Damn you mathematicians

[Borodog]

Started rotational kinematics in class today. Was again reminded of this bollocks:

s = r*theta

This, and all that follows from it, drives me [CENSORED] NUTS. Do you see why?

/images/graemlins/mad.gif

[flipdeadshot22]

So you're more of a linear motion type guy I take it? What course/level is this btw?

[Duke]

First instinct: you hate radians.

Second thought: Typically this is where students start getting greek letters shoved up their asses, and you don't like phi, so you use theta instead.

Third: Kids are used to degrees, and will start screwing everything up because they aren't yet comfortable with radians.

[Siegmund]

It could be worse. Borodog could have been one of those crazy chemistry teachers who think there is some merit in using the base-10 logarithm.

[Duke]

[ QUOTE ]
It could be worse. Borodog could have been one of those crazy chemistry teachers who think there is some merit in using the base-10 logarithm.

[/ QUOTE ]

I'm still not convinced that any of my possible explanations are correct, though I'm pretty sure that it involves the teaching aspect and not just the material itself.

[holmansf]

'cause s doesn't have units length*angle. Or at least that's my guess.

[thylacine]

[ QUOTE ]
Started rotational kinematics in class today. Was again reminded of this bollocks:

s = r*theta

This, and all that follows from it, drives me [CENSORED] NUTS. Do you see why?

/images/graemlins/mad.gif

[/ QUOTE ]

No.

BTW what are s, r, \theta here?

[Borodog]

[ QUOTE ]
'cause s doesn't have units length*angle. Or at least that's my guess.

[/ QUOTE ]

Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[Borodog]

[ QUOTE ]
[ QUOTE ]
Started rotational kinematics in class today. Was again reminded of this bollocks:

s = r*theta

This, and all that follows from it, drives me [CENSORED] NUTS. Do you see why?

/images/graemlins/mad.gif

[/ QUOTE ]

No.

BTW what are s, r, \theta here?

[/ QUOTE ]

Arc length, radius, and angle in radians.

[thylacine]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Started rotational kinematics in class today. Was again reminded of this bollocks:

s = r*theta

This, and all that follows from it, drives me [CENSORED] NUTS. Do you see why?

/images/graemlins/mad.gif

[/ QUOTE ]

No.

BTW what are s, r, \theta here?

[/ QUOTE ]

Arc length, radius, and angle in radians.

[/ QUOTE ]

Of course my first guess would be that you are worried about units, but that couldn't be it because angles are obviously dimensionless. In fact this equation makes this clear. So I really can't guess what the problem could be. Please enlighten me.

[Duke]

So you don't like angles not having dimensions?

[MelchyBeau]

Don't angles by definition have 2 dimensionality. Though in physics it is best to define it in R3. Otherwise, saying you are going to rotate something does not make sense unles you know by what axis you are going to rotate something about

[Neuge]

[ QUOTE ]
Don't angles by definition have 2 dimensionality. Though in physics it is best to define it in R3. Otherwise, saying you are going to rotate something does not make sense unles you know by what axis you are going to rotate something about

[/ QUOTE ]
Angles measure the difference in slope between two lines or vectors, which is dimensionless.

[Phil153]

[ QUOTE ]
[ QUOTE ]
'cause s doesn't have units length*angle. Or at least that's my guess.

[/ QUOTE ]

Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[/ QUOTE ]
Borodog,

You should let the market decide the units, rather than imposing your centrally planned bureaucratic monopolist unit system.

Off topic non sequiturs about the evil government are annoying, no?

[flipdeadshot22]

I don't understand. radians are dimensionless units of angular displacement. what can be so hard about teaching a bunch of 14 year olds about this? I didn't have much difficulty in learning about the concept of radians (not trying to brag.) Just break down the scientific justification of the radian measurement to these kids, i'm sure they'll follow.

[bxb]

All the formulas are the same as they are for regular kinematics cept different letters.

[bluesbassman]

[ QUOTE ]
[ QUOTE ]
'cause s doesn't have units length*angle. Or at least that's my guess.

[/ QUOTE ]

Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[/ QUOTE ]

I still don't understand what is your problem with the given expression. That must make me pretty dumb, since I "do" rotational kinematics for a living. (I design attitude control systems for spacecraft.)

FWIW, where possible, I prefer to express 3-dimensional rotational kinematics in terms of the rate of change of a quaternion: (see equation 9, p. 14)

Quaternion dynamics (http://www.math.unm.edu/~vageli/papers/rrr.pdf)

[Borodog]

Ok, I shouldn't have said "dimensionally incorrect." However, just because angles are physically dimensionless does not imply that they are unitless. The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

I stress to my students over and over that they should pay attention to the units, that the units will not let them down. Until we get to rotational kinematics, and the units of radians suddenly vanish for little better than handwaving reasons. This could all have been avoided by choosing a constant of proportionality of 1 radian^-1 rather than simply 1.

[Borodog]

[ QUOTE ]
I don't understand. radians are dimensionless units of angular displacement. what can be so hard about teaching a bunch of 14 year olds about this?

[/ QUOTE ]

They grow college kids young where you live.

[ QUOTE ]
I didn't have much difficulty in learning about the concept of radians (not trying to brag.) Just break down the scientific justification of the radian measurement to these kids, i'm sure they'll follow.

[/ QUOTE ]

You are missing the point. I beat them over the head with units for 6 weeks, and then suddenly tell them, "Oh, yeah, those units of radians? They magically disappear."

[Borodog]

Phil,

[ QUOTE ]
You are ignoring this troll.

[/ QUOTE ]

Congratulations.

[Fels krone]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
'cause s doesn't have units length*angle. Or at least that's my guess.

[/ QUOTE ]

Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[/ QUOTE ]

I still don't understand what is your problem with the given expression. That must make me pretty dumb, since I "do" rotational kinematics for a living. (I design attitude control systems for spacecraft.)

FWIW, where possible, I prefer to express 3-dimensional rotational kinematics in terms of the rate of change of a quaternion: (see equation 9, p. 14)

Quaternion dynamics (http://www.math.unm.edu/~vageli/papers/rrr.pdf)

[/ QUOTE ]

Attitude Control Systems, heh heh

Im not sure where the confusion is, radians are unitless. If you dont like that, go back to geometry and convince yourself of why they have no units. I learned why a long time ago, but I dont worry about it every time I have to work a problem with radians.

[Magic_Man]

[ QUOTE ]
The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

[/ QUOTE ]

To be fair, it's only "correct" for one system of LINEAR measurement also. S has to be the same units as R. Doesn't seem any more arbitrary than saying that theta has to be in rads.

[Borodog]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
'cause s doesn't have units length*angle. Or at least that's my guess.

[/ QUOTE ]

Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[/ QUOTE ]

I still don't understand what is your problem with the given expression. That must make me pretty dumb, since I "do" rotational kinematics for a living. (I design attitude control systems for spacecraft.)

FWIW, where possible, I prefer to express 3-dimensional rotational kinematics in terms of the rate of change of a quaternion: (see equation 9, p. 14)

Quaternion dynamics (http://www.math.unm.edu/~vageli/papers/rrr.pdf)

[/ QUOTE ]

Attitude Control Systems, heh heh

Im not sure where the confusion is, radians are unitless. If you dont like that, go back to geometry and convince yourself of why they have no units. I learned why a long time ago, but I dont worry about it every time I have to work a problem with radians.

[/ QUOTE ]

A radian is emphatically NOT unitless; it is physically dimensionless. You can tell that it is not unitless because it has a unit, i.e. 1 radian is distinct from 2 radians.

[bluesbassman]

Okay, now I understand what you are saying. I would simply teach that, by definition, all variables which represent angles in mathematical expressions are radians.

I agree though that you need to be picky about units. If you want to really be a jerk (like me), check your bill for units next time you eat out at a restaurant. Often you will find there is a number shown which apparently represents what you owe for the meal, but includes no monetary units or dollar symbol. It's then great fun to sincerely ask the server whether that number represents amount owed in dollars (or whatever). If you are dining with a wife/girlfriend, she will get REALLY mad at you. Yes, I have actually done this. /images/graemlins/blush.gif

(But at least I tipped the server extremely generously for putting up with me.)

[Borodog]

[ QUOTE ]
[ QUOTE ]
The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

[/ QUOTE ]

To be fair, it's only "correct" for one system of LINEAR measurement also. S has to be the same units as R. Doesn't seem any more arbitrary than saying that theta has to be in rads.

[/ QUOTE ]

In fact, no. For example, 1mi = 1609m.

[Duke]

Boro,

So you're really annoyed by unit-less angles in principle, or you're just annoyed that it makes the teaching job harder? I'm actually having a difficult seeing it through your eyes, since I guess I internalized the concepts much differently.

I'm fairly certain that introducing a magic constant of 1/rad would be far more confusing to everyone than just realizing that a radian is just a pure number.

[Borodog]

[ QUOTE ]
Boro,

So you're really annoyed by unit-less angles in principle,

[/ QUOTE ]

Angles are emphatically NOT UNITLESS; rather they are physically dimensionless.

[ QUOTE ]
or you're just annoyed that it makes the teaching job harder? I'm actually having a difficult seeing it through your eyes, since I guess I internalized the concepts much differently.

[/ QUOTE ]

a) It undermines everything that I have taught the students about units and dimensional analysis to have units that magically disappear when no other units have this property, and

b) I find it extremely aesthetically displeasing.

[ QUOTE ]
I'm fairly certain that introducing a magic constant of 1/rad would be far more confusing to everyone than just realizing that a radian is just a pure number.

[/ QUOTE ]

I doubt it. We introduce "magic constants" all the time in physics, and we always construct them so that the units work properly. For example, G = 6.67x10^-11 Nm^2/kg^2.

[Duke]

Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

If you tell someone that you want 3/5 of a pizza, and the Pizza has 8 slices, you'll be getting 4.8 slices. Certainly not 4.8 FRAC*slices*1/FRAC.

Perhaps the equivalence I see between these is illegitimate.

EDIT: Fixed middle paragraph so that the right guy is getting the pizza.

[gumpzilla]

[ QUOTE ]
Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

[/ QUOTE ]

This is the definition of a radian - the angle such that the arc length subtended by that angle on a circle is equal to the radius.

Boro - I see what you're saying. But I think if you just explicitly mention that radians are essentially just 2*pi times a fraction of a circle, it becomes pretty clear that they are dimensionless, AND that this is what you'd want for expressing lengths.

[Borodog]

[ QUOTE ]
Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel.

[/ QUOTE ]

Fine. Where then do the magical radians come from? It's simply inconsistent and hence annoying.

[ QUOTE ]
I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

If you tell someone that you want 3/5 of a pizza, and the Pizza has 8 slices, you'll be getting 4.8 slices. Certainly not 4.8 FRAC*slices*1/FRAC.

Perhaps the equivalence I see between these is illegitimate.


[/ QUOTE ]

I don't see any equivalence at all.

[Borodog]

[ QUOTE ]
[ QUOTE ]
Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

[/ QUOTE ]

This is the definition of a radian - the angle such that the arc length subtended by that angle on a circle is equal to the radius.

Boro - I see what you're saying. But I think if you just explicitly mention that radians are essentially just 2*pi times a fraction of a circle, it becomes pretty clear that they are dimensionless, AND that this is what you'd want for expressing lengths.

[/ QUOTE ]

Yes, I know. Perhaps I'm not making myself clear. The problem that I have with it is this:

meters = meter radians

The radians magically vanish when I have told the students to explicitly and carefully handle all units. And they only vanish under certain circumstances; only when converting from angular to linear measure, never when converting from one system of angular measure (radians) to another (degrees or revolutions, for example).

As a consequence, one can apparently magically put radians whereever one likes. My new house is going to be 3000 square footradians. My car gets 25 radianmiles/gallon on the highway, and 21 miles/gallonradian in the city.

Now if you'll excuse me, I'm going to go scramble up a couple of eggradians for my lunchradian.

/images/graemlins/tongue.gif

[Siegmund]

If you don't like the definition of a radian, I am not sure there is much we can do for you.

The arc length of a circle is in meters (or whatever), the radius of a circle is in meters (or whatever), so when we take the ratio of arc length to radius, the units had damn well better cancel out.

There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments. We have the word to help people understand why angles are measured that way, but I wouldn't call it the name of a unit.

An order of magnitude less confusing, IMO, than insisting that torques shall have units of newton*meters but not Joules.

Come to think of it, the two issues are quite similar. Two sides of a coin, or rather, the same sides of two coins (as the Player says to Rosencrantz.) Torques have directions and energies don't. Angles have a physical interpretation to them that garden variety real numbers don't.

[Borodog]

[ QUOTE ]
If you don't like the definition of a radian, I am not sure there is much we can do for you.

[/ QUOTE ]

Sigh. I give up. Read my posts again. That isn't what I said. I said that I dislike inconsistent handling of units.

[ QUOTE ]
The arc length of a circle is in meters (or whatever), the radius of a circle is in meters (or whatever), so when we take the ratio of arc length to radius, the units had damn well better cancel out.

[/ QUOTE ]

No [censored]. That isn't the problem.

[ QUOTE ]
There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments.

[/ QUOTE ]

Ok, as far as I know this is not common. Trigonometric functions take angles as arguments, else I could not take the sin of 30 degrees, which I can: sin(30 degrees) = 0.5.

[ QUOTE ]
We have the word to help people understand why angles are measured that way, but I wouldn't call it the name of a unit.

[/ QUOTE ]

It is absolutely the name of a unit. /images/graemlins/confused.gif If it weren't for units of angular measurement you couldn't convert between different systems of angular measurement, nor even make a measurement in the first place.

[ QUOTE ]
An order of magnitude less confusing, IMO, than insisting that torques shall have units of newton*meters but not Joules.

Come to think of it, the two issues are quite similar. Two sides of a coin, or rather, the same sides of two coins (as the Player says to Rosencrantz.) Torques have directions and energies don't. Angles have a physical interpretation to them that garden variety real numbers don't.

[/ QUOTE ]

Which is exactly my point. "The ratio of the arclength to the radius" is NOT the angle subtended, it is not an angle at all. It is proportional to the angle. They're two different things; one is a ratio of lengths, the other is a rotation. The two are proportional to each other, and the constant of proportionality defines a particular system of angular measure. Making that constant a unitless number like 1 makes no more sense to me than setting weight equal to mass.

[Magic_Man]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

[/ QUOTE ]

To be fair, it's only "correct" for one system of LINEAR measurement also. S has to be the same units as R. Doesn't seem any more arbitrary than saying that theta has to be in rads.

[/ QUOTE ]

In fact, no. For example, 1mi = 1609m.

[/ QUOTE ]

But the numbers aren't right unless you know that conversion factor. You can just pretend that 1m_rad = 1m, if it's a big issue. Following this convention, you can use s = r*theta for ALL units, if you simply know the conversion factor. For example, 1m_deg ~= .0175m, which is as arbitrary as 1mi = 1609m, no?

Edit: Of course, thinking this way is silly; I'm just pointing out that "the fact that the expression is correct for only one system of units tells you that there is monkey business going on" doesn't seem like a valid excuse. I agree that it's kind of strange and off-putting that radians are unitless, though.

[thylacine]

[ QUOTE ]

[ QUOTE ]
There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments.

[/ QUOTE ]

Ok, as far as I know this is not common. Trigonometric functions take angles as arguments, else I could not take the sin of 30 degrees, which I can: sin(30 degrees) = 0.5.


[/ QUOTE ]

What are the units of x^n in the following?

sin x = \sum{k=0}^{\infty} (-1)^k x^{2k+1} / ((2k+1)!)

I'm putting this here as food for thought, boro. I am starting to think you have enough of a point, so as to be not completely wrong.

[Borodog]

[ QUOTE ]
[ QUOTE ]

[ QUOTE ]
There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments.

[/ QUOTE ]

Ok, as far as I know this is not common. Trigonometric functions take angles as arguments, else I could not take the sin of 30 degrees, which I can: sin(30 degrees) = 0.5.


[/ QUOTE ]

What are the units of x^n in the following?

sin x = \sum{k=0}^{\infty} (-1)^k x^{2k+1} / ((2k+1)!)

I'm putting this here as food for thought, boro. I am starting to think you have enough of a point, so as to be not completely wrong.

[/ QUOTE ]

Yes, I know; this point is in the wiki article on radians. I would argue that there is simply a great deal of inherent sloppiness in the subject, due to the inherent sloppiness of mathematicians when it comes to units. Hence the title of the thread. /images/graemlins/tongue.gif

If I concede, for example, that the arguments of trigonometric functions are not angles but pure numbers, that means that I cannot actually do things like this: y = r*sin(30 degrees), yet everyone knows what this means. If true, then there is some handwaving going on in the middle.

On the other hand, if you concede that the arguments of trigonometric functions are indeed angles, then expansions like this: sin(theta) = theta - 0.5*theta^2 + . . . must be seen to contain the handwaving.

Where the sloppiness comes in is that if we use the radian system of angular measure and look the other way at the appropriate times and sweep the units of radians under the rug, it all works out in the end. While that is certainly enough justification for most people, it still offends my delicate sensibilities.

[holmansf]

[ QUOTE ]
Making that constant a unitless number like 1 makes no more sense to me than setting weight equal to mass.


[/ QUOTE ]

These aren't comparable at all, in my opinion. Setting weight equal to mass would imply that the force of gravity is the same everywhere. While I personally have never performed an experiment to contradict this, I'm pretty well convinced that it's not true.

Defining angle as the ratio of arclength to radius makes calculations easy.

[Siegmund]

Hmm. Didn't cross my mind that the notion that trig functions take pure numbers as arguments was a questionable one. They are useful for a lot of things to do with angles, but they make nice snaky pictures weaving back and forth across the x-axis endlessly, too.

Here's a counter-question for you, borodog: is a "dozen" a unit of measurement?

Six dozen can be converted to, or is another name for, 72, depending on your mindset. Most of us are happy to say that 72 is a pure number.

One approach is to say these are pure constants, not units. "Six dozen" is evaluated just like any other instance of 6k where k happens to be 12. "Forty-five degrees" is evaluated just like any other instance of 45k where k happens to be pi/90.

Physicists (it seems to me) are in the habit of having to specify their standard unit for everything they measure - as when they choose whether to take the meter or the Planck length or some other convenient amount to be "1 unit of distance." Maybe you'd find it logical to add to your list of fundamental units the "one," to be used as a yardstick for comparing all of those things that the rest of us were happy to refer to as pure numbers. Now you can add "1 dozen = 12 ones", "1 degree = pi/90 radians", and something like "1 radian = 1 one travelled along the perimeter of a unit circle" to your definition list. (People don't usually explicitly put, for instance, 1 Hertz = 1 Becquerel on such lists... but you've gotta have SOME notion in physics of things being formally equal but given different names to keep them clear in context.)
SI puts radians and steradians in a special category of "dimensionless derived units" but does not give a name to the dimensionless base unit. Is that the problem?

From the mathematical perspective, of course, it doesn't make a hoot of difference to us, since we treat all the units as if they are constants bouncing along for the ride in equations.

[Metric]

This may amount to using an elephant gun where a pocket knife would do, but here is one way to think of it:

We're really trying to make sense of the following equation:

ds^2 = g_i_j dx^i dx^j (summation notation implied)

The left hand side is the square of a physical distance -- a directly measurable thing, with units of meters^2, if you like.

The right hand side consists of a metric, and coordinates. The coordinates dx, etc. can be thought of as having "dimensionless units" (if that makes any sense). Degrees, radians, etc. Basically, they are just labels for which coordinate system you are using. The metric, then, has units of distance^2/(dimensionless units)^2.

Now you can transform back and forth between any system of coordinates that you like -- the metric will look different in different coordinates, of course, but the dimensionality will always work out.

[Duke]

After careful consideration, I agree with Borodog that it's retarded. This hinges on the fact that a radian is technically called an SI derived unit. I think that in that respect, someone was being an idiot.

If they want to call it a unit, then I'll even support the addition of a conversion constant called the "Borodog" or something to make the units work out right.

The best solution, though, would be to just stop calling it a unit. I don't internalize it as a unit, and neither should you!

[Borodog]

[ QUOTE ]
Hmm. Didn't cross my mind that the notion that trig functions take pure numbers as arguments was a questionable one. They are useful for a lot of things to do with angles, but they make nice snaky pictures weaving back and forth across the x-axis endlessly, too.

Here's a counter-question for you, borodog: is a "dozen" a unit of measurement?

[/ QUOTE ]

No, a dozen is not a unit of measurement. A dozen what? Now, "a dozen eggs" could be a unit of measurement.

[ QUOTE ]
Six dozen can be converted to, or is another name for, 72, depending on your mindset. Most of us are happy to say that 72 is a pure number.

[/ QUOTE ]

True, but a dozen is also a pure number.

[ QUOTE ]
One approach is to say these are pure constants, not units. "Six dozen" is evaluated just like any other instance of 6k where k happens to be 12. "Forty-five degrees" is evaluated just like any other instance of 45k where k happens to be pi/90.

[/ QUOTE ]

This seems ad hoc and somehow misses the point. While the radian is certainly a "natural" unit for angular measure, it is still, without a doubt, a unit. You can't measure if you don't have a unit. Period.

[ QUOTE ]
Physicists (it seems to me) are in the habit of having to specify their standard unit for everything they measure - as when they choose whether to take the meter or the Planck length or some other convenient amount to be "1 unit of distance." Maybe you'd find it logical to add to your list of fundamental units the "one," to be used as a yardstick for comparing all of those things that the rest of us were happy to refer to as pure numbers. Now you can add "1 dozen = 12 ones", "1 degree = pi/90 radians", and something like "1 radian = 1 one travelled along the perimeter of a unit circle" to your definition list. (People don't usually explicitly put, for instance, 1 Hertz = 1 Becquerel on such lists... but you've gotta have SOME notion in physics of things being formally equal but given different names to keep them clear in context.)
SI puts radians and steradians in a special category of "dimensionless derived units" but does not give a name to the dimensionless base unit. Is that the problem?

[/ QUOTE ]

But there is a name given to the dimensionless base unit--it's called the radian. I think we're going around in circles. Pun intended. /images/graemlins/wink.gif

[ QUOTE ]
From the mathematical perspective, of course, it doesn't make a hoot of difference to us, since we treat all the units as if they are constants bouncing along for the ride in equations.

[/ QUOTE ]

Which, in my opinion, is a weakness that math instructors inflict on their unsuspecting students. /images/graemlins/wink.gif Mathematicians often totally ignore units, or pretend that they have little importance, or even no real meaning! Famously at a school I used to teach at there was a math teacher who said that units were meaningless and gave this example as an illustration: "Take a carton of eggs. 6 eggs by 2 eggs. How many eggs do you have? 12 eggs squared? Nonsense!" /images/graemlins/tongue.gif

Riddle me this. If you launch a projectile with speed v at an angle theta above the horizontal, how far will it go? Before you even work out the problem you know that the answer must be proportional to v^2/g. Why? Because that's the only possible way to produce the proper units. Furthermore, if angles are truly "pure numbers" without units, it would be conceivable that the answer could be proportional to some power of the angle--but that can't be. It can only be proportional to some trigonometric functions of the angle, because the outputs of _those_ are truly unitless pure numbers.

[holmansf]

Shouldn't the dx^i's be in meters?

[Borodog]

[ QUOTE ]
After careful consideration, I agree with Borodog that it's retarded.

[/ QUOTE ]

Yay!

[Metric]

As an example, take our coordinate system to be polar coordinates (in radians), and take the displacement to be a pure rotation. The only term is:

ds^2 = r^2 dtheta^2

Adding units, we have:

ds^2 (meters squared) = r^2 (meters squared)/(radians squared) dtheta^2 (radians squared)

Now let's say we want to transform to degrees. The metric has a well defined transformation property -- basically you multiply it by dx(old units)/dx(new units) twice (ignoring index gymnastics). So in the new "degrees" coordinate system, you simply end up multiplying the old "radian" metric by (radians/degrees)^2 to get the new "degrees" metric, which accounts exactly for the conversion factor.

Of course, radians are "more natural" simply because in those coordinates the metric looks extra simple, with no numerical factor besides "1," which happens to make ignoring units extra convenient as well (which is what is driving you crazy).

[Metric]

[ QUOTE ]
Shouldn't the dx^i's be in meters?

[/ QUOTE ]
Not really -- it's just convention. In relativity (where I learned geometry), people put emphasis on the fact that coordinates have no intrinsic meaning, whereas physical distances do. Hence the metric gets the "concrete" units, and coordinates merely get "lables."

[_brady_]

[ QUOTE ]
Famously at a school I used to teach at there was a math teacher who said that units were meaningless and gave this example as an illustration: "Take a carton of eggs. 6 eggs by 2 eggs. How many eggs do you have? 12 eggs squared? Nonsense!" /images/graemlins/tongue.gif

[/ QUOTE ]

(6 eggs/row)(2 rows) = 12 eggs

[thylacine]

[ QUOTE ]
[ QUOTE ]
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[ QUOTE ]
There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments.

[/ QUOTE ]

Ok, as far as I know this is not common. Trigonometric functions take angles as arguments, else I could not take the sin of 30 degrees, which I can: sin(30 degrees) = 0.5.


[/ QUOTE ]

What are the units of x^n in the following?

sin x = \sum{k=0}^{\infty} (-1)^k x^{2k+1} / ((2k+1)!)

I'm putting this here as food for thought, boro. I am starting to think you have enough of a point, so as to be not completely wrong.

[/ QUOTE ]

Yes, I know; this point is in the wiki article on radians. I would argue that there is simply a great deal of inherent sloppiness in the subject, due to the inherent sloppiness of mathematicians when it comes to units. Hence the title of the thread. /images/graemlins/tongue.gif

If I concede, for example, that the arguments of trigonometric functions are not angles but pure numbers, that means that I cannot actually do things like this: y = r*sin(30 degrees), yet everyone knows what this means. If true, then there is some handwaving going on in the middle.

On the other hand, if you concede that the arguments of trigonometric functions are indeed angles, then expansions like this: sin(theta) = theta - 0.5*theta^2 + . . . must be seen to contain the handwaving.

Where the sloppiness comes in is that if we use the radian system of angular measure and look the other way at the appropriate times and sweep the units of radians under the rug, it all works out in the end. While that is certainly enough justification for most people, it still offends my delicate sensibilities.

[/ QUOTE ]

ROFLMAO those sloppy handwaving under-the-rug-sweeping mathematicians, it's a wonder they can prove anything. Just as well those physicist are around to defend Fort Rigor.

Borodog, do you know what a semigroup graded ring is?

[holmansf]

[ QUOTE ]

Not really -- it's just convention. In relativity (where I learned geometry), people put emphasis on the fact that coordinates have no intrinsic meaning, whereas physical distances do. Hence the metric gets the "concrete" units, and coordinates merely get "lables."


[/ QUOTE ]

Hmm, actually I think for things to work out in all cases the units of the dx^i's must be the same as the units of the x^i's.

[Duke]

The contents of this thread could be made into a physics/math joke. You'd utilize both the sloppiness of mathematicians, and the joy that a physicist has when experiment comes within a few orders of magnitude of the theoretical result.

[flipdeadshot22]

[ QUOTE ]
The contents of this thread could be made into a physics/math joke. You'd utilize both the sloppiness of mathematicians, and the joy that a physicist has when experiment comes within a few orders of magnitude of the theoretical result.

[/ QUOTE ]

i'm just relieved no one found a way to sneak religion into this one.

[jason1990]

[ QUOTE ]
On the other hand, if you concede that the arguments of trigonometric functions are indeed angles, then expansions like this: sin(theta) = theta - 0.5*theta^2 + . . . must be seen to contain the handwaving.

[/ QUOTE ]
It seems, then, that any series expansion would pose difficulties for dimensional analysis. Suppose, for example, you have a dynamical system described by some complicated ODE which you solve by representing its solution as a series. Or suppose you are considering heat flow or particle diffusion and you represent the evolving temperature/density function as an eigenfunction expansion, such as

u(t,x) = \sum_{n=0}^\infty c_n \phi_n(x) e^{-\lambda_n t},

where \lambda_n are the eigenvalues, \phi_n are the eigenfunctions, and c_n is the L^2 inner product of \phi_n with u(0,x). In this case, the only quantities involved are time, space, and temperature or particles per unit volume. Does this equation make sense in terms of dimensional analysis?

I wonder if the "handwaving" is in dimensional analysis itself, i.e. in the treatment of units as though they were algebraic quantities capable of being canceled, multiplied, etc. I realize that this is a very powerful heuristic tool, but it is certainly not mathematical rigorous. (At least, I have never seen a rigorous presentation of it.) So perhaps it is not surprising that an occasional "contradiction" will pop up, or that there will be situations in which it simply does not make sense to apply it.

[Borodog]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

[ QUOTE ]
There is a reason that in trigonometry classes we always say "the sin of pi/4" NOT "the sin of pi/4 radians." The trig functions take pure numbers as arguments.

[/ QUOTE ]

Ok, as far as I know this is not common. Trigonometric functions take angles as arguments, else I could not take the sin of 30 degrees, which I can: sin(30 degrees) = 0.5.


[/ QUOTE ]

What are the units of x^n in the following?

sin x = \sum{k=0}^{\infty} (-1)^k x^{2k+1} / ((2k+1)!)

I'm putting this here as food for thought, boro. I am starting to think you have enough of a point, so as to be not completely wrong.

[/ QUOTE ]

Yes, I know; this point is in the wiki article on radians. I would argue that there is simply a great deal of inherent sloppiness in the subject, due to the inherent sloppiness of mathematicians when it comes to units. Hence the title of the thread. /images/graemlins/tongue.gif

If I concede, for example, that the arguments of trigonometric functions are not angles but pure numbers, that means that I cannot actually do things like this: y = r*sin(30 degrees), yet everyone knows what this means. If true, then there is some handwaving going on in the middle.

On the other hand, if you concede that the arguments of trigonometric functions are indeed angles, then expansions like this: sin(theta) = theta - 0.5*theta^2 + . . . must be seen to contain the handwaving.

Where the sloppiness comes in is that if we use the radian system of angular measure and look the other way at the appropriate times and sweep the units of radians under the rug, it all works out in the end. While that is certainly enough justification for most people, it still offends my delicate sensibilities.

[/ QUOTE ]

ROFLMAO those sloppy handwaving under-the-rug-sweeping mathematicians, it's a wonder they can prove anything. Just as well those physicist are around to defend Fort Rigor.

Borodog, do you know what a semigroup graded ring is?

[/ QUOTE ]

Nope.

And just in case you think I am unfairly slandering (or libeling; I forget which) mathematicians, I defy you to find a high school mathematics text that correctly treats units. In fact, I have never seen one that treats them at all.

[Borodog]

[ QUOTE ]
[ QUOTE ]
On the other hand, if you concede that the arguments of trigonometric functions are indeed angles, then expansions like this: sin(theta) = theta - 0.5*theta^2 + . . . must be seen to contain the handwaving.

[/ QUOTE ]
It seems, then, that any series expansion would pose difficulties for dimensional analysis. Suppose, for example, you have a dynamical system described by some complicated ODE which you solve by representing its solution as a series. Or suppose you are considering heat flow or particle diffusion and you represent the evolving temperature/density function as an eigenfunction expansion, such as

u(t,x) = \sum_{n=0}^\infty c_n \phi_n(x) e^{-\lambda_n t},

where \lambda_n are the eigenvalues, \phi_n are the eigenfunctions, and c_n is the L^2 inner product of \phi_n with u(0,x). In this case, the only quantities involved are time, space, and temperature or particles per unit volume. Does this equation make sense in terms of dimensional analysis?

I wonder if the "handwaving" is in dimensional analysis itself, i.e. in the treatment of units as though they were algebraic quantities capable of being canceled, multiplied, etc. I realize that this is a very powerful heuristic tool, but it is certainly not mathematical rigorous. (At least, I have never seen a rigorous presentation of it.) So perhaps it is not surprising that an occasional "contradiction" will pop up, or that there will be situations in which it simply does not make sense to apply it.

[/ QUOTE ]

To my knowledge dimensional analysis is totally rigorous until you get to the problem of angles; and even then it could be handled completely rigorously. In the example you gave, any expansion should be dimensionally correct in every term, try it.

For example, the Taylor series expansion for the position of a particle: x = x_0 + x'_0*t + (1/2!)*x''_0*t^2 + (1/3!)*x'''_0*t^3 + . . . (where the ' denotes a time derivative rather than the normal spacial derivative), which gives the undergrad student his first kinematics equation x = x_0 + v_0*t + (1/2)a*t^2, is dimensionally correct in every term, as it must be.

In your example, every term of the expansion will have units of temp or density or whatever, and the lambda must always have units of inverse time (to make the argument of the exponential dimensionless).

[jason1990]

[ QUOTE ]
For example, the Taylor series expansion for the position of a particle: x = x_0 + x'_0*t + (1/2!)*x''_0*t^2 + (1/3!)*x'''_0*t^3 + . . . (where the ' denotes a time derivative rather than the normal spacial derivative), which gives the undergrad student his first kinematics equation x = x_0 + v_0*t + (1/2)a*t^2, is dimensionally correct in every term, as it must be.

[/ QUOTE ]
I agree that Taylor expansions seem to work out fine. I thought about that after posting.

[ QUOTE ]
In your example, every term of the expansion will have units of temp or density or whatever, and the lambda must always have units of inverse time (to make the argument of the exponential dimensionless).

[/ QUOTE ]
In the one-dimensional diffusion equation on a finite interval of length L with Dirichlet boundary conditions, \lambda_n = n^2\pi^2/L^2, if I am remembering right. Doesn't that mean the eigenvalues have units of (space)^{-2}? Or, thinking about it another way, if the lambda units were inverse time, then wouldn't they be invariant under a change of spatial units, whereas we know that they are not? And if their units really are (space)^{-2}, then what kind of units are

exp{(time)/(space)^2}?

[jason1990]

[ QUOTE ]
To my knowledge dimensional analysis is totally rigorous until you get to the problem of angles; and even then it could be handled completely rigorously.

[/ QUOTE ]
I suppose we could have different interpretations of "rigorous." As a mathematician, a rigorous presentation of dimensional analysis for me would involving working in some product space R x U, where R is your number system and U is your set of units, equipped with some kind of algebraic operations. I would imagine that in order to do calculus in this product space, one might have to use nonstandard analysis on the R space. It would be in this context that one could prove theorems about, for example, the consistency of dimensional analysis. I have never seen anything like this.

[gumpzilla]

[ QUOTE ]

True, but a dozen is also a pure number.

[/ QUOTE ]

Well, so are radians. Radians are just 2*pi times a fraction of a circle - what's that if not a pure number? It's just a fancy, dressed-up name for a fraction.

[gumpzilla]

[ QUOTE ]
[ QUOTE ]


ROFLMAO those sloppy handwaving under-the-rug-sweeping mathematicians, it's a wonder they can prove anything. Just as well those physicist are around to defend Fort Rigor.

Borodog, do you know what a semigroup graded ring is?

[/ QUOTE ]

Nope.

[/ QUOTE ]

I believe the graded part refers to, essentially, units. Everything has some factor epsilon that multiplies at first, and you can keep track of the various orders of epsilon as you perform the multiplications.

[Borodog]

jason1990,

I'd have to see the full analysis because it's been so long, but suffice it to say, if it isn't dimensionally correct, it's wrong. Your problem was probably non-dimensionalized, which is a fairly standard thing to do.

As an example that I am a little more familiar with, the time evolution of the state of a quantum harmonic oscillator:

|psi(t)> = e^{-i*(n+1/2)*omega*t} [ c_n*|n> + c_(n+1)*e^{-i*omega*t}*|n + 1> ]

omega has dimensions of inverse time, making the argument of the exponents dimensionless.

[Borodog]

[ QUOTE ]
[ QUOTE ]

True, but a dozen is also a pure number.

[/ QUOTE ]

Well, so are radians. Radians are just 2*pi times a fraction of a circle - what's that if not a pure number?

[/ QUOTE ]

Something else. I'm not sure how to describe it. Rotations happen in the real world, like masses, distances, times. Pure numbers don't.

[jason1990]

I think my expansion involves a constant which has been set to 1. This constant has just the right dimensions to make the argument of the exponential dimensionless. But this makes me wonder something else...

You seemed to use a "theorem" (or perhaps a "metatheorem") of dimensional analysis, which says informally that whatever appears as an argument in an exponential must be dimensionless. This led me to think along the following lines:

Suppose x has units "meters" (m). Then x^2 has units m^2, and 1/x has units 1/m. But what are the units of e^x? Is the expression e^x meaningless in this case, since x is not dimensionless? What about this argument:

y = e^x iff
x = ln(y) iff
x = int_1^y (1/u) du

In the last line, y and u have the same units; and x and 1/u have the same units. So y and 1/x have the same units. In other words, if x is in meters, then e^x is in 1/meters. Is this right? If not, then where is the flaw in my argument?

[jason1990]

[ QUOTE ]
y = e^x iff
x = ln(y) iff
x = int_1^y (1/u) du

In the last line, y and u have the same units; and x and 1/u have the same units.

[/ QUOTE ]
Nevermind. In the last line, x would have the same units as y*(1/u), making x dimensionless. Very interesting...

[Neuge]

[ QUOTE ]
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True, but a dozen is also a pure number.

[/ QUOTE ]

Well, so are radians. Radians are just 2*pi times a fraction of a circle - what's that if not a pure number?

[/ QUOTE ]

Something else. I'm not sure how to describe it. Rotations happen in the real world, like masses, distances, times. Pure numbers don't.

[/ QUOTE ]
Angles are used to describe how masses, distances, and time change relative to their rotational axis. Rotations are reference frame constructs, not a separate quantity to be measured.

[Borodog]

If you can figure out how to raise something to the power of a meter, or take the logarithm of a meter, let me know.

In fact, while I could be wrong (I haven't thought about it extensively), I doubt there is any physical quantity in the real world that requires units of, for example, the square root of distance, length, time, etc. (i.e. the fundamental units).

[almostbusto]

[ QUOTE ]
If you can figure out how to raise something to the power of a meter, or take the logarithm of a meter, let me know.

In fact, while I could be wrong (I haven't thought about it extensively), I doubt there is any physical quantity in the real world that requires units of, for example, the square root of distance, length, time, etc. (i.e. the fundamental units).

[/ QUOTE ]

people take the log of a distance in meters all the time. there are obvious reasons why you would want to do this.

This is kind of unrelated, but its one of my favorite 'laws'.
http://en.wikipedia.org/wiki/Benford's_law
[ QUOTE ]

More precisely, Benford's law states that the leading digit d (d ∈ {1, ..., b − 1} ) in base b (b ≥ 2) occurs with probability proportional to logb(d + 1) − logb(d)

[/ QUOTE ]

there is an instance where someone is taking the logs of meters, feet, time.

EDIT: i guess the quote didn't come through great, just read the link.

[Borodog]

Uh, d is a digit? It's a pure number.

And no, people do not take the log of distance in meters all the time. They might take the log of a non-dimensionalized distance: log(d/d0), where d0 = 1m, but they emphatically do not take the log of a meter, because it is nonsensical to do so. And if they do, they are being sloppy.

[Metric]

[ QUOTE ]
[ QUOTE ]

Not really -- it's just convention. In relativity (where I learned geometry), people put emphasis on the fact that coordinates have no intrinsic meaning, whereas physical distances do. Hence the metric gets the "concrete" units, and coordinates merely get "lables."


[/ QUOTE ]

Hmm, actually I think for things to work out in all cases the units of the dx^i's must be the same as the units of the x^i's.

[/ QUOTE ]

How does one translate a coordinate point with no additional structure into a physically measurable distance?

[almostbusto]

[ QUOTE ]
Uh, d is a digit? It's a pure number.

And no, people do not talk the log of distance in meters all the time. They might take the log of a non-dimensionalized distance: log(d/d0), where d0 = 1m, but they emphatically do not take the log of a meter, because it is nonsensical to do so. And if they do, they are being sloppy.

[/ QUOTE ]

i guess i don't understand what you are talking about at all. i am a math major and a number is a number for the most part.

are you saying "log (100 meters)" is working with a "pure number"? because benfords law is stating that if you measure the lengths of rivers in meters, the probability of a certain leading digit, in meters, is proportional to its log.

of course in basic formulation of the law, you are only concerned with one signifcant digit.


for one, i see the time to complete algorithms expressed in logs all the time. how is that sloppy? its transforming the data so it is much much easier to work with.

[Borodog]

[ QUOTE ]
[ QUOTE ]
Uh, d is a digit? It's a pure number.

And no, people do not talk the log of distance in meters all the time. They might take the log of a non-dimensionalized distance: log(d/d0), where d0 = 1m, but they emphatically do not take the log of a meter, because it is nonsensical to do so. And if they do, they are being sloppy.

[/ QUOTE ]

i guess i don't understand what you are talking about at all. i am a math major and a number is a number for the most part.

are you saying "log (100 meters)" is working with a "pure number"? because benfords law is stating that if you measure the lengths of rivers in meters, the probability of a certain leading digit, in meters, is proportional to its log.

of course in basic formulation of the law, you are only concerned with one signifcant digit.

[/ QUOTE ]

I'm saying it is meaningless to take the log of 100m because there is no physical interpretation of the log of a meter. However, log (100m/1m) is a perfectly reasonable thing to do.

[Borodog]

[ QUOTE ]
for one, i see the time to complete algorithms expressed in logs all the time. how is that sloppy? its transforming the data so it is much much easier to work with.

[/ QUOTE ]

"log(t)" is sloppy. log(t/t0) is not.

[almostbusto]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Uh, d is a digit? It's a pure number.

And no, people do not talk the log of distance in meters all the time. They might take the log of a non-dimensionalized distance: log(d/d0), where d0 = 1m, but they emphatically do not take the log of a meter, because it is nonsensical to do so. And if they do, they are being sloppy.

[/ QUOTE ]

i guess i don't understand what you are talking about at all. i am a math major and a number is a number for the most part.

are you saying "log (100 meters)" is working with a "pure number"? because benfords law is stating that if you measure the lengths of rivers in meters, the probability of a certain leading digit, in meters, is proportional to its log.

of course in basic formulation of the law, you are only concerned with one signifcant digit.

[/ QUOTE ]

I'm saying it is meaningless to take the log of 100m because there is no physical interpretation of the log of a meter. However, log (100m/1m) is a perfectly reasonable thing to do.

[/ QUOTE ]

well in physics that might make sense because you are often working with different units simultaneously. but if you working in a context where everything in the same units, it doesn't matter. at least i don't see why it would.


an analogy. some nit could say 2+2=4 is sloppy, what you really mean is (2+0i+0j+0k+0l) + (2+0i+0j+0k+0l) = 4+0i+0j+0k+0l

but if are working in a context where i,j,k,l don't exist then it's worthless.

like if you are dealing with dollars, and you want to calculate the increase in your checking account, you don't need to convert those dollars to 'pure numbers' to calculate a percentage.

[Borodog]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Uh, d is a digit? It's a pure number.

And no, people do not talk the log of distance in meters all the time. They might take the log of a non-dimensionalized distance: log(d/d0), where d0 = 1m, but they emphatically do not take the log of a meter, because it is nonsensical to do so. And if they do, they are being sloppy.

[/ QUOTE ]

i guess i don't understand what you are talking about at all. i am a math major and a number is a number for the most part.

are you saying "log (100 meters)" is working with a "pure number"? because benfords law is stating that if you measure the lengths of rivers in meters, the probability of a certain leading digit, in meters, is proportional to its log.

of course in basic formulation of the law, you are only concerned with one signifcant digit.

[/ QUOTE ]

I'm saying it is meaningless to take the log of 100m because there is no physical interpretation of the log of a meter. However, log (100m/1m) is a perfectly reasonable thing to do.

[/ QUOTE ]

well in physics that might make sense because you are often working with different units simultaneously. but if you working in a context where everything in the same units, it doesn't matter. at least i don't see why it would.


an analogy. some nit could say 2+2=4 is sloppy, what you really mean is (2+0i+0j+0k+0l) + (2+0i+0j+0k+0l) = 4+0i+0j+0k+0l

but if are working in a context where i,j,k,l don't exist then it's worthless.

like if you are dealing with dollars, and you want to calculate the increase in your checking account, you don't need to convert those dollars to 'pure numbers' to calculate a percentage.

[/ QUOTE ]

Ugh. You are SO making my point from the title of the thread . . . /images/graemlins/tongue.gif

[almostbusto]

[ QUOTE ]


Ugh. You are SO making my point from the title of the thread . . . /images/graemlins/tongue.gif

[/ QUOTE ]

i am sure i am, i am just trying to understand your position.

i see absolutely no upshot of formally converting $ to numbers before messing around with your checkbook. THIS WOULD NEVER HAPPEN in a proof involving the increase in your checkbook. it would be suppressed because a monkey could realize what you are doing.

so whats the upshot other than the satisfaction nitty people get from being nits.

i don't read physics literature, but at the graduate level, [censored] like this has to be suppressed all the time. who would want to take the time to notify the reader than its the log (t/t0) rather than log (t) because one is just "silly"

regarding your OP though, i mean radians are inherently dimensionless so that shouldn't be what gets you worked up. the examples i mentioned might make some sense. but i don't get your point, yet.

[jamzfive]

[ QUOTE ]
Ok, I shouldn't have said "dimensionally incorrect." However, just because angles are physically dimensionless does not imply that they are unitless. The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

I stress to my students over and over that they should pay attention to the units, that the units will not let them down. Until we get to rotational kinematics, and the units of radians suddenly vanish for little better than handwaving reasons. This could all have been avoided by choosing a constant of proportionality of 1 radian^-1 rather than simply 1.

[/ QUOTE ]


There is no monkey business going on. All angles measurements are ratios, whether degrees, radians, or gradians. They all just represent what fraction of an entire circle has been swept out by an angle. The different names given aren't "units", but descriptors of how the ratio was taken, so you can be sure to use the right type of ratio in the right calculation.

The difference between angle measure and other types of measurements that really do have units is that, in angle measure, there's some quantity that can be considered "the whole". For mass, length, volume, electrical charge, etc. there is no "whole". There is some arbitrarily defined base quantity, and everything else is multiples of that. But angles are different. "One radian" and "One degree" weren't defined first, and then everything else is just a multiple of that. The "whole" is the circle, and then an arbitrary number of subdivisions was selected (a circle can be subdivided into 360 pieces, or 2*pi pieces, or 400 pieces), and then the angle measure is just a ratio of what part of the entire circle is swept out, expressed in how ever many subdivisions were selected for your application.

Clearly you've given this a lot of thought, and you're pretty well convinced of what you believe (even though most people are saying you're wrong), but try this: instead of talking about angles in terms of degrees or radians, think about them JUST in terms of percents. We can agree that percents are unitless right? Well, don't think 90 "degrees" or pi/2 "radians" - think about "25%" of the circle.

jb

[jason1990]

Okay, I think I am beginning to see this. In calculus, one way to define the logarithm function is as the integral from 1 to x of (1/x)dx. But this endows the point 1 with a "special" status.

If we regard the half-line (0,\infty) as a smooth manifold with no natural coordinate system, then we would be unable to define ln(x). We could only define ln(x/x_0). Does this sound right? It sounds right to me, and it's pretty cool. I've never thought about this, but then again I'm not a differential geometer.

Maybe your thread should have been titled, "Damn you mathematicians, except for differential geometers."

[Borodog]

[ QUOTE ]
[ QUOTE ]


Ugh. You are SO making my point from the title of the thread . . . /images/graemlins/tongue.gif

[/ QUOTE ]

i am sure i am, i am just trying to understand your position.

i see absolutely no upshot of formally converting $ to numbers before messing around with your checkbook. THIS WOULD NEVER HAPPEN in a proof involving the increase in your checkbook. it would be suppressed because a monkey could realize what you are doing.

[/ QUOTE ]

There's no need to "convert $ to numbers" because a) You can't, and b) All the units in my checkbook are identical. When I add and subtract dollars to and from dollars I get dollars. When the number of dollars is increased by 10% they are still dollars.

[ QUOTE ]
so whats the upshot other than the satisfaction nitty people get from being nits.

[/ QUOTE ]

Uh, a deeper understanding of the physical world? Being able to know if your solution is possibly correct? Predicting the form of the solution before you start? Not having your bridge fall down or your Martian lander crash because you didn't pay attention to your units?

What's the force exerted on a fireman by the firehose? I don't even have to work the problem to know that it must be proportional to rho*v^2*d^2, where rho is the density of the water, v is its speed, and d is the diameter of the stream, because that's the only dimensionally correct way to combine those factors to get units of force.

[ QUOTE ]

i don't read physics literature, but at the graduate level, [censored] like this has to be suppressed all the time.

[/ QUOTE ]

You are 100% wrong.

[ QUOTE ]
who would want to take the time to notify the reader than its the log (t/t0) rather than log (t) because one is just "silly"

[/ QUOTE ]

One is not silly, one is simply wrong. Not only would I never be caught dead trying to take the logarithm of a meter or a second, such an egregious mistake would never make it past peer review.

[ QUOTE ]
regarding your OP though, i mean radians are inherently dimensionless so that shouldn't be what gets you worked up. the examples i mentioned might make some sense. but i don't get your point, yet.

[/ QUOTE ]

The point is simple. Units shouldn't magically disapper. Writing "s = r*theta" requires the units of radians to magically disappear. This could have been avoided with a simple constant of proportionality: s = c*r*theta, where c = 1/radian.

[gumpzilla]

[ QUOTE ]

There is no monkey business going on. All angles measurements are ratios, whether degrees, radians, or gradians.

[/ QUOTE ]

This is true of all measurements with a unit, not just angular measurements.

[Borodog]

[ QUOTE ]
[ QUOTE ]
Ok, I shouldn't have said "dimensionally incorrect." However, just because angles are physically dimensionless does not imply that they are unitless. The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

I stress to my students over and over that they should pay attention to the units, that the units will not let them down. Until we get to rotational kinematics, and the units of radians suddenly vanish for little better than handwaving reasons. This could all have been avoided by choosing a constant of proportionality of 1 radian^-1 rather than simply 1.

[/ QUOTE ]


There is no monkey business going on. All angles measurements are ratios, whether degrees, radians, or gradians. They all just represent what fraction of an entire circle has been swept out by an angle. The different names given aren't "units", but descriptors of how the ratio was taken, so you can be sure to use the right type of ratio in the right calculation.

[/ QUOTE ]

You're begging the question. That same argument would apply to all measurements.

[ QUOTE ]
The difference between angle measure and other types of measurements that really do have units is that, in angle measure, there's some quantity that can be considered "the whole". For mass, length, volume, electrical charge, etc. there is no "whole".

[/ QUOTE ]

Uh, according to whom? There are fundamental units of charge (e, or 1/3 e if you want to talk about quarks), distance (the Planck length), and time (the Planck time).

[ QUOTE ]
There is some arbitrarily defined base quantity, and everything else is multiples of that. But angles are different. "One radian" and "One degree" weren't defined first, and then everything else is just a multiple of that. The "whole" is the circle, and then an arbitrary number of subdivisions was selected (a circle can be subdivided into 360 pieces, or 2*pi pieces, or 400 pieces), and then the angle measure is just a ratio of what part of the entire circle is swept out, expressed in how ever many subdivisions were selected for your application.

[/ QUOTE ]

This is absolutely as arbitrary. I'm surprised you can't see that. What makes 1 turn less arbitrary than a half turn or 2 turns? Nothing other than that it is convenient for us.

And even if it were somehow not arbitrary, that still does not magically make it not a unit. You can't make a measuremtn without a unit to refer to, whether it be a distance, a rotation, or whatever. The "whole" you refer to could be a unit: 1 rotation.

[ QUOTE ]
Clearly you've given this a lot of thought, and you're pretty well convinced of what you believe (even though most people are saying you're wrong),

[/ QUOTE ]

Read the thread again and recount. And what I "believe"? All I believe is that units should be treated consistently and that having things like:

meters = meter radians

is embarrassing. If angles do not have units, why can't I have s = r*theta with theta measured in degrees? Or rotations? Or gradians? It is _only_ because in the radian system of measure that it happens to work out if I magically drop the unit of angular measure that I get the correct numerical result that let's people get away with this kind of sloppiness.

[ QUOTE ]
but try this: instead of talking about angles in terms of degrees or radians, think about them JUST in terms of percents. We can agree that percents are unitless right? Well, don't think 90 "degrees" or pi/2 "radians" - think about "25%" of the circle.

[/ QUOTE ]

Great. Wonderful. Except "25% of a circle" is not an angle anymore than the ratio of the arclength to the radius is. In fact, I could USE % as a measure of angle, wherein 90 degrees = 25%, and "%" would have to be understood as shorthand for "25% of a rotation". But could I then use my new angular measurement in the equation s = r*theta? It's dimensionless, after all! Of course I couldn't. Again, it is ONLY because you happen to get the correct answer that allows people to magically drop the radians.

[Borodog]

[ QUOTE ]
[ QUOTE ]

There is no monkey business going on. All angles measurements are ratios, whether degrees, radians, or gradians.

[/ QUOTE ]

This is true of all measurements with a unit, not just angular measurements.

[/ QUOTE ]

Bingo. Succinct.

[Siegmund]

This thread has been an interesting read - though I have to say I think that all the damning needs to happen on the physicists' side of the river. A few random thoughts:

About dimensional analysis "without calculation"... as a practitioner in the field, you find this to be a convenient way of explaining it. If one actually writes out all the steps in the dimensional analysis, it turns out to take exactly the same amount of work as solving the actual problem does, if you're an "outsider" who has to see why each of those things has the units it does. It's a useful way to check an answer, or to intuit what form an answer should have, yes... I don't see it as fundametally different from a mathematician seeing a particular form of problem and immediately saying, well, if a solution exists, it must come from such and such family of curves.

It has certain limitations.... as you've said, even if you know the units of the final answer, you don't know to what power, if at all, various dimensionless quantities will appear in the calculation. The part of your argument that makes no sense is where you pull, out of thin air, the claim that its preposterous that an angle or a square of an angle or anything else like that could ever appear in the formula. It CAN, and it DOES - and you started this thread by posting a simple example where an angle and not a trigonometric function of an angle appears in a formula.

You seem to believe that log and exp (and gamma and bessel functions and every other function you've ever seen) take pure numbers as arguments. Why do you think trig functions are somehow different, and can take anything except a pure number as an argument? Dont forget exp(i*theta) = cos(theta)+i sin(theta), either -- whatever kind of argument you think exp, cos, and sin take it HAS to be the SAME kind of argument!

I was also a bit bothered by your condemnation of high school math textbooks for not teaching much about units. I have news for you: in math classes, we teach math. What do you want us to do, teach them physics instead? We teach manipulation of pure numbers because THAT IS WHAT THE CLASS IS ABOUT. Yes, sometimes applications to the "real world" are shown, with some discussion about how to transform real-world questions into equations to be solved, and yes, *in the process of that transformation*, one of the things that sometimes comes up is dealing with units.

Someone mentioned graded rings, and we've touched on adding meters to meters but not being able to take exps of meters. Whether units can be carried along for the ride depends almost entirely on whether stray x'es can be carried along for the ride. (All the linear operators are OK, pretty much, and almost everything else either has to be dimensionless or has some special restrictions on it.)

I also wanted to touch briefly on the assertion that one cannot measure anything without first choosing a unit. Does borodog admit counting is a measurement? I suppose he can say that "one of whatever you are counting" is your unit. If he'll call 1 "the unit by which pure numbers are measured" I suppose I won't disagree too violently. But if he'll admit to THAT he could write s = r * theta * 1 (or maybe it's s *1 = r* theta ) and be happy.

[Borodog]

[ QUOTE ]
This thread has been an interesting read - though I have to say I think that all the damning needs to happen on the physicists' side of the river. A few random thoughts:

About dimensional analysis "without calculation"... as a practitioner in the field, you find this to be a convenient way of explaining it. If one actually writes out all the steps in the dimensional analysis, it turns out to take exactly the same amount of work as solving the actual problem does, if you're an "outsider" who has to see why each of those things has the units it does. It's a useful way to check an answer, or to intuit what form an answer should have, yes... I don't see it as fundametally different from a mathematician seeing a particular form of problem and immediately saying, well, if a solution exists, it must come from such and such family of curves.

[/ QUOTE ]

While I disagree with much of that (that dimensional analysis takes as much work as solving the problem; not), so what?

[ QUOTE ]
It has certain limitations.... as you've said, even if you know the units of the final answer, you don't know to what power, if at all, various dimensionless quantities will appear in the calculation. The part of your argument that makes no sense is where you pull, out of thin air, the claim that its preposterous that an angle or a square of an angle or anything else like that could ever appear in the formula. It CAN, and it DOES - and you started this thread by posting a simple example where an angle and not a trigonometric function of an angle appears in a formula.

[/ QUOTE ]

Good one. Conceded.

[ QUOTE ]
You seem to believe that log and exp (and gamma and bessel functions and every other function you've ever seen) take pure numbers as arguments. Why do you think trig functions are somehow different, and can take anything except a pure number as an argument? Dont forget exp(i*theta) = cos(theta)+i sin(theta), either -- whatever kind of argument you think exp, cos, and sin take it HAS to be the SAME kind of argument!

[/ QUOTE ]

I've already conceded this. Did you even read the thread? /images/graemlins/confused.gif

[ QUOTE ]
I was also a bit bothered by your condemnation of high school math textbooks for not teaching much about units. I have news for you: in math classes, we teach math. What do you want us to do, teach them physics instead? We teach manipulation of pure numbers because THAT IS WHAT THE CLASS IS ABOUT. Yes, sometimes applications to the "real world" are shown, with some discussion about how to transform real-world questions into equations to be solved, and yes, *in the process of that transformation*, one of the things that sometimes comes up is dealing with units.

[/ QUOTE ]

And most often it doesn't. It's simply neglected, leading to huge problems later on.

[ QUOTE ]
Someone mentioned graded rings, and we've touched on adding meters to meters but not being able to take exps of meters. Whether units can be carried along for the ride depends almost entirely on whether stray x'es can be carried along for the ride. (All the linear operators are OK, pretty much, and almost everything else either has to be dimensionless or has some special restrictions on it.)

I also wanted to touch briefly on the assertion that one cannot measure anything without first choosing a unit. Does borodog admit counting is a measurement?

[/ QUOTE ]

Does Siegmund admit addressing someone in the third person is a condescending douchebag move?

[ QUOTE ]
I suppose he can say that "one of whatever you are counting" is your unit.

[/ QUOTE ]

Of course. 12 eggs. The unit is an egg, you have measured the number of eggs compared to the unit, and there are 12 of those units.

[ QUOTE ]
If he'll call 1 "the unit by which pure numbers are measured" I suppose I won't disagree too violently. But if he'll admit to THAT he could write s = r * theta * 1 (or maybe it's s *1 = r* theta ) and be happy.

[/ QUOTE ]

I have no idea what you're trying to say; 1*r*theta = r*theta, which still has units of meter radians.

[Siegmund]

Sigh. I am trying to say, "if it's so easy for you to see that newtons are kilogram*meter/seconds, what is so hard about seeing that a radian is a derived SI unit that, in terms of base units, is "1"? Really! It's IN the SI unit list described that way! Replacing "radian" with "1" is exactly the same operation as replacing "N" with "kg*m/s"!

The third-person thing wasn't trying to be condescending. I wrote the first half of the post addressed to you, then wandered off talking to whomever brought up the rings, and then got back to you after forgetting how I had phrased the first half of the post.

I did indeed read the thread. But if you conceded that trig functions took pure numbers as arguments, I somehow missed you conceding anything about radian meters.

[holmansf]

First let me say I learned geometry as well, geometry, so I don't know all the general relativity stuff. But for example, the Euclidean metric on R^2 is:

dx^2 + dy^2

Here the g_ij is just the identity matrix, which you presumably don't wish to say has units m^2 . If the coordinates are measuring distance in meters, then it would appear dx has units m^2. In polar coordinates:

dx^2 +dy^2 = dr^2 + r^2 dtheta^2.

The dr should have units m, and the dtheta should be dimensionless. Of course there are examples of coordinate systems on parts of R^2 that could not have meaningful units, but with the above examples in mind it appears inappropriate to give the matrix g_ij units m^2 arbitrarily. In fact the entire metric g_ij dx^i dx^j should have units m^2. I don't think you can create a rule for saying in every case what the units of g_ij and dx^i are.

[jamzfive]

[ QUOTE ]
The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

[/ QUOTE ]

OK, I'll put away the “radians isn't a unit” argument for a minute and see if I can get you more comfortable with the equation you're upset about.

Let's use phi to represent “the fraction of the circle subtended by an angle”, a pure ratio. That is,

phi = s / C.

We can substitute for C and rearrange to get

s = 2 * pi * r * phi.

This is the general form of your equation. To calculate phi, we just take the measure of the angle and divide by the angle measure of the entire circle.

Radians:
s = 2 * pi * r * (theta radians) / ((2 * pi) radians) = r * theta.

Degrees:
s = 2 * pi * r * (theta degrees) / (360 degrees) = (1/180) * r * pi * theta.

Gradians:
s = 2 * pi * r * (theta gradians) / (400 gradians) = 0.05 * pi * r * theta.

So the form of the equation you've been railing against only works for radians because it was simplified from the general form FOR radians. That’s why the equation always notes that theta be measured in radians. What’s not described well is that theta in the equation is just the “magnitude” of the angle measure, as the short version of the equation is what you get after units have already been cancelled out! Note that there ARE equivalent equations for the other units, however.

This probably makes you happy enough.


HOWEVER, there is another possibility (here comes the “radians isn't a unit” argument again), and that is that angle measurements are just magnitudes in the first place.

Here are two links, one to the National Institute of Standards and Technology (http://www.physics.nist.gov/cuu/Units/units.html), and the other to the International Bureau of Weights and Measures (http://www.bipm.org/en/si/si_brochure/chapter2/2-2/table3.html) (the folks who decide all this stuff about measurements and units for the scientific community). If anyone knows what’s up, it’s them.

Both pages indicate that radians are a derived unit, with an “Expression in terms of SI base units” of “m/m”, or “m*m^-1”. The BIPM site goes on to note that “The radian ... [is a] special name for the number one.”

As Wikipedia puts it, “Although the radian is a unit of measure, anything measured in radians is dimensionless,” where “dimensionless” is explained as “a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all units cancel.”

So I guess you can think of a radian as a ghostly type of unit: it’s tacked on when the other units in the calculation have already been cancelled. How to explain this to students? I would start from first principles, showing how the radian was applied out of thin air to start with, and making sure to note that radian is the only SI unit that is special in this way.

I honestly can’t understand it completely myself, but my approach is to defer to the experts, and then come up with thought exercises to approach the issue from a direction that helps me try to make some sort of sense of it. That’s what I was attempting to do in my earlier post, although apparently not doing it so well. Let me continue to beat the dead horse for a moment, however, with possibly a new approach. Consider the following situations:

A) If you’ve got a function f(x) that does some calculation where “x” is measured in “meters”, but what you know is how many you've got in “inches”, you can’t just plug that in for “x” and expect to get the right answer. You either have to convert the number of inches into the corresponding number of meters, or re-work the equation to come up with an equivalent one that can take “inches” as input.

B) If you’ve got a function f(x) that does some calculation where “x” is measured in “pairs”, but what you know is how many you've got in “dozens”, you can’t just plug that in for “x” and expect to get the right answer. You either have to convert the number of dozens into the corresponding number of pairs, or re-work the equation to come up with an equivalent equation that can take “dozens” as input.

Now, neither “pairs” nor “dozens” is a unit, but they sure sound like them, don't they? I think the same thing applies to units of angle measure. Just think of the unit name of an angle being shorthand for what the denominator of the ratio is. They same way “pair” means “multiply by 2” and “dozen” means “multiply by 12”, “radians” means “divide by 2 * pi”.

Using this approach, I would now modify the earlier statement "What’s not described well is that theta in the equation is just the “magnitude” of the angle measure, as the short version of the equation is what you get after units have already been cancelled out" to read "What's not described well is that theta in the equation is just the numerator of the angle measure, as the short version of the equation is what you get after after a particular denominator has already been assumed.

Maybe that's an interesting "angle" to think about it from?


jb

[Daisydog]

Let x=20 meters
Let y=10 meters
Let z=x/y

What are the units for z?

[Siegmund]

As an equal opportunity nitpicker, I have to point out that the steradian, to measure solid angles (ratio of surface area of a portion of a sphere to r^2) is also a dimensionless SI derived unit.

[Neuge]

[ QUOTE ]
[ QUOTE ]

There is no monkey business going on. All angles measurements are ratios, whether degrees, radians, or gradians.

[/ QUOTE ]

This is true of all measurements with a unit, not just angular measurements.

[/ QUOTE ]
How so?

[Metric]

[ QUOTE ]
First let me say I learned geometry as well, geometry, so I don't know all the general relativity stuff. But for example, the Euclidean metric on R^2 is:

dx^2 + dy^2

Here the g_ij is just the identity matrix, which you presumably don't wish to say has units m^2 . If the coordinates are measuring distance in meters, then it would appear dx has units m^2. In polar coordinates:

dx^2 +dy^2 = dr^2 + r^2 dtheta^2.

The dr should have units m, and the dtheta should be dimensionless. Of course there are examples of coordinate systems on parts of R^2 that could not have meaningful units, but with the above examples in mind it appears inappropriate to give the matrix g_ij units m^2 arbitrarily. In fact the entire metric g_ij dx^i dx^j should have units m^2. I don't think you can create a rule for saying in every case what the units of g_ij and dx^i are.

[/ QUOTE ]
Yes, I'm saying that the metric for Euclidean space in Cartesian coordinates can carry the units of meters^2 just fine. Being proportional to the identity matrix doesn't spoil this at all.

Borodog's original problem was that of allowing the coordinate differentials "dx^i" to carry units of physical length -- e.g. meters -- in one coordinate system (cartesian), but not in another (polar). This goes against his intuition as horribly inconsistent.

But we can get around the inconsistency just fine if we realize that no coordinate differentials carry dimensionful units -- that is the job of the metric, which always carries the "length squared" units. All coordinate differentials merely carry dimensionless labels (such as radians or degrees) which remind us which coordinate system we are using.

So in his original equation ds^2 = r^2 dtheta^2, we should write:

ds^2 (meters)^2 = r^2 (meters/radians)^2 dtheta^2 (radians)^2

"r" is contributed by the metric, and so doesn't carry units of meters only -- it carries units of "meters per radian".

In cartesian coordinates, we can do the same thing:

ds^2 = 1 (meters/cordinate length)^2 dx^2 (coordinate length)^2 + 1 (meters/coordinate length)^2 dy^2 (coodinate length)^2

This form of keeping track of units is coordinate system invariant and perfectly consistent. But it just seems like overkill when working in cartesian coordinates, so we ditch the dimensionless labels and get a little sloppy for the sake of convenience.

[Metric]

[ QUOTE ]
[ QUOTE ]

i don't read physics literature, but at the graduate level, [censored] like this has to be suppressed all the time.

[/ QUOTE ]

You are 100% wrong.



[/ QUOTE ]
I wouldn't say "100% wrong," exactly. Relativists often use "natural units" such that c = h = G = 1, which is basically like supressing dimensions. At the end of the calculation, if you want your units back, there is only one combination of c, h, and G which will give you the correct dimensions, so you simply multiply the dimensionless answer by that combination.

[siegfriedandroy]

you arent bright.

[jason1990]

[ QUOTE ]
This form of keeping track of units is coordinate system invariant and perfectly consistent.

[/ QUOTE ]
I think this is the heart of the matter and is related to what I was getting at when I said we cannot define ln(x) is a way which is independent of the coordinate system on (0,\infty). We can only define ln(x/x_0), which supports the idea that the logarithm can only take a dimensionless argument. I said earlier that a rigorous formulation of dimensional analysis might involve some product space, etc. and that I had never seen that before. I realize now that my comment was off-base. The rigorous formulation of dimensional analysis is encoded in the language of differential geometry. Dimensional analysis is all about the rules we must follow when we change coordinate systems. Those rules are the subject of differential geometry.

For the record, I'll repeat that I'm not a differential geometer, so I could be wrong about all this. But it sure seems to be the only thing that makes sense.

[Daisydog]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

[/ QUOTE ]

This is the definition of a radian - the angle such that the arc length subtended by that angle on a circle is equal to the radius.

Boro - I see what you're saying. But I think if you just explicitly mention that radians are essentially just 2*pi times a fraction of a circle, it becomes pretty clear that they are dimensionless, AND that this is what you'd want for expressing lengths.

[/ QUOTE ]

Yes, I know. Perhaps I'm not making myself clear. The problem that I have with it is this:

meters = meter radians

The radians magically vanish when I have told the students to explicitly and carefully handle all units. And they only vanish under certain circumstances; only when converting from angular to linear measure, never when converting from one system of angular measure (radians) to another (degrees or revolutions, for example).

As a consequence, one can apparently magically put radians whereever one likes. My new house is going to be 3000 square footradians. My car gets 25 radianmiles/gallon on the highway, and 21 miles/gallonradian in the city.

Now if you'll excuse me, I'm going to go scramble up a couple of eggradians for my lunchradian.

/images/graemlins/tongue.gif

[/ QUOTE ]

Let X = number of poker tournaments you play
Let Y = number of poker tournaments you win
Let P = the percent of poker tournaments you win

Then Y = X*P

tournaments = tournament percents

Oh my god! The units aren't working out!

Now if you'll excuse me, I'm going to go scramble up a couple of eggpercents for my lunchpercent.

[holmansf]

I agree that the metric carries units, but g_ij is not the metric. The metric is g_ij dx^i dx^j. You need the differentials to make it a 2-tensor- otherwise you just have an array of functions.

When we're dealing in the Euclidean space (and thinking of it as modeling a plane in which you measure distances) the coordinates (x,y) actually are measuring the distance in certain directions from a reference point, and thus have units.

The formula

ds^2 = r^2 dtheta

is saying that the round metric on the circle is equal in two different coordinate systems. On the left you have the coordinate given by arclength from some reference point. In this system the coordinate has units meters since it is actually measuring a distance. On the right you are using the angle as a coordinate. Consider a different coordinate system on (a part of) the circle given by arclength squared: t = s^2 (obviously this is only a valid coordinate system away from the reference point from which arclength is measured). Then:

dt^2 = 4s^2 ds^2 => 1/(4s^2) dt^2 = ds^2.

Now you're saying 1/(4s^2) has units m^2. By a similar argument you could get almost any function of s to have units m^2.

Are you just saying that when writing metrics you should should ignore any units on the coordinates and then write, for example, m^2 after the metric? This doesn't seem like a good idea since the whole point of dimensional analysis is to check your formulas (which relate different coordinates) using units. If you just define that all g_ij have the same coordinates it will naturally be consistent, but you're just throwing out all information about the units of the coordinates.

[Metric]

[ QUOTE ]
I agree that the metric carries units, but g_ij is not the metric. The metric is g_ij dx^i dx^j. You need the differentials to make it a 2-tensor- otherwise you just have an array of functions.

[/ QUOTE ]
Obviously all reference too coordinates goes away if you express the metric in coordinate-independent fashion -- in that case, its just a geometrical object "g". But here we're talking about going back and forth between coordinate systems, in which case g_i_j are the components of the metric tensor, carrying appropriate units to keep everything under control and comprehensible.

[ QUOTE ]
When we're dealing in the Euclidean space (and thinking of it as modeling a plane in which you measure distances) the coordinates (x,y) actually are measuring the distance in certain directions from a reference point, and thus have units.

[/ QUOTE ]
I disagree -- you're taking an example of a particular coordinate system where the metric has a very simple form, and then claiming that the coordinate differentials have certain dimensions based on the fact that you can almost get away with pretending the components of the metric aren't there.

Once again, in every coordinate system, the length element takes the form ds^2 = g_i_j dx^i dx^j -- it doesn't matter if its cartesian, polar, or whatever else you dream up. The left hand side is a length only -- the right hand side is made up of coordinates and the metric.

[ QUOTE ]
The formula

ds^2 = r^2 dtheta

is saying that the round metric on the circle is equal in two different coordinate systems. On the left you have the coordinate given by arclength from some reference point.

[/ QUOTE ]

NO! The left hand side is a length only!

[ QUOTE ]
In this system the coordinate has units meters since it is actually measuring a distance.

[/ QUOTE ]
I have no idea why you're dreaming up such a contorted system. The metric handles all dimensional analysis in any coordinates -- you don't need two coordinate systems (one for coordinates, and other to describe "arc length") to write down a simple distance, which is just a scalar quantity.

[ QUOTE ]
On the right you are using the angle as a coordinate. Consider a different coordinate system on (a part of) the circle given by arclength squared: t = s^2 (obviously this is only a valid coordinate system away from the reference point from which arclength is measured). Then:

dt^2 = 4s^2 ds^2 => 1/(4s^2) dt^2 = ds^2.

Now you're saying 1/(4s^2) has units m^2. By a similar argument you could get almost any function of s to have units m^2.

[/ QUOTE ]
You're confusing yourself by naming one coordinate "s", which is also the lable used for the arc length. I will call it instead "x" and call its units "nats" (because it is a very "natural" coordinate).

So your first coordinate system is simple:
ds^2 (meters)^2 = 1 (meters/nats)^2 dx^2 (nats)^2

Now you define a new coordinate "t" whose units I will call "nuts". t (nuts) = s^2 (nats^2) Note that NO DIMENSIONS OF DISTANCE ARE YET INVOLVED. Nats and nuts are dimensionless units, like radians and degrees.

We start out with g = 1 (meter/nat)^2 in the first coordinate system, and then transform it to the new one g' via the standard rule:

g' = dx/dt dx/dt g

Now adding units, we note that the new metric is:
g' (meters/nuts)^2 = (dx/dt)^2 (nats/nuts)^2 g (meters/nats)^2

I.E. g' = 1/4t (meters/nuts)^2

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nats)^2 dt^2 (nats)^2

Please tell me that you see it now.

[ QUOTE ]
Are you just saying that when writing metrics you should should ignore any units on the coordinates and then write, for example, m^2 after the metric? This doesn't seem like a good idea since the whole point of dimensional analysis is to check your formulas (which relate different coordinates) using units. If you just define that all g_ij have the same coordinates it will naturally be consistent, but you're just throwing out all information about the units of the coordinates.

[/ QUOTE ]

Where in your example did we lose any information??? You can use "dimensionless units" all you want -- radians, degrees, nats, nuts, etc. etc. etc. and keep yourself consistent with those. They carry information, but they do not carry dimensions of length -- nor do any coordinate differentials. By no means should you ever have a coordinate differential magically turn into something with dimensions of distance without involving the metric -- that is what it is for!

[Metric]

Made one typo:

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nats)^2 dt^2 (nats)^2

should read:

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nuts)^2 dt^2 (nuts)^2

[jstnrgrs]

I haven't read the whole thread, but I agree with boro here. The solution here is to abandon the radian unit entierly and just to treat angle measurements as pure numbers.

As a consequence, the degree unit would be equal to pi/180.

Trig functions would take pure numbers as their arguments. This wouldn't be a problem since, acording to what I said about degrees above, sin(30degrees)=sin(pi/6).

Just eliminate the term radian, define the term degree as being equal to pi/180, and there are no more problems.

[holmansf]

[ QUOTE ]
I have no idea why you're dreaming up such a contorted system. The metric handles all dimensional analysis in any coordinates -- you don't need two coordinate systems (one for coordinates, and other to describe "arc length") to write down a simple distance, which is just a scalar quantity.


[/ QUOTE ]

I didn't dream up anything. This is the problem we were talking about. Remember Borodog's formula:

s = r*theta

s and theta are different coordinates on the circle of radius r, and this is the formula that relates them. I understand now that in physics you often write ds^2 for the metric. Serendipitously, in these coordinates the round metric on the circle actually is ds^2.

I think we will have to agree to disagree on this issue. However, let me just say that I don't understand the formula

[ QUOTE ]
ds^2 (meters)^2 = 1 (meters/nats)^2 dx^2 (nats)^2


[/ QUOTE ]

Where does the last nats^2 come from?

Also, I appear to have made a typo in the last paragraph. It should have been, "If you just define that all g_ij have the same units...." (not same coordinates) Maybe that makes it clearer.

[Metric]

dx^2 has dimensionless units of nats^2 just as dtheta^2 has dimensionless units of radians^2 or degrees^2, depending on your choice of coordinate system.

But this is not my thread and I can only show how a system is consistent in so many ways before I get bored with dimensional analysis. If you want to believe that coordinates have dimensions of length "except for when they don't," be my guest. I doubt the future of democracy hangs in the balance.

[Mano]

Can't you just explain it using units if you choose?

s (units of length) = r (units of length / radius) * theta (radians) * (1 radius/radian)

[holmansf]

The object of dimensional analysis is to check your answers. Refering to your nat/nut example, your final answer for the metric in t coordinates is:

ds^2 = 1/4t dt^2

By your reasoning 1/4t should have units meters^2/nuts^2. However it appears to only have units 1/nut (or meters^2/nut?) since t has units nuts. This is the information you have lost by defining g_ij to have appropriate units in all cases. Note that when you replace nats by meters (and nuts by meters^2 since x^2 = t) the dimensional analysis does work to confirm that the answer is correct.

You're example just boils down to the fact that if in the formula

g dx^2 = g dx/dt dx/dt dt^2

you "cancel the dt's" it works out. This is obvious (and thus your system is consistent), but it in no way helps you check your answers. I too tire of explaining dimensional analysis, and will therefore not post in this thread anymore.

[surftheiop]

Am i getting leveled or dont the units work out ?
tournaments won = tournies played * (tournieswon/100 tournies played)
tournies played cancels and it seems units work out for me?