how do you interpret divided units?

[michel123456]

This thread paraphrasing an old interesting one in this same forum.

 

As always fascinated by the power of units and after a post of dear Swansont, here are some thoughts:

 

_as an Architect, I use quite commonly a method to measure the slope of a piece of land, or the slope of a ramp, expressed as a percentage.

 

For example say you have a ramp 10 meters long going one meter up, that gives a slope of 10%

 

What did you do?

 

You took the eight divided by the length (the horizontal projection), that is:

 

1m/10m=0,10 or 10%

 

In fact you divided meters by meters and ended up with a unitless number.

 

This unitless number represents something physical though: it is a slope or if you prefer it is an angle. An angle of 10% that can be translated in degrees.(arctangent 0,10=5,71degrees approx)

 

And degrees are a kind of unit.

 

So, dividing meters by meters you did not end up with something unitless.

 

Or do I miss something?

[swansont]

An angle not considered a proper unit, in the sense that meters, kilograms or Coulombs represent something physical. A rotation of 1 radian/sec has units of inverse seconds. The radians (or degrees) is a helpful label.

[michel123456]

An angle represents something physical too.

and has several units. See Wiki.

 

And any unitless number is not necessary an angle.

[Arnaud Antoine ANDRIEU]

Hi Michel. The question is "how do you interpret divided units?"

Divided unit and compartment :

 

a little bit words in french..

Edited November 29, 2012 by Arnaud Antoine ANDRIEU

[Daedalus]

In fact you divided meters by meters and ended up with a unitless number.

 

This unitless number represents something physical though: it is a slope or if you prefer it is an angle. An angle of 10% that can be translated in degrees.(arctangent 0,10=5,71degrees approx)

 

And degrees are a kind of unit.

 

So, dividing meters by meters you did not end up with something unitless.

 

Or do I miss something?

The proper term you are looking for is dimensionless units or dimensionless quantity. The measurement representing the angle still has units, which can be in degrees, radians, or whatever you want to call it.

 

 

Hi Arnaud Antoine ANDRIEU. I just wanted to point out that there are [math]2\pi[/math] radians per [math]360[/math] degrees or 1 revolution. As for the rest, I have no idea what you are talking about.

Edited November 30, 2012 by Daedalus

[Arnaud Antoine ANDRIEU]

Hi Arnaud Antoine ANDRIEU. I just wanted to point out that there are [math]2\pi[/math] radians per [math]360[/math] degrees or 1 revolution. As for the rest, I have no idea what you are talking about.

Hi Daedaus. These [math]2\pi[/math] more represent the anti-matter phases. They are twin's fonctions.

if you want more info about that system (theoretical), please cick the following link : Thank you. http://www.scienceforums.net/topic/70967-gravity-and-the-higgs/

Edited November 30, 2012 by Arnaud Antoine ANDRIEU

[CaptainPanic]

10% slope is dimensionless, but it is perfectly OK to write it as 10% m/m (meter per meter), although it might confuse the hell out of an unsuspecting construction worker...

 

In fact, in chemical engineering, we often calculate the fraction of a certain chemical in a mixture. A well known fraction (and very yummy fraction) is the percentage alcohol in beer. It is quite a big difference if we are talking about 5% weight/weight, or 5% volume/volume or 5% mol/mol.

 

This is why on beer bottles it is specified which kind of percentage we're talking about (usually %vol/vol, also written as %vol).

 

Now, when it comes to slopes of a hill, there is little doubt that you're talking about length/length... so specifying it is a little less important.

 

And now for something completely different

Arnaud Antoine ANDRIEU, your link seems 100% (information per information) off topic.

It should me mentioned that in the case of 100%, the specification of which kind of percentage you're dealing with is usually unimportant, because 100% is the whole thing no matter what you're talking about. Anyway... your post doesn't make any sense to me.

[michel123456]

The proper term you are looking for is dimensionless units or dimensionless quantity. The measurement representing the angle still has units, which can be in degrees, radians, or whatever you want to call it.

(...)

 

Great reply! Thanks

 

But in the long list of this article:

 

No Angle (because angles do have units) although angles are treated in the text.

 

No Percentage (which is a ratio, treated several times in the right column of the list----after some thinking any number is a ratio, isn't it?)

 

 

Maybe I am looking too far, but I find it very amazing that a ratio (no unit) is the same thing as an angle (or is it not?)

--------------

(cross edited -the + rep was for the unedited part above, thank you)

 

10% slope is dimensionless, but it is perfectly OK to write it as 10% m/m (meter per meter), although it might confuse the hell out of an unsuspecting construction worker...

 

In fact, in chemical engineering, we often calculate the fraction of a certain chemical in a mixture. A well known fraction (and very yummy fraction) is the percentage alcohol in beer. It is quite a big difference if we are talking about 5% weight/weight, or 5% volume/volume or 5% mol/mol.

 

This is why on beer bottles it is specified which kind of percentage we're talking about (usually %vol/vol, also written as %vol).

 

Now, when it comes to slopes of a hill, there is little doubt that you're talking about length/length... so specifying it is a little less important.

(...)

 

Bolded mine.

O.K.

that is 5% volume of alcohol by volume of beer. These are 2 measurements of 2 different things.

And that's maybe the answer to my question: when measuring a slope, it is a ratio of vertical measurement by horizontal measurement, again 2 different things.

Edited November 30, 2012 by michel123456

[CaptainPanic]

that is 5% volume of alcohol by volume of beer. These are 2 measurements of 2 different things.

And that's maybe the answer to my question: when measuring a slope, it is a ratio of vertical measurement by horizontal measurement, again 2 different things.

Of course: if you take 10% of horizontal measurement over another horizontal measurement then it's a different thing altogether. Still dimensionless though.

[michel123456]

Of course: if you take 10% of horizontal measurement over another horizontal measurement then it's a different thing altogether. Still dimensionless though.

 

Right.

But no smiley. Different numbers representing different physical things ought to have different units. Imagine you make unit cancelation in some weird new Theory, you may obtain totally unphysical results.

[D H]

Different numbers representing different physical things ought to have different units.

That begs the question, how many different physical things are there? I'll start with the International System of Units (SI), which has seven fundamental units. One of those, the mole, is obviously dimensionless; it's just the number of items that constitutes a mole of such items. Another, the candela, is in reality a derived unit that pertains to how the human eye works. That leaves five. Is this the right number?

 

Before answering this question, I'll first look back to Newton's second law prior to the SI. Newton's second law does not say [imath]F=ma[/imath] (force is mass times acceleration). It says that force is proportional to the product of mass and acceleration: [imath]F=kma[/imath], where k is some constant of proportionality. Is this proportionality constant k a dimensionful or dimensionless quantity?

 

The early view in the development of the metric system was that this proportionality constant is dimensionful. From this perspective, constructing the units of force, mass, length, and time so that this proportionality constant has a numeric value of one was merely a convenient mathematical trick that simplified the math and removed a source for error. The modern view is that [imath]F=ma[/imath] is not just a trick that hides this proportionality constant. This proportionality constant is dimensionless, and the proper value is one. [imath]F=ma[/imath] is the "right" way to define force. From this point of view, the need to express Newton's second law as [imath]F=kma[/imath] with [imath]k\ne1[/imath] means that the underlying definitions of force, mass, length, and time are inconsistent.

 

Now let's look at velocity: Is velocity dimensionful or dimensionless? If time and distance are fundamentally different quantities, the interpretation of the Lorentz transformation as a hyperbolic rotation of space-time coordinates has to be viewed as a trick that just happens to get the math right. Viewing the Lorentz transformation as a rotation is not a trick if one views time and distance as fundamentally the same thing.

 

Some physicists prefer to work in some set of normalized units, systems where the speed of light has a numeric value of one. If velocity is a dimensionful quantity, choosing units of length and time such that the speed of light has a numeric value of one is just a mathematical trick that makes some calculations easier. If velocity is a dimensionless quantity, this choice not a trick. There is one universally agreed-upon velocity by all observers of some phenomenon, and that universally agreed-upon velocity is the speed of light. This is the natural value that makes a system of units consistent. From this perspective, time and distance only appear to be different because the SI ultimately is an inconsistent set of units, just as is the English system with its inconsistent representations of force, mass, length, and time.

 

So instead of five fundamental units, perhaps there are only four? Another one of those fundamental units, temperature, bites the dust quite readily. Temperature is essentially energy per unit mass. That leaves but three: time/distance, mass, and charge. Is this the right answer?

 

Rather than go any deeper, I'll let three physicists make their cases.

 

Michael J. Duff et al, Trialogue on the number of fundamental constants, JHEP03(2002)023 doi:10.1088/1126-6708/2002/03/023

Preprint at http://arxiv.org/abs/physics/0110060

 

In this paper, Lev Okun argues that the correct number is three, Gabriele Veneziano argues that the correct number is two, and Michael Duff argues that the correct number is zero. More and more physicists are taking Duff's point of view, that there are no dimensionful quantities.

[CaptainPanic]

Right.

But no smiley. Different numbers representing different physical things ought to have different units. Imagine you make unit cancelation in some weird new Theory, you may obtain totally unphysical results.

 

I personally think that you ask for too many units. It makes more sense to give a proper explanation (in a piece of text) what something represents. The sheer number of possible dimensionless numbers is mind boggling. I would not want to design an "SI system" for that.

 

But a proper description is essential, and in many cases also provided.

[michel123456]

Fantastic DH. +10 if I could.

 

--------------------

Commenting the arxiv: http://arxiv.org/abs/physics/0110060

 

It is stated that in some practical unit system, C =1 (dimensionless number)

It is also stated that

In 1983, the Conf´erence G´en´erale des Poids et Mesures declared c to have the value

299,792,458 metres/second exactly, by definition, thus emphasizing its role as a nothing

but a conversion factor.

(bolded mine)

 

But a "conversion factor" (a ratio) is a way to convert some unit into some other unit (usually C converts metres in seconds and vice-versa).

It may be a dimensionless number but it applies to some unit. Whithout any unit in the first place you cannot obtain a second one.

IOW I miss the supporters of the Single Constant Party (paraphrasing Michael J. Duff).

Edited November 30, 2012 by michel123456

[Phi for All]

Hi Daedaus. These [math]2\pi[/math] more represent the anti-matter phases. They are twin's fonctions.

if you want more info about that system (theoretical), please cick the following link : Thank you. http://www.sciencefo...-and-the-higgs/

!

Moderator Note

Arnaud Antoine ANDRIEU, you've been warned three times now to stop hijacking threads with your own speculative ideas. Please take the next week to reconsider this behavior and familiarize yourself with our rules.

[Daedalus]

Great reply! Thanks

 

But in the long list of this article:

 

No Angle (because angles do have units) although angles are treated in the text.

 

No Percentage (which is a ratio, treated several times in the right column of the list----after some thinking any number is a ratio, isn't it?)

 

 

Maybe I am looking too far, but I find it very amazing that a ratio (no unit) is the same thing as an angle (or is it not?)

Numbers are just numbers until we give them meaning. So it is not proper to think of any number as a ratio. However, a ratio can have any numerical value : ) Furthermore, angles are not ratios. The trigonometric operations applied to angles are ratios. For instance, the sin of an angle is the height divided by the hypotenuse, but the angle itself is just a measurement.

Edited December 1, 2012 by Daedalus

[michel123456]

Numbers are just numbers until we give them meaning. So it is not proper to think of any number as a ratio. However, a ratio can have any numerical value : ) Furthermore, angles are not ratios. The trigonometric operations applied to angles are ratios. For instance, the sin of an angle is the height divided by the hypotenuse, but the angle itself is just a measurement.

 

from wiki http://en.wikipedia.org/wiki/Category:Units_of_angle

 

Pages in category "Units of angle"The following 12 pages are in this category, out of 12 total. This list may not reflect recent changes (learn more).

Minute of arc

Arcsecond

Binary radians

Clock position

Degree (angle)

Grade (slope)Gradian

Angular mil

Radian

Square degree

Steradian

Turn (geometry)

Bolded mine

 

And under Wiki's page Slope

 

There are several systems for expressing slope:

 

1.as an angle of inclination to the horizontal. (This is the angle α opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)

2.as a percentage, the formula for which is which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., this percentage "grade" is the most commonly used unit for communicating slopes in transportation (streets, roads, highways and rail tracks), surveying, construction, and civil engineering.

3.as a per mille figure, the formula for which is which could also be expressed as the tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway.

4.as a ratio of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20. (The word "in" is normally used rather than the mathematical ratio notation of "1:20"). This is generally the method used to describe railway grades in Australia and the UK.

 

Bolded mine.

[Klaynos]

Please indulge my slightly odd post, hopefully it might aid understanding...

 

Ratios can all be considered angles in the correct space. Let us take beer as the example. First beer space, define the x angle as volume of non alcohol and the y axis as volume of alcohol. We'll consider only positive values as they are the only ones with a physical mapping (although I suspect there is a psyclogcal mapping for negative alcahol but that's a whole other topic). If we now take a test sample of 250 ml of beer, we measure that there is 10 ml of alcohol ad 240 ml Pfeiffer other. A ratio of 4%. Plot this point on the graph, we now have two points, as a sample of 0 will be 0, 0 an we know it is linear. So we have a line which must have an angle between it and the axis. If we extend the line we can see how much alcohol we''d have for a given quantity. So the angle only depends on the ratio. Ratio's are therefore angles in the correct space.

 

Let us go further and consider any angle, and the trig identities, sin(¥)=(opposite/hypotinuse) etc... we can easily see that what we have are directionless ratios, angles are just ratio's. It's often easier for humans to understand angles than pure ratios and vice versa depending on the context, an alcoholic drink of 3 degrees wouldn't mean much to most people.

[michel123456]

Doesn't that mean that a dimensionless number (as a result of unit cancelation) keeps a "meaning", although unitless?

[Klaynos]

Doesn't that mean that a dimensionless number (as a result of unit cancelation) keeps a "meaning", although unitless?

 

It must have a meaning, else why would it exist? Depending how you define meaning I guess.

[Daedalus]

Perhaps an angle can be considered a ratio in certain space. However, slope can be obtained by taking the tangent of an angle. For instance, [math]\text{tan} \left(\pi / 4\right)=1[/math], which demonstrates that trigonometric functions are ratios while angles are measurements.

[Klaynos]

Perhaps an angle can be considered a ratio in certain space. However, slope can be obtained by taking the tangent of an angle. For instance, [math]\text{tan} \left(\pi / 4\right)=1[/math], which demonstrates that trigonometric functions are ratios while angles are measurements.

 

I'd describe the trig functions as a mapping function between units.

[michel123456]

I'd describe the trig functions as a mapping function between units.

Yes.

[John Cuthber]

I wouldn't.

Sine (x) is always dimensionless. (ditto for the others)

And the "x" of which you are calculating the sine is also dimensionless.

Edited December 2, 2012 by John Cuthber

[Klaynos]

Dimensionless units.

[John Cuthber]

Doesn't that mean that they are mappings between numbers and, in that case, does it not apply to any unary function- logs for example?