Worrying about dimensionlessnes

[Relative]

I've just stumbled over an article where somebody stated, that "the number one is not a rate, so is certainly not the rate at which anything (the flow of time included) happens".
Well, I would like to ask this person to wallpaper his flat. He will need a certain amount of squaremeters of wallpaper to do so. If he devides this by the squaremeters of his flat, he will get a dimensionless number (squaremeters per squaremeters =1), which is dependent on a.) the number of walls (each wall will separate the room in two spaces, the next wall can be a either part of it (three rooms) or dividing it into four rooms, and so on...
Of course, it is also depending on b.) the height of the room as well, another nice question.
BUT!!:
There will be (a lot of) flats, which have a "rate" of exactly one, i.e. the same amount of squaremeter wallpaper is needed like the ground floor of that flat exhibits, so the value will be an all dimensionless "one"!

[grapes]

I've moved this thread into OTB since there doesn't seem to be an ATM connection.

[grapes]

I've just stumbled over an article where somebody stated, that "the number one is not a rate, so is certainly not the rate at which anything (the flow of time included) happens".

Do you have a link to the article?

[Strange]

I would say that your "dimensionless" 1 has dimensions of wallpaper/flat.

[Relative]

Yes,
here it is web.mit.edu/bskow/www/research/sec-per-sec.pdf

[Relative]

@Strange
Yeah, but a "flats" unit is m^2 and the wallpapers unit is m^2 as well.. So wallpaper/flat unit is 1.
Don't misunderstand: I'm not arguing against that! But 1 second per second does make the same sense then, and this person refuses to..

[swampyankee]

"Rate" is the time derivative of something. The derivative of time with respect to time is 1. The derivative of anything with respect to itself is 1.

[grapes]

Ha, Bradford Skow does not like dimensional analysis. Or, maybe, he just doesn't like treating dimensional analysis as mathematics.

"Rate" is the time derivative of something.

Rate can also be some other ratio: the going rate for a pound of coffee, for instance.

[Relative]

"Rate" is the time derivative of something. The derivative of time with respect to time is 1. The derivative of anything with respect to itself is 1.

But the amount of wallpaper is dependent on the flat's area. So it is a derivate of an independent value

[Relative]

Ha, Bradford Skow does not like dimensional analysis. Or, maybe, he just doesn't like treating dimensional analysis as mathematics.

Rate can also be some other ratio: the going rate for a pound of coffee, for instance.

Thanks, Grapes. I guess that's it. He was about to convince me.. :-)

[swampyankee]

But the amount of wallpaper is dependent on the flat's area. So it is a derivate of an independent value

To a first approximation -- I've hung wallpaper -- the area of wallpaper is equal to the area of the walls in a given room or set of rooms, so you're putting 1 square meter of wallpaper per square meter of wall. Your "rate" of wallpaper is 1 m²/m².

Rate can also be some other ratio: the going rate for a pound of coffee, for instance.

Point taken, but I just put on my physics teacher's hat

[Moose]

To a first approximation -- I've hung wallpaper -- the area of wallpaper is equal to the area of the walls in a given room or set of rooms, so you're putting 1 square meter of wallpaper per square meter of wall. Your "rate" of wallpaper is 1 m²/m².

And a ratio of 1.

[korjik]

I would say that your "dimensionless" 1 has dimensions of wallpaper/flat.

except that wallpaper and flat are no more dimensions than radians are....

[HenrikOlsen]

but it's worth to note that radians are rather important.

[Paul Beardsley]

except that wallpaper and flat are no more dimensions than radians are....

When did radians stop being dimensions? How do they measure angular speed nowadays?

[korjik]

When did radians stop being dimensions? How do they measure angular speed nowadays?

Radians arent a proper unit because they arent conserved. The proper unit for angular velocity is just s^-1. You can see this when you multiply angular velocity by a radius and get a velocity. If radians were a unit you would have:

Length*Radians/second=Length-radians/second.

Thing is, you only get a velocity.

It is like what Henrik said, the radians are important, but they arent a unit.

[Moose]

Wouldn't it be easier to say that "radians", like "flats" and "wallpapers", aren't dimensions because they're not constants? (My flat's bigger than yours, so a comparison of wallpapers/flats isn't meaningful without standardizing the term.)

[swampyankee]

Radians are constant; they're defined as the "A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, approximately 57°17′44.6″. "http://www.answers.com/topic/radian.

More importantly, they're convenient in that they get rid of that annoying 2π that would show up in the equations for angular motion.

[Paul Beardsley]

Radians arent a proper unit because they arent conserved. The proper unit for angular velocity is just s^-1. You can see this when you multiply angular velocity by a radius and get a velocity. If radians were a unit you would have:

Length*Radians/second=Length-radians/second.

Thing is, you only get a velocity.

It is like what Henrik said, the radians are important, but they arent a unit.

Good response, but aren't they units rather than dimensions?

[HenrikOlsen]

They are units of angle, but they're dimensionless units, which can make them a bit tricky to think about.

[Paul Beardsley]

They are units of angle, but they're dimensionless units, which can make them a bit tricky to think about.

Yes, that's what I mean. If you are doing dimensional analysis you can ignore the radians, along with 2*pi, a half, and selected other constants and certain values such as the number of wavelengths.

[Tobin Dax]

Yes, that's what I mean. If you are doing dimensional analysis you can ignore the radians, along with 2*pi, a half, and selected other constants and certain values such as the number of wavelengths.

Exactly. I started thinking about number as a unit, too, as I started reading your post. The mole, for example, is another a dimensionless unit since it is just a measure of the number of objects in the system being considered.

[korjik]

Alot of times unit and dimension are treated as synonymous. You can have units of anything, but dimensions have to be preserved. So radians are units of angle, but not a dimension.

From what I have seen (and done) the useage is pretty sloppy tho.

[HenrikOlsen]

What do you expect?
It's physics, not mathematics

[korjik]

Exactly!