned angular measures. Or we can use a normalized measure using a chosen reference angle, such as the angle for which the subtended arclength is equal to the radius of the arc. For that last one we have a pure number because it is normalized.

For that reason, it seems nonsense to say that normalized length has unit and also to conclude that Planck length is dimensional.

Who said that a normalized length has a unit. I didn't and it doesn't -- it is normalized and normalized measures are dimensionless. But it is still perfectly valid to convert between a normalized measure and a dimensioned measure by using the reference measure to which the normalized measure was normalized.

And no where am I "concluding" that the Planck length is dimensional -- it IS dimensioned, pure and simple.

 

Okay, fine. So what?

So the factor conversion between radian and degree is also dimensionless.

We can measure a distance in meters, inches, feet, yards, or a dimensionless number that is normalized to a chosen reference length, such as the Planck length. All of these are measures of length. All but the last one have dimensions of length, but the last one is just a pure number precisely because it is normalized.

The same is true with angular measure. We can measure it in degrees, grads, or a host of other dimensioned angular measures. Or we can use a normalized measure using a chosen reference angle, such as the angle for which the subtended arclength is equal to the radius of the arc. For that last one we have a pure number because it is normalized.

Yes, no problem here.

Who said that a normalized length has a unit. I didn't and it doesn't -- it is normalized and normalized measures are dimensionless. But it is still perfectly valid to convert between a normalized measure and a dimensioned measure by using the reference measure to which the normalized measure was normalized.

And no where am I "concluding" that the Planck length is dimensional -- it IS dimensioned, pure and simple.

Sorry but it was meant to anyone, just a general statement.
Just one thing confusing. I see no difference in the way radian and degree are defined. Both are ratio of length, so if radian is dimensionless then degree should also be dimensionless.

 

Again, where is this ratiometric definition for degree?

It must be of the form: The degree measure of an angle is defined as the ratio between A and B.

What is A?

What is B?

The degree is defined as 1/360th of the angle the constitutes a full circle. Just as (at one time) the meter was defined as 1/40,000,000th of the distance between the equator and the north pole on the prime meridian.

 

It must be of the form: The degree measure of an angle is defined as the ratio between A and B.

What is A?

What is B?

A is the circumference of the circle (equal to 2πr, where r is radius).
B is the 1/360 of circumference.

 

In Laplace transform s = σ + jω is complex number. e^(st) should be dimensionless. So s should have the unit of frequency (1/time).
Does it make sense to say that a complex number has unit?

Then z may or may not have units? Can it be complex or imaginary units?

 

A is the circumference of the circle (equal to 2πr, where r is radius).
B is the 1/360 of circumference.

That doesn't define an angle at all.

Neither of the two things you mention has any relation to the angle being measured.

In order to have a ratiometric measure, you need to draw an angle. Measure two things associated with that angle. Define the angle in terms of the ratio of those two things.

For radian measure this is:

Given an angle, the measure of the angle is the ratio of the length of an arc subtended by THAT angle to the length of the radius of THAT arc.

What is the corresponding definition of the ratiometric measure of an angle in degrees?

 

For radian measure this is:

Given an angle, the measure of the angle is the ratio of the length of an arc subtended by THAT angle to the length of the radius of THAT arc.

What is the corresponding definition of the ratiometric measure of an angle in degrees?

For a given angle, first draw a circle with vertex of the angle as a center. Then measure the circumference of that circle. From this we can get 1/360 circumference, say for example A.
Now measure the length of an arc subtended by that angle called B.
Degree = B/A.

 

\Then z may or may not have units? Can it be complex units?

Just like a non-complex quantity, say x, may or may not have units, a complex quantity, say z, may or may not have units. I'm not aware of anything that would qualify as "complex units". In the case of s = σ + jω, the units are inverse-seconds. The ω in particular can best be thought of as radians/second while σ is probably just best left as inverse-seconds.

 

For a given angle, first draw a circle with vertex of the angle as a center. Then measure the circumference of that circle. From this we can get 1/360 circumference, say for example A.
Now measure the length of an arc subtended by that angle called B.
Degree = B/A.

You're just brute forcing a ratio and not defining the measure in terms of a ratio. We could do the same thing with any measure. Measure the duration of a day and divide that by 86,400. Measure the duration of some event. Now divide the time of the event by 1/86,400th of the duration of a day and, poof, time is now dimensionless! Doesn't work that way.

 

For a given angle, first draw a circle with vertex of the angle as a center. Then measure the circumference of that circle. From this we can get 1/360 circumference, say for example A.
Now measure the length of an arc subtended by that angle called B.
Degree = B/A.

Keep in mind that in math and physics, angles are formally measured in radians -not degrees- and radian is expressed as a "pure number" with no units or dimensions.

 

You're just brute forcing a ratio and not defining the measure in terms of a ratio. We could do the same thing with any measure. Measure the duration of a day and divide that by 86,400. Measure the duration of some event. Now divide the time of the event by 1/86,400th of the duration of a day and, poof, time is now dimensionless! Doesn't work that way.

Well, when you divide the time of the event by 1/86,400th of the duration of a day the number you get is dimensionless and not called time now. It is normalized time as you called.
The same thing goes for degree. The number got by dividing the arclength by 2*pi*r/360 is dimensionless and called degree.

Keep in mind that in math and physics, angles are formally measured in radians -not degrees- and radian is expressed as a "pure number" with no units or dimensions.

Yes, I see.

 

Well, when you divide the time of the event by 1/86,400th of the duration of a day the number you get is not dimensionless and not called time now. It is normalized time as you called.
The same thing goes for degree. The number got by dividing the arclength by 2*pi*r/360 is dimensionless and called degree.

So we are both agreed that my made-up definition of time measure is not the definition of a second.

The only remaining question is whether your made-up definition of angular measure is the definition of a degree.

My guess is that you will find a mixed bag, with most defining the degree as 1/360th of a full rotation and others defining it as π/180 radians. The former (done according to its historical definition) argues that it is dimensioned and the latter (done as a means of tying it to SI units) that it is not. So it is probably a question without a definitive answer. However, I doubt you will find many, if any, places that define it as you have done.

 

I think the definition is relating to central angle.
https://books.google.com.vn/books?i...redir_esc=y#v=onepage&q=central angle&f=false

 

I get an error saying that that page is not available for viewing.

 

I get an error saying that that page is not available for viewing.

I am not sure why but I just clicked the link and it is OK.
The books is The Mathematics that Every Secondary School Math Teacher Needs to Know By Alan Sultan, Alice F. Artzt.
Section 5.6.1 Inscribed and Central Angles (page 180).
Here is the page I am referring to.

http://i.imgur.com/OkBZNBk.png

 

I see that as defining (and keeping in mind that any "definition" in a textbook is not definitive) what is meant by certain terms, such as central angle and arc subtended by a central angle. I don't see anything there that defines what a degree is -- it uses a prior definition of a degree and rather strongly implies that that definition is 1/360th of a complete rotation.