conversion from frequency to angular frequency

[alainstgt]

the way Mcad is handling the conversion from frequencies to angular frequencies is not correct and also not coherent, since rad and deg are handled differently - see attached document.

[TomGutman]

>>Tom, are you making the difference between unit and dimension? probably there is a confusion.<<

Yes, I am. Angle is a dimension, a type of quantity. Radians, degrees, grades, revolutions, right angles, straight angles, etc. are all units, particular angles, that can be used to measure (or express) angles.

Exactly the same as with other dimensions. Length is a dimension, a type of quantity. Meter, kilometer, foot, inch, mile, light-year, etc. are all units, specific lengths, which can be used to measure (or express) lengths.

� � � � Tom Gutman

[ptc-1368288]

On 3/14/2007 8:27:04 PM, Tom_Gutman wrote:
>>>Tom, are you making the difference between unit and dimension? probably there is a confusion.<<
>
>Yes, I am. Angle is a
>dimension, a type of quantity.
>Radians, degrees, grades,
>revolutions, right angles,
>straight angles, etc. are all
>units, particular angles, that
>can be used to measure (or
>express) angles.
>
>Exactly the same as with other
>dimensions. Length is a
>dimension, a type of quantity.
>Meter, kilometer, foot, inch,
>mile, light-year, etc. are all
>units, specific lengths, which
>can be used to measure (or
>express) lengths.
>
>� � � � Tom Gutman
___________________

"Angle is a dimension, a type of quantity".

Yes it is, but unfortunately many collabs can't understand "dimension" and "unit". They should go to bed and repeat until sleeping ... "qualify the quantities, quantify the qualities"

If a cow is an angle, an angle is a cow. A flock of cows (though they don't fly) of 100 is a unit flock ... etc.
Think in term of an angle being defined complex.

jmG

[ptc-1368288]

... that's reminds me the crappy Mathcad qs of using units resulting with Reynolds number (dimensionless) coming out as 123456789/mbr

mbr = "monkeys on the branch" or another like idiosyncrasy.

jmG

[StuartBruff]

On 3/14/2007 8:27:04 PM, Tom_Gutman wrote:
== Yes, I am. Angle is a dimension, a type of quantity. Radians, degrees, grades, revolutions, right angles, straight angles, etc. are all units, particular angles, that can be used to measure (or express) angles.

Angle is a Quantity.

Angle is not a base quantity in the International System of Quantities (ISQ), it is a derived quantity and has dimension one.

What you mean is that the System of Quantities (ISQ) is based upon a false premiss, namely that an angle is simply the ratio of two lengths and is, therefore, dimensionless.

Consequently, angle should be a base quantity within the ISQ. Hence, the dimension of angle would be itself (to the power of one).

Stuart

From the VIM.
----------------------
dimension: dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities
----------------------

[TomGutman]

>>Angle is not a base quantity in the International System of Quantities (ISQ), it is a derived quantity and has dimension one.<<

That is true and irrelevant to my premise. I am talking about reality, not what the SI commitee produced. I also see little point in changing from the term "dimension", which I learned way back when first introduces to dimensional analysis and which is the terminology used in Mathcad's unit system to the term "base quantity".

>>What you mean is that the System of Quantities (ISQ) is based upon a false premiss, namely that an angle is simply the ratio of two lengths and is, therefore, dimensionless.<<

Didn't I already say I don't accept that a camel is a horse? How much clearer could I have been.

>>Consequently, angle should be a base quantity within the ISQ. Hence, the dimension of angle would be itself (to the power of one).<<

Maybe, maybe not. Depends on what SI (or ISQ, or, perhaps next week, IQS) is intended to do. If it is to match reality and provide for consistent units and dimensions, something that can be used to implement an automated system of unit handling and checking, then yes, it should. If it is just intended to follow the most common practice, whether or not consistent, and even if it ensures that any automated unit system will be in error, then no, it shouldn't.

And remember, there's more to the SI inconsistencies. There's the redefinition of the Hertz, making the Hertz and the radian per second have the same definition, 1/second, even though any beginning EE knows very well that one Hertz is 2π radians per second. They have not only made angles dimensionless (contrary to fact) but also defined two different angles, the cycle (by making the Hertz, which was one cycle per second, be just one per second) and the radian as dimensionless unities. This does not compute.

� � � � Tom Gutman

[StuartBruff]

On 3/15/2007 4:26:09 AM, Tom_Gutman wrote:
>>Angle is not a base quantity in the International System of Quantities (ISQ), it is a derived quantity and has dimension one.<<
== That is true and irrelevant to my premise.

No it is not irrelevant to your premise. You (and I) are talking about amending the system by which Mathcad handles 'unit's. To understand the problem and justify the solution you have to know where you're coming from.

== I am talking about reality, not what the SI commitee produced. I also see little point in changing from the term "dimension", which I learned way back when first introduces to dimensional analysis and which is the terminology used in Mathcad's unit system to the term "base quantity".

No, you're talking about the definitions in force when you learned whatever you learned. We're discussing metrology here. The terms are well defined and understood in the metrology community. I do not believe for one second that a product offering an 'alternative' set of definitions is going to be more credible or acceptable. If that's what Mathcad is stating a dimension is, then it needs correcting.

Much of the debate I've seen arises because people do not fully appreciate the differences between quantities, dimensions and units. There is already enough confusion without adding to it.

>>What you mean is that the System of Quantities (ISQ) is based upon a false premiss, namely that an angle is simply the ratio of two lengths and is, therefore, dimensionless.<<

== Didn't I already say I don't accept that a camel is a horse? How much clearer could I have been.

See above 🙂

>>Consequently, angle should be a base quantity within the ISQ. Hence, the dimension of angle would be itself (to the power of one).<<
== Maybe, maybe not. Depends on what SI (or ISQ, or, perhaps next week, IQS) is intended to do. If it is to match reality and provide for consistent units and dimensions, something that can be used to implement an automated system of unit handling and checking, then yes, it should. If it is just intended to follow the most common practice, whether or not consistent, and even if it ensures that any automated unit system will be in error, then no, it shouldn't.

I've got no problem with that.

== And remember, there's more to the SI inconsistencies. There's the redefinition of the Hertz, making the Hertz and the radian per second have the same definition, 1/second, even though any beginning EE knows very well that one Hertz is 2� radians per second. They have not only made angles dimensionless (contrary to fact) but also defined two different angles, the cycle (by making the Hertz, which was one cycle per second, be just one per second) and the radian as dimensionless unities. This does not compute.

'hertz' and 'radian per second' do not have the same definition within the SI:

-----
SI Booklet Section 2.2.2
A derived unit can often be expressed in different ways by combining the names of base units with special names for derived units. This, however, is an algebraic freedom to be governed by common-sense physical considerations. Joule, for example, may formally be written newton meter, or even kilogram meter
squared per second squared, but in a given situation some forms may be more helpful than others.

In practice, with certain quantities preference is given to the use of certain special unit names, or combinations of unit names, in order to facilitate
the distinction between different quantities having the same dimension. For example, the SI unit of frequency is designated the hertz, rather than the
reciprocal second, and the SI unit of angular velocity is designated the radian per second rather than the reciprocal second (in this case retaining the word radian emphasizes that angular velocity is equal to 2pi times the rotational frequency). Similarly the SI unit of moment of force is designated the newton meter rather than the joule.

In the field of ionizing radiation, the SI unit of activity is designated the becquerel rather than the reciprocal second, and the SI units of absorbed dose
and dose equivalent the gray and sievert, respectively, rather than the joule per kilogram. In the field of catalysis, the SI unit of catalytic activity is
designated the katal rather than the mole per second.� The special names becquerel, gray, sievert and katal were specifically introduced because of the
dangers to human health which might arise from mistakes involving the units reciprocal second, joule per kilogram and mole per second.
-----

Section 1.4 of the SI Booklet states:

For a detailed exposition of the system of quantities used with the SI units see ISO 31, Quantities and units (ISO Standards Handbook, 3rd edition, ISO, Geneva, 1993).

The ISO 31-2 defines the terms 'frequency' and 'angular frequency'. It states that frequency is the inverse of period and has unit Hz; it defines period as the duration of 1 cycle. It also states explicitly that angular frequency has unit rad/s and is defined as 2pi times frequency

It is true that a 'cycle per second' and a 'radian per second' both have dimension s^-1 within the SI, which gives rise to the perceived identity problem when using a naive units system. The 'system' relies upon the author and audience of a document keeping track of the actual quantities and derived units by textual annotation or implication - which is fine until automated units systems get involved, because the context is then lost; the unit information is no longer carried with the data when the system of quantities is restricted to the ISQ.


for example, in Mathcad the author might write:

Define Angle : theta := 1.2 rad

which might appear later as

Scale the Angle: stheta := scl*theta

Scaled Angle : stheta = 2.3

The reader can readily infer that the stheta has unit 'radian' from the document context; indeed the author might clarify this by adding the unit as text

Scaled Angle : stheta = 2.3 rad

If the the underlying automation implements strict dimensional analysis, however, it has no way of tracking this. Which is why a system that deals with this must somehow track the unit or quantity (as a data attribute, say).

Stuart

[TomGutman]

>>You (and I) are talking about amending the system by which Mathcad handles 'unit's. <<

No, I'm discussing the reality of angles. Whether Mathsoft chooses to have Mathcad follow reality or SI or ISQ or some combination thereof or various combinations thereof (as options) is quite another matter.

>>We're discussing metrology here. The terms are well defined and understood in the metrology community. <<

They may be well defined, and even possibly understood, in the metrology community. But I am not a member of that community, and Mathcad does not seem to be a member either. The dialog where I can control the dimensions for Mathcad is labeled dimensions, not base quantities. I prefer terminology that I understand, that real people use and understand, and that relates to the software that eventually relates to the topic over terminology from a group that claims that angles are ratios of lengths.

>>SI Booklet Section 2.2.2<<

Huge amount of blathering and double talk, which seems to boil down to an acknowledgement that the formal definitions are wrong and should be actually be used, and provides other definitions for use.

>>Section 1.4 of the SI Booklet states:<<

Not much better. What is the value of one radian? A dimensionless unit is just a number, and must have a fixed numeric value. Similarly, a hertz has the dimensions of inverse time. So one hertz times one second is a dimensionless number. As such, it must have a dfinitive value. What is that value?

>>Which is why a system that deals with this must somehow track the unit or quantity (as a data attribute, say). <<

Piffle. The angular unit can, and should, be tracked the same way as any other dimension, like length or mass. I see no reason to complicate things by inventing another version of the wheel which will, in the end, boil down to doing exactly what the existing wheel does, just with different verbiage.

� � � � Tom Gutman

[PhilipOakley]

I think I've seen this all before.

Philip Oakley

[StuartBruff]

On 3/15/2007 5:12:12 PM, philipoakley wrote:
== I think I've seen this all before.

You have, Philip, you have.

Stuart

(Either that, or you've got a bad case of deja vu!)

[ptc-1368288]

On 3/15/2007 5:38:01 PM, stuartafbruff wrote:
>On 3/15/2007 5:12:12 PM, philipoakley
>wrote:
>== I think I've seen this all before.
>
>You have, Philip, you have.
>
>Stuart
>
>(Either that, or you've got a bad case
>of deja vu!)
___________________

Philip has probably seen many episodes of "war flames" about units.
As a surviving bird of the "war flames", don't fly above fire.

jmG

[TomGutman]

>>I think I've seen this all before.<<

Well, of course. Angles haven't changed their nature since at least Euclid. And commitees haven't changed their nature for at least as long (when were camels invented?)

� � � � Tom Gutman

[ptc-1368288]

"when were camels invented ?"
__________

A question for dimensioning the IQ ?

Before, before, before ... Peddy the parrot .

jmG

[StuartBruff]

On 3/15/2007 6:44:00 PM, Tom_Gutman wrote:
== (when were camels invented?)

28 October, 4004 BC? [1]

Stuart

Reference:
[1] Ussher, J. Annales veteris testamenti, a prima mundi origine deducti (1650)

[PhilipOakley]

You may remember I proposed a unit/quantity/kind/flag that I called the 'ara' (short for At Right Angles') to add to the unit of Torque N.m so that it was separated from the Force times Distance moved (Energy N.m).

Now as Tom will point out Torque time Angle turned also is Energy. [and the angle is in measured radians in this case],

So my (N.m.ara)*(rad) = (N.m) == Energy.

So ara*rad == 1

So how do we explain that a right angle is 1/rad? 😉

Now the angle turned is usually considered as a 'small' (infinitessimal?) so we can easily have
that the 'ara' is 1/(+0) = +Inf.

Or we can have that Length, Width and Depth are not commensurate and the ara is pointing that out almost as if the ara = Cotan('small'angle).

The units of volume of a cube may be the same as the length^3 but they are not the same quantity, even if they have the same value, so what is the missing 'unit' and how do we relate it to 'angle'

Are they duals of each other (electronic engineers are happy to use both magnitude and phase, and real and imaginary components of a frequency). One with a rotating (readjustable) frame of reference, the other with a fixed frame of reference.

Any other thoughts on how to explain that ara*rad == 1

Philip Oakley

[StuartBruff]

On 3/16/2007 8:10:42 AM, philipoakley wrote:
== The units of volume of a cube may be the same as the length^3 but they are not the same quantity, even if they have the same value, so what is the missing 'unit' and how do we relate it to 'angle'

Here's one thought ...

It's not so much a missing unit as missing quantity. The distinction lies in the fact that you are measuring different attributes. In this instance, they could be described as, say:

Length*Breadth*Width, where Length, Breadth and Width imply the 'kind' length but oriented in orthogonal directions.

We can further state that they are, again for example,

length-in-x_direction*length-in-y_direction*length-in-z_direction

which can, if required, be broken down into further attributes

(length).(x_direction)*(length).(y_direction)*(length).(z_direction).

If you wanted to name these extra attributes, you could call them xness, yness and zness (*)

Under the ISQ, the three 'directions' would map to dimension one, leaving the 'normal' length³

Furthermore, you could take account of orthogonality relationship and replace each attribute with an angular equivalent.

Define a Rotation operator (details = reader exercise)

Rot(θ,φ)
where
θ = angle.azimuth
φ = angle.elevation

and azimuth and elevation (or azimuth/altitude, yaw/pitch) are further attributes expressing the direction of the rotation.

xness ↔ Rot(0 rad,0 rad)
yness ↔ Rot(π/2 rad,0 rad)
zness ↔ Rot(0 rad,π/2 rad)

Stuart

(*) Similarly, it is well known that many people's spending gets heavier in December because
of Xmass

[PhilipOakley]

Intersting idea the rotation bit. Normal cartesian axes are at 90 degrees to each other to define their orthogonality.

This is not necessary to span a space. As long as the axial dircetions are all linearly independent that those axes will fully span the cartesian space (i.e. you can get to any point with a linear combination of the base vectors) [for the casual reader, Google "linear algebra span" for more, I'm sure Stuart already knows this 😉 ]

In terms of using axes and angles, such as in gimballed sensor systems, the standard azimuth then elevation gimbal method always has a gimbal lock at 90 deg.

One long standing idea I have had relates to using three rotation axes at 60 degrees to each other so that lock never happens.

Back at the xness * yness, my thought (which tries to summarise the common man's view) is that they would think of how many right angle axes have I in my system. so I'm still supportive of the 'ara'.

However, after the drive home, I now think that what we have is a mixed 'dual system'. so we want (using an Argand complex number diagram as an exemplar) to describe a location P by both

P = x+i.y and by P = r .angle. theta

neither is wrong, both are valid, but the unit of length for x, y, and r causes confusion.

More thoughts: The first term approximation for trig relations is sin(t)=t, tan(t)=t, cos(t)=1, so the differentials are 1, 1, and 0.
[note that the usual cos(t) = 1-(t^2)/2 is two terms and order two]. It is the use of this very simplified level that is used to create these 'radian' confusions
[i.e. that ara.rad == 1].

Philip Oakley

[PhilipOakley]

More:
Ultimately, this would need a tick box for 'use small angle units approximation'.

This would then allow the simplifications of:
# Sterad = rad^2
# ara*rad = 1
and other consequents of that (swept steradian calcs etc)

I've been wondering about how to add the (sterad/rad^2)=1 small angle approximation. Now we have a simplification rule for those cases when it is chosen to apply.

Philip Oakley

[TomGutman]

Because you've chosen to include the angle dimension by something you call ARA.

Torque is the derivative of energy with respect to an angle. Therefore it has the dimensions of energy per unit angle, and the natural units of joules/radian. The usual formula for torque, force times lever arm, is incorrect. It is an empirical formula that works only of you consider the radian to be a dimensionless unity. The correct formular is force times lever arm divided by one radian.

Factors of a one radian angle show up in quite a few formulae involving circular motion, when they are derived rigorously keeping the angle dimension. They are a blasted nuisance, which is why common usage is to ignore them, treating the one radian angle as a dimensionless unity. As a general rule, people prefer convenience to rigor.

� � � � Tom Gutman

[Fred_Kohlhepp]

On 3/16/2007 2:39:09 PM, Tom_Gutman wrote:
The usual
>formula for torque, force
>times lever arm, is incorrect.
>It is an empirical formula
>that works only of you
>consider the radian to be a
>dimensionless unity. The
>correct formular is force
>times lever arm divided by one
>radian.

My head is starting to hurt. Can you explain the rationale behind that statement?



Fred Kohlhepp
fkohlhepp@sikorsky.com

[ptc-1368288]

"The units of measurement of the covariance cov(X, Y) are those of X times those of Y. By contrast, correlation, which depends on the covariance, is a dimensionless measure of linear dependence.
Random variables whose covariance is zero are called uncorrelated".

jmG

[TomGutman]

Sure.

� � � � Tom Gutman

[StuartBruff]

On 3/15/2007 1:59:16 PM, Tom_Gutman wrote:
>> You (and I) are talking about amending the system by which Mathcad handles 'unit's. <<
== No, I'm discussing the reality of angles.

My apologies.

>>We're discussing metrology here. The terms are well defined and understood in the metrology community. <<
== They may be well defined, and even possibly understood, in the metrology community. But I am not a member of that community, and Mathcad does not seem to be a member either.

You may not be, but PTC certainly claim that Mathcad conforms to the standards set by the metrological community.

In a certain sense, everyone is a part of the metrologial community whether they know it or not. Weights and measures have been important aspects of commercial and personal life for millenia. People are likely to quantify attributes using the units laid down by such organizations as the BIPM and as directed by their national governments. As examples of the impact that the metrological community has on peoples' lives ...

http://ts.nist.gov/WeightsAndMeasures/Metric/mpo_home.cfm
http://ts.nist.gov/WeightsAndMeasures/Metric/pub814.cfm#act
http://physics.nist.gov/cuu/pdf/SIFedReg.pdf

== The dialog where I can control the dimensions for Mathcad is labeled dimensions, not base quantities.

... and doesn't actually accord with what is stated in the Mathcad xml and mpl files.

== I prefer terminology that I understand, that real people use and understand,

I'm perfectly sure that lots of people feel more comfortable with the terminology they use and understand. Unfortunately, a lot of the errors I see (mine included) are because people didn't really understand the terminology, or its underlying rationale, in the first place. Furthermore, as time goes on, our understanding of issues changes and we see that previous terminology may no longer accurately reflects our current understanding (indeed, it can be quite misleading[*])

So, all in all, I prefer terminology that actually relates more accurately to the subject at hand, and lets me have some chance of understanding why things are described the way they are. I'd rather reeducate myself and have some reasonable chance of addressing inconsistencies and errors in a manner compatible with that terminology.

== and that relates to the software that eventually relates to the topic over terminology from a group that claims that angles are ratios of lengths.

See above. (BTW, formally, it's the ISO that think that, not the BIPM (where it has been a subject for debate for several years. the trouble is that they keep banging their heads up against apparently intractible theoretical aspects))

>>SI Booklet Section 2.2.2<<
== Huge amount of blathering and double talk, which seems to boil down to an acknowledgement that the formal definitions are wrong and should be actually be used, and provides other definitions for use.

Is that a round about way of saying you were wrong about the hertz and the radian/second being the same thing under the SI?

>>Section 1.4 of the SI Booklet states:<<
== Not much better. What is the value of one radian?

1 rad

== A dimensionless unit is just a number, and must have a fixed numeric value. Similarly, a hertz has the dimensions of inverse time. So one hertz times one second is a dimensionless number. As such, it must have a dfinitive value. What is that value?

1

>>Which is why a system that deals with this must somehow track the unit or quantity (as a data attribute, say). <<
== Piffle. The angular unit can, and should, be tracked the same way as any other dimension, like length or mass.

You mean base quantity.

== I see no reason to complicate things by inventing another version of the wheel which will, in the end, boil down to doing exactly what the existing wheel does, just with different verbiage.

Because people have a greater probablilty of knowing why they get certain results, how a system should operate, and how they can circumvent problems if they understand and use the correct terminology 😕

Stuart

[*] As a minor example, it is common in the UK for real people to refer to a degree of pain as 'chronic' ("How yer doin', Alfie? Yer not lookin' too good.", "It's me back, Reg, fell over a couple of minutes ago. 'Urts summat chronic"). Now both Alfred and Reginald understand that by 'chronic' Alfred means "It is extremely painful, Reginald". In technical medical terms, however, 'chronic' simply means its been going on for a long time and says nothing about the degree of pain. However, if Alf went to his doctor about it and the doctor said "Ah! So it's acute pain?", he'd be likely to get a thump in certain parts of the country ("Cute? What's cute about it?" .. thump .. "Bet you don't think that's cute, do you Doc?").

I would be reluctant to change medical terminology simply because lots of people have a different understanding of the terms.

[TomGutman]

>>Not much better. What is the value of one radian?

1 rad
<<

Thereby denying the unit of radian the status of being dimensionless. Indeed, taking it as a dimension (or base quantity). That statement says that the radian cannot be expressed in any simpler terms. Making it a base quantity. If you were to treat it as truly dimensionless, it would have to be expressible as just a number, a number with no units or qualifications attached.

As another question, do you accept the relationship that one hertz is equal to two π radians per second?

� � � � Tom Gutman

[StuartBruff]

On 3/16/2007 2:48:58 PM, Tom_Gutman wrote:
== >>Not much better. What is the value of one radian?

1 rad
<<

== Thereby denying the unit of radian the status of being dimensionless.

No. The radian is the unit of the quantity angle. In the ISQ angle has dimension one. The quantity, not the unit, is 'dimensionless'.

== Indeed, taking it as a dimension (or base quantity). That statement says that the radian cannot be expressed in any simpler terms. Making it a base quantity.

No. radian is the unit. In similar terms I cannot express energy any more simply than 'joule'. However, it has dimension L².M.T^-2, and is alternatively expressed in SI units as m².kg.s^-2. It is not unique in having this dimension, and angle is not unique in having dimension one.

This is a consequence of any system of quantities that fails to distinguish between quantities of different kinds.

== If you were to treat it as truly dimensionless, it would have to be expressible as just a number, a number with no units or qualifications attached.

If it were truly 'dimensionless', then yes it would. However, 'dimensionless' is a bit of a misnomer and misleading, which is why the BIPM prefer the term 'of dimension one'.

It's also why I suggested that it may offer a partial way round *some* of the problems that Valery, inter alia, identified. By including the dimension one in the system, it would be possible to distinguish a quantity from just a number, thus making '1 rad + 1' flag up a unit mismatch error. Of course, it would still fail to flag up equating 'dimensionless' quantities of different kinds.

Which is why I've also suggested that there should be a general user facility for adding base quantities (radiation, angles, currency, whatever) to address problem domains where this lack of distinction is critical or just plain annoying.

== As another question, do you accept the relationship that one hertz is equal to two π radians per second?

No. In certain domains there is an equivalence, but in general there is no identity.

Stuart

[TomGutman]

>>No. The radian is the unit of the quantity angle. In the ISQ angle has dimension one. The quantity, not the unit, is 'dimensionless'.<<

A unit is just a specific quantity (of a particular type) used as the reference quantity for expressing quantities of that type. The radian is a specific angle (defined mathematically by one of its properties, just like the meter is a specific length, until recently defined by a couple of scratches) and is one unit that can be used to measure angles. A unit has, perforce, the same dimensions as the underlying quantity.

What is the value of one meter divided by two meters?

>> In similar terms I cannot express energy any more simply than 'joule'.<<

The joule is defines as a meter multiplied by a newton, a force of one newton acting over a distance of one meter. It can properly be written as a meter-newton (not much used by users of SI, but the cognate formulation of foot-pound is in common use), using "simpler" (closer to the base units) units.

>>No. In certain domains there is an equivalence, but in general there is no identity.<<

So at least one of hertz, π, radian, and second is not well defined but depends on the domain. Which one of these is it? Can you categorize or characterize the domains within which specific meanings apply? Are there any domains within which you do accept that one hertz is equal to two π radians per second?

I suppose it should be obvious, but in light of some of your answers I need to ask: do you accept that one radian per second times one second is one radian?

� � � � Tom Gutman

[StuartBruff]

On 3/17/2007 1:12:03 AM, Tom_Gutman wrote:
>>No. The radian is the unit of the quantity angle. In the ISQ angle has dimension one. The quantity, not the unit, is 'dimensionless'.<<
== A unit is just a specific quantity (of a particular type) used as the reference quantity for expressing quantities of that type. The radian is a specific angle (defined mathematically by one of its properties, just like the meter is a specific length, until recently defined by a couple of scratches) and is one unit that can be used to measure angles. A unit has, perforce, the same dimensions as the underlying quantity.

Oops! You're quite right, a unit is a quantity. That's what comes from using an inaccurate definition. <beg>

From the VIM:
-----------------------------------------------------
unit of measurement

scalar quantity, defined and adopted by convention, with which other quantities of the
same kind are compared in order to express their magnitudes

NOTES
1 Units are designated by conventionally assigned names and symbols.
2 Units of quantities of the same dimension may be designated by the same name and
symbol even when the quantities are not of the same kind. For example the joule per
kelvin, J/K, is the name and symbol of both a unit of heat capacity and a unit of entropy,
which are generally not considered to be quantities of the same kind.
3 Units of quantities of dimension one are simply numbers. In some cases these
numbers are given special names, e.g. radian and steradian, or are expressed by
quotients such as millimole per mole
-----------------------------------------------------

>> In similar terms I cannot express energy any more simply than 'joule'.<<
== The joule is defines as a meter multiplied by a newton, a force of one newton acting over a distance of one meter. It can properly be written as a meter-newton (not much used by users of SI, but the cognate formulation of foot-pound is in common use), using "simpler" (closer to the base units) units.

Well you can define words to mean whatever suits your position; closer yet to the base units is the base unit expression itself, or we could have the zeno-isaac, where 'isaac' is my new unit for momentum and 'zeno' is my new unit for velocity. Personally, I think 'joule' is "simpler" than 'metre-newton' because it has less letters.

== What is the value of one meter divided by two meters?

1/2 in simplified form.

Although I might, depending upon circumstance, express it as 1/2 m/m (per sim the millimole per mole example quoted above), or 1/2 one.

== I suppose it should be obvious, but in light of some of your answers I need to ask: do you accept that one radian per second times one second is one radian?

Yes.

OK, now we've got the fun stuff over with, I think you're trying to address several different issues here but wrapping them up as one: the handling of dimensionless quantities within *any* systems of quantities, the reality of angles, and the equivalence of an angle and a 'cycle'. I'll deal with the second issue first because it's the 'easy' one.

Reality of Angles

I fully and completely agree that angle should be a base quantity.

I have held, and maintained, this position since shortly after we first started discussing this matter a couple of years ago. I may disagree with your notion of exactly what an angle is, as discussed in another thread, but that's more to do with the scope of an angle than its 'reality'.

'Dimensionless' Quantities

== The radian is a specific angle (defined mathematically by one of its properties, just like the meter is a specific length, until recently defined by a couple of scratches) and is one unit that can be used to measure angles. A unit has, perforce, the same dimensions as the underlying quantity.

You mean has the same dimension - note the singular.

Unfortunately for your argument, it doesn't really change much.

If I decide to express Mach number 1.2 in unit form, I can define the unit to be the 'mach' and write "1.2 mach", instead of the more conventional "Mach 1.2" or "M 1.2". We can also define other useful ratios and assign the unity ratio to be the unit for each ratio; for example, the 'reynold' for Reynold Number. Now clearly I know I mean that the unit refers to speed ratios even though it shares the same dimension as any other 'dimensionless' quantity, and I wouldn't equate mach = reynold on purely (simplified) dimensional grounds - they are different kinds of quantity.

The same problem arises with any set of quantities of different kinds that have the same dimension.

In the field of ionizing radiation, for example, activity has the unit bequerel rather than reciprocal second (or hertz). However, this is pretty useless information under strict dimensional analysis, as the context (quantity kind) is lost, and and activity multiplied by time is 'dimensionless'.

Which is why I've suggested a general facility to add base quantities, rather than just addressing the specific example of the angle.

Cycles and Radians

== As another question, do you accept the relationship that one hertz is equal to two p radians per second?
>> No. In certain domains there is an equivalence, but in general there is no identity.<<
== So at least one of hertz, p, radian, and second is not well defined but depends on the domain. Which one of these is it?

None. They are well defined within the ISQ. See the bit above on dimensionless quantities, though.

== Can you categorize or characterize the domains within which specific meanings apply?

I said 'equivalence' not 'identity' because ...

== Are there any domains within which you do accept that one hertz is equal to two p radians per second?

... Not without a suitable set of connecting definitions.

If one wishes to pin down the meaning of 'cycle' then that could result in identities. For example, IF the cycle was defined to be a unit of angle AND frequency was redefined to be 'cycles per second' THEN one hertz == two p radians per second.

As it stands, I do not know what the formal definition of a cycle is. The SI Brochure is not explicit, and I believe that the ISO 31 draws a distinction between frequency and angular frequency.

In the absence of a formal definition, I tend towards what I think is Richard's view, that a 'cycle' is rather a vague concept and merely indicates that some pattern is repeating. The IUPAC Gold Book, which I believe accords with the ISO, defines frequency as the reciprocal of period, and period as the time for one cycle of a periodic phenomenon; no mention of angle. On this basis, I might draw an analogy between cycles and angles, but I wouldn't say they are the same thing.

Stuart

[TomGutman]

I quite understand that truly dimensionless quantities (quantities which are inherently ratios of like quantities) is a problem. While they have the same dimensions (none) they are clearly not all the same. Yet they all have some behaviours in common, such as the ability to take arbitrary powers of them and to multiply by other quantities and keeping the dimensions.

But I restrict myself to the case of angles, because it is a quite different problem. The problem here is that angles are not dimensionless, and pretending that they are inevitably leads to various forms of confusion and inconsistencies.

There are other cases where SI's treatment of something as dimensionless is questionable. That is when the quantity is not a ratio of two like quantities (like a sine or a refractive index or a mach number) but rather involves a count. SI treats all counts as dimensionless quantities. That is questionable. A pure count is just a number. But I would think that a count of something would have the dimensionality of that something. But I haven't worked that through, have not seen it actually causing problems (this thread started by a collab pointing out the inconsisties engendered by the angle fiction), and prefer not to get into that at this time.

>>3 Units of quantities of dimension one are simply numbers<<

Therefore if angles are of dimension one the radian, being an angle, must be simply a number. What is that number? To deny that the radian can be expressed as just a number is, by the point (definition? corolary?) quoted above, to deny that the radian is of dimension one.

>>== What is the value of one meter divided by two meters?

1/2 in simplified form.<<

Why not 1/2 radian? If angles are lengths/lengths, then this should be the definition of an angle.

>>You mean has the same dimension - note the singular.<<

I consider force to have the dimensions of mass times length times two inverse times. In general a unit has multiple dimensions. So I use the plural in general.

>>== So at least one of hertz, p, radian, and second is not well defined but depends on the domain. Which one of these is it?

None. They are well defined within the ISQ. See the bit above on dimensionless quantities, though.<<

If they are well defined (and a meaningless, incomprehensible, inconsistent definition does not result in something being well defined, so just the existence of text that claims to be, and has the form of, a definition does not mean that something is well defined) the relationship "one hertz is equal to two π radians per second" is well defined and is either true or not. True meaning for all values, not just coincidentally for particular isolated values. So, is it true or is it not true? Your little digression on cycle is not relevant here, as none of the terms involved have definitions involving the cycle (some should, but they don't). If you cannot say if the relationship is true or not, then you cannot have a definite definition for all the terms. Well, there is one possible out -- while if the terms are all well defined the relation will be true or false, it is possible that it is not provably true or false (Godel's incompleteness theorem). Do you wish to take the position that the truth or falsity of this relationship is undecidable?

>>I tend towards what I think is Richard's view, that a 'cycle' is rather a vague concept and merely indicates that some pattern is repeating. The IUPAC Gold Book, which I believe accords with the ISO, defines frequency as the reciprocal of period, and period as the time for one cycle of a periodic phenomenon; no mention of angle. On this basis, I might draw an analogy between cycles and angles, but I wouldn't say they are the same thing.<<

I take a somewhat different approach. A cycle is always associated with an angle of 2πradians, and fractions of a cycle can be represented by fractions of that angle. The angle may be a real angle, if rotating machinery is involved, or it may be a metaphoric angle, as commonly used by EEs (and others). Whether the angle is real or metaphoric it may be used (as in doing an analysis using Fourier series) or it may be ignored (doing a purely algebraic analysis of the system). But it will not be contradicted.

� � � � Tom Gutman

[StuartBruff]

On 3/17/2007 7:16:40 PM, Tom_Gutman wrote:
== TG reply -1
== SB reply -1
++ TG reply 0

++ I quite understand that truly dimensionless quantities (quantities which are inherently ratios of like quantities) is a problem. While they have the same dimensions (none) they are clearly not all the same. Yet they all have some behaviours in common, such as the ability to take arbitrary powers of them and to multiply by other quantities and keeping the dimensions.

"While they have the same dimensions (none)" is not true; they have dimension 1.

++ There are other cases where SI's treatment of something as dimensionless is questionable. That is when the quantity is not a ratio of two like quantities (like a sine or a refractive index or a mach number)

How about height-to-width ratio v height-to-height ratio?

++ but rather involves a count. SI treats all counts as dimensionless quantities. That is questionable. A pure count is just a number. But I would think that a count of something would have the dimensionality of that something.

I agree with that.

++ But I haven't worked that through, have not seen it actually causing problems (this thread started by a collab pointing out the inconsisties engendered by the angle fiction),

I gave you an example. From the SI Brochure:

----------------
In the field of ionizing radiation , the SI unit of activity is designated the becquerel rather than the reciprocal second, and the SI units of absorbed dose and dose
equivalent the gray and sievert , respective ly, rather than the joule per kilogram . The special names becquerel , gray and sievert were specifically introduced
because of the dangers to human health which might arise from mistakes
----------------

But the quantity becquerel*second is dimensionless (and therefore indistinguishable from any other dimensionless quantity), thus opening the floodgates wide to mistakes and somewhat negating the reason for having it in the first place.

The mole is an interesting unit in this respect (counting):

1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its
symbol is �mol�.
2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. (my highlight)

AFAIA, Mathcad's unit system does not offer any automatic method for tracking such a specification.

In Section 2.2.2 of the SI Brochure, following Table 4, it states?
--------------------------
A single SI unit may correspond to several different quantities, as noted in paragraph 12 (p. 92). In the above table, which is not exhaustive, there are several examples. Thus the joule per kelvin (J/K) is the SI unit for the quantity heat capacity as well as for the quantity entropy; also the ampere (A) is the SI unit for the base quantity electric current as well as for the derived quantity magnetomotive force. It is therefore important not to use the unit alone to specify
the quantity. This rule applies not only to scientific and technical texts but also, for example, to measuring instruments (i.e. an instrument should indicate both
the unit and the quantity measured ) .
--------------------------

Based on this, it's not too unreasonable to suggest that Mathcad should provide an automated facility for tracking, and displaying, quantities.

++ ... But I restrict myself to the case of angles, because it is a quite different problem. The problem here is that angles are not dimensionless, and pretending that they are inevitably leads to various forms of confusion and inconsistencies.
++ ... and prefer not to get into that at this time.

I would like to get into it at this time, because otherwise any solution might be so tailored to dealing with the angle problem that it makes the generalized problem more difficult than need be.

>>3 Units of quantities of dimension one are simply numbers<<
++ Therefore if angles are of dimension one the radian, being an angle, must be simply a number. What is that number?

That number is 1. However, ...

++ To deny that the radian can be expressed as just a number is, by the point (definition? corolary?) quoted above, to deny that the radian is of dimension one.

That's also the conclusion I drew when I first saw that statement. I think it is incorrect (that is, a defect in the VIM) and believe that a distinction should be made between a number and dimension one.

>> You mean has the same dimension - note the singular.<<
++ I consider force to have the dimensions of mass times length times two inverse times. In general a unit has multiple dimensions. So I use the plural in general.

A quantity has but one dimension. Therefore, a unit has exactly one dimension if that unit uniquely attaches to a quantities of the same kind.

VIM
dimension
dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities
a) In the ISQ, where L, M and T denote the dimensions of the base quantities length, mass, and time, the dimension of force is LMT^-2. (my highlight)

== So at least one of hertz, p, radian, and second is not well defined but depends on the domain. Which one of these is it?
>> None. They are well defined within the ISQ. See the bit above on dimensionless quantities, though.<<
++ If they are well defined (and a meaningless, incomprehensible, inconsistent definition does not result in something being well defined, so just the existence of text that claims to be, and has the form of, a definition does not mean that something is well defined)

They are 'well' defined within the system.

++ the relationship "one hertz is equal to two π radians per second" is well defined and is either true or not.
++ True meaning for all values, not just coincidentally for particular isolated values.
++ So, is it true or is it not true?

It is false. As I said about 4 messages back .. or is the word 'No' insufficiently precise an answer to the question "do you accept the relationship that one hertz is equal to two π radians per second?"?

== What is the value of one meter divided by two meters?
>> 1/2 in simplified form.<<
++ Why not 1/2 radian? If angles are lengths/lengths, then this should be the definition of an angle.

Replace 'should' with 'could'. (+2)²=4 and (-2)²=4, why 'should' the square root of 4 map to -2?

>>I tend towards what I think is Richard's view, that a 'cycle' is rather a vague concept and merely indicates that some pattern is repeating. The IUPAC Gold Book, which I believe accords with the ISO, defines frequency as the reciprocal of period, and period as the time for one cycle of a periodic phenomenon; no mention of angle. On this basis, I might draw an analogy between cycles and angles, but I wouldn't say they are the same thing.<<
++ I take a somewhat different approach. A cycle is always associated with an angle of 2πradians, and fractions of a cycle can be represented by fractions of that angle. The angle may be a real angle, if rotating machinery is involved, or it may be a metaphoric angle, as commonly used by EEs (and others). Whether the angle is real or metaphoric it may be used (as in doing an analysis using Fourier series) or it may be ignored (doing a purely algebraic analysis of the system). But it will not be contradicted.

A 'metaphoric' angle? I thought you were discussing the 'reality' of angles? Now you're introducing 'metaphoric' ones? What's the domain of these metaphoric angles? Are there any other kinds of angle we should know about (allegorical?, metonymic?, parabolic?)? How 'real' are they? Can I use a protractor to measure them?

Stuart

[TomGutman]

>>"While they have the same dimensions (none)" is not true; they have dimension 1.<<

Dimension one means no dimensions. The dimensions of a quantity are the various base dimensions raised to appropriate powers. Zero powers are elided. When all basic dimensions cancel out (all the exponents work out to zero) the quantity is dimensionless, it has no dimensions. In contexts where this is unacceptable (say a table where there is a column for dimensions that is not allowed to be blank) the numer one is used, presumably on the basis that anything, even if not a number, raised to the zeroth power is one. Factors of one can be added or deleted from any product and are completely irrelevant. The concept of "dimension one" is meaningless -- it means something is dimensionless, just a pure number.

>>Based on this, it's not too unreasonable to suggest that Mathcad should provide an automated facility for tracking, and displaying, quantities.<<

Without some specification of what that means, and how (logically) this is to be done, it is meaningless. Just a general pious hope. I know what needs to be done for angles, and have worked out how it works. I have no idea what it would mean for moles, where three moles of oxygen might, under various physical conditions, turn into two moles of ozone or six moles of atomic oxygen.

>>++ the relationship "one hertz is equal to two � radians per second" is well defined and is either true or not.
++ True meaning for all values, not just coincidentally for particular isolated values.
++ So, is it true or is it not true?

It is false.<<

So all these EEs who take a 10hz signal and make it 20πradians per second are wrong, at least if SI is accepted as correct.

That may well be true. They are not wrong in making the conversion, they just know that SI is really wrong and ignore it, doing what they know is right, regardless of what SI says.

>>Replace 'should' with 'could'. (+2)²=4 and (-2)²=4, why 'should' the square root of 4 map to -2?<<

What does that have to do with the price of tea in China? The term "square root" is not well defined, and it is well known that it is not well defined. At various times various authors have used the expressions √x and x^½ as expressly diffent.

>>A 'metaphoric' angle? ... Can I use a protractor to measure them?<<

I've been using this term for years now. You just noticed? No, you cannot use a protractor to measure them (at least, not a real protractor, possibly a metaphoric protractor). But never fear. Just go find the nearest EE -- he will have no trouble measuring those angles.

� � � � Tom Gutman

[StuartBruff]

On 3/18/2007 9:59:03 PM, Tom_Gutman wrote:
>>"While they have the same dimensions (none)" is not true; they havedimension 1.<<
== Dimension one means no dimensions. The dimensions of a quantity are the various base dimensions raised to appropriate powers. Zero powers are elided. When all basic dimensions cancel out (all the exponents work out to
zero) the quantity is dimensionless, it has no dimensions. In contexts where this is unacceptable (say a table where there is a column for dimensions that is not allowed to be blank) the numer one is used, presumably on the basis that anything, even if not a number, raised to the zeroth power is one. Factors of one can be added or deleted from any product and are completely irrelevant. The concept of "dimension one" is meaningless -- it means something is dimensionless, just a pure number.

I disagree. Having a dimension is the thing that distinguishes a quantity from a pure number and makes sense of giving it a unit.

>>Based on this, it's not too unreasonable to suggest that Mathcad should provide an automated facility for tracking, and displaying, quantities.<<
== Without some specification of what that means, and how (logically) this is to be done, it is meaningless.

Correct; without some specification it is meaningless. And this is relevant how?

What's meaningless about "Mathcad should provide an automated facility for tracking, and displaying, quantities.", given that I'd provided in-context reference to the SI Brochure? Taking the thread as a whole, I have referenced the VIM, the ISO-31 and the SI Brochure to provide definitions of the various terms (ie, quantity, dimension), which any reasonably competent requirements engineer should be able to convert from a high-level user need to a system requirement specification.

I have no intention of telling any product engineer how they should logically do anything. Why impose the limits of my imagination on them? If they say it can't be done, *then* I might tell them how, but not before.

== Just a general pious hope.

You mean like your general pious hope that I'll know what you mean by 'metaphorical' angle without a specification 🙂

== I know what needs to be done for angles, and have worked out how it works.

Uh huh. If you manage to properly define an angle, I might have a degree more confidence in that statement.

== I have no idea what it would mean for moles,

That's OK, we'll ask a domain expert.

== where three moles of oxygen might, under various physical conditions, turn into two moles of ozone or six moles of atomic oxygen.

One possible solution is to create a quantity 'number of oxygen atoms' and declare it to be a base quantity. Now define a molecule (I presume that's what you meant) of oxygen as having an oxygen count of 2. A quantity equation will determine the actual transformation , and balance the number of oxygen atoms.

That's one of the underlying reasons why I've asked for a general user capability to add base quantities.

>>++ the relationship "one hertz is equal to two � radians per second" is well defined and is either true or not.
++ True meaning for all values, not just coincidentally for particular isolated values.
++ So, is it true or is it not true?
-- It is false.<<
== So all these EEs who take a 10hz signal and make it 20πradians per second are wrong, at least if SI is accepted as correct.

Yes.

== That may well be true. They are not wrong in making the conversion, they just know that SI is really wrong and ignore it, doing what they know is right, regardless of what SI says.

I have my doubts that most of 'all these EEs' have ever considered the rights and wrongs of the SI. I'll bet they've been told something amounting to '1 hertz = 1 cycle per second' and 'a cycle is a unit of angle such that 1 cycle == 2 π radian', and never put a moments thought into what the SI and ISO actually say.

As, most of the time, they manually carry out unit manipulation, they keep mental track of the quantities involved and write down the correct units for angles. However, because they are unaware of the actual definitions and do not understand the limitations of dimensional analysis[*], they have problems when SI-compliant dimensional analysis tools don't give the answer they are expecting.

[*] or haven't stepped back far enough from the problem to think about it.

>>Replace 'should' with 'could'. (+2)�=4 and (-2)�=4, why 'should' the square root of 4 map to -2?<<
== What does that have to do with the price of tea in China? The term "square root" is not well defined, and it is well known that it is not well defined. At various times various authors have used the expressions ?x and x� as expressly different.

The term "square root" seems to be sufficiently well defined that many mathematicians, physicists, and engineers (including the EE subtype) are happy to use it and know that 4 has two "square roots". They also can (and do) make use of any problem-related constraints to identify which "square root" is applicable to their problem.

http://mathworld.wolfram.com/SquareRoot.html

So if it's good enough for general usage, then it's good enough for me to use it as an analogy to point out that merely saying that "angle is defined by length/length" does not imply that all ratios of length/length are angles. You can add contextual material to identify them as such, but without this information, there is no immediate justification for making that identity.

>>A 'metaphoric' angle?
== I've been using this term for years now. You just noticed?

How unobservant of me.

But I'm a little puzzled. If you've been using it for years, then I would expect that you have come up with a formal definition of a metaphoric angle by now.

>> ... Can I use a protractor to measure them?<<
== No, you cannot use a protractor to measure them (at least, not a real protractor, possibly a metaphoric protractor).

A 'metaphorical protractor'. Is that a complex device or purely imaginary (given that it's not real)?

== But never fear. Just go find the nearest EE -- he will have no trouble measuring those angles.

Not until you can provide a formal definition that allows him (or her) to be sure they're measuring the thing you mean.

Stuart