The radian
Epiphany
and I know that the radian is convenient for trigonometric calculations
and for calculating things like the work done by a torque turning
through a given angle. So, why is the radian so *absent* from common
use? Has anyone, for example, published a terrestrial or celestial
atlas with latitude-longitude or right ascension-declination given in
radians? Does anyone make magnetic compasses with azimuth marked in
radians?
It occurs to me that however primary the radian may be to angular
measure, it does not give a full revolution as a nice decimal value.
Thinking along that line, I have wondered why no one ever suggested
using "millirevolutions" and "microrevolutions" to measure angles, where
there 1000 millirevolutions = 1 revolution and 1000000 microrevolutions
= 1 revolution. I know that there is the grade, with 400 grades per
revolution, but seeing as even the degree is defined as 360 degrees per
revolution, it seems to me like it is the *revolution* that is
fundamental and that it would be most logical to use a metric-like
decimalization of the revolution if we cannot stomach the irrational
number of radians per revolution using the radian entails.
How do others here view this angle?
\ | /
-Epiphany-
/ | \
Erik Max Francis
> I know that the radian is the unit of angular measure adopted by SI
> and I know that the radian is convenient for trigonometric calculations
> and for calculating things like the work done by a torque turning
> through a given angle. So, why is the radian so *absent* from common
> use? Has anyone, for example, published a terrestrial or celestial
> atlas with latitude-longitude or right ascension-declination given in
> radians? Does anyone make magnetic compasses with azimuth marked in
> radians?
> It occurs to me that however primary the radian may be to angular
> measure, it does not give a full revolution as a nice decimal value.
Not to mention its comparative obscurity if you're not introduced to
trigonometry, the reason it's not in common use is probably for
precisely that reason.
> I know that there is the grade, with 400 grades per
> revolution, but seeing as even the degree is defined as 360 degrees per
> revolution, it seems to me like it is the *revolution* that is
> fundamental and that it would be most logical to use a metric-like
> decimalization of the revolution if we cannot stomach the irrational
> number of radians per revolution using the radian entails.
> How do others here view this angle?
The radian is the only natural unit of angular measure. You use it or
you don't. You can't hide it but measuring submultiples of a radian
(which you already can under SI, anyway).
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
Blood is the god of war's rich livery.
-- Christopher Marlowe
Jukka K. Korpela
> Epiphany wrote:
- -
>> I know that there is the grade, with 400 grades per
>> revolution, but seeing as even the degree is defined as 360 degrees
>> per revolution, it seems to me like it is the *revolution* that is
>> fundamental and that it would be most logical to use a metric-like
>> decimalization of the revolution if we cannot stomach the irrational
>> number of radians per revolution using the radian entails.
>> How do others here view this angle?
>
> The radian is the only natural unit of angular measure.
Epiphany's point was that revolution is a fundamental and natural concept.
For example, the sum of the angles of a triangle is 0.5 revolutions.
Mesopotamian cultures used a revolution as the basis of angle measurements
and divided it into parts using their number system. Defining a revolution
as the unit of angular measure would surely be the most natural approach,
and naturally if we did that, we would use decimal submultiples and
multiples thereof.
Now, there might be mathematical reasons, related to analytic geometry, that
make some people think that the radian is more "natural", or even that it is
_the_ natural unit. But this would be based on concepts and formulas that
are just a few hundred years old and on simplicity of some _formulas_, not
what is natural to a human being.
And even in mathematics, _some_ things would be easier and more natural when
using the revolution. For example, the period of the sine function would be
simply 1.
--
Yucca, http://www.cs.tut.fi/~jkorpela/
Erik Max Francis
If you want a coherent unit system, then radians are the only possible
choice, because they're the only unit of a plane angle that's actually
dimensionless. The power of SI is not only that it is metric, but that
it is coherent -- that is, the conversion factor between all fundamental
and derived units (and supplementary units, like the radian) is 1.
Same argument goes with steradians for solid angles.
If you want to think in cycles/revolutions, then you can use the unit
hertz (defined as s^-1), which is what it's designed for.
You may well be right that it may not be obvious why the radian is the
best choice of unit for plane angle. But once you understand the
reasons (dimensional analysis, trigonometry, calculus, etc.), it's a bit
hard to deny. And, at any rate, since this newsgroup is about the
metric system, it's hard to avoid acknowledging that the only possible
choice for the plane angle is the radian.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
Wars may be fought with weapons, but they are won by men.
-- Gen. George S. Patton
Jukka K. Korpela
> If you want a coherent unit system, then radians are the only possible
> choice, because they're the only unit of a plane angle that's actually
> dimensionless.
Why would a quantity need a dimensionless unit?
> If you want to think in cycles/revolutions, then you can use the unit
> hertz (defined as s^-1), which is what it's designed for.
The issue is angles, not frequencies.
> You may well be right that it may not be obvious why the radian is the
> best choice of unit for plane angle.
Is is the best, the only possible, the most natural one, or the only natural
unit? You are not very consistent in your statements.
> But once you understand the
> reasons (dimensional analysis, trigonometry, calculus, etc.), it's a
> bit hard to deny.
Dimensional analysis, trigonometry, calculues, and coherence are rather far
from the natural meaning of "natural".
> And, at any rate, since this newsgroup is about the
> metric system, it's hard to avoid acknowledging that the only possible
> choice for the plane angle is the radian.
Not true. People measured angles long before the radian was invented. If you
wish to prove that it is the only/best/most natural one _given some specific
requirements_, then you should list down those requirements and outline the
proof, the deduction from those requirements.
--
Yucca, http://www.cs.tut.fi/~jkorpela/
Erik Max Francis
> Erik Max Francis wrote:
>
>> If you want a coherent unit system, then radians are the only possible
>> choice, because they're the only unit of a plane angle that's actually
>> dimensionless.
>
> Why would a quantity need a dimensionless unit?
Because it's the only way to measure a plane angle as a coherent unit,
and SI is a system of coherent units. If you don't know what a coherent
unit system is, it's in the part you clipped out while replying.
>> You may well be right that it may not be obvious why the radian is the
>> best choice of unit for plane angle.
>
> Is is the best, the only possible, the most natural one, or the only
> natural unit? You are not very consistent in your statements.
I didn't even use the word _natural_ in that sentence, so what supposed
inconsistency are you complaining about?
Read it again: "You may well be right that it may not be obvious why
the radian is the best choice of unit for plane angle." But knowledge
of the things I list below indicate it is.
>> But once you understand the
>> reasons (dimensional analysis, trigonometry, calculus, etc.), it's a
>> bit hard to deny.
>
> Dimensional analysis, trigonometry, calculues, and coherence are rather
> far from the natural meaning of "natural".
Well, they are precisely the things that make you realize that if you
use anything other than radians, then you're going to get yourself into
trouble. Those things are precisely why the radian was chosen as the
plane angle SI unit. You can ignore the reasons why it was chosen, but
still they're there, licking their chops.
You seem to be getting into a self-set semantic trap about the meaning
of the word _natural_. You don't deny that the official SI unit of
plane angle is the radian, right? Well, do you know why it was chosen?
If not, are you asking why, given the reasoning from the various
subjects I've listed? If you don't know, say so, so it can be
explained, rather than trying to get in a pointless semantic quibble
about what you think of the word _natural_ when applied to units.
>> And, at any rate, since this newsgroup is about the
>> metric system, it's hard to avoid acknowledging that the only possible
>> choice for the plane angle is the radian.
>
> Not true. People measured angles long before the radian was invented. If
> you wish to prove that it is the only/best/most natural one _given some
> specific requirements_, then you should list down those requirements and
> outline the proof, the deduction from those requirements.
I already did. SI is a coherent unit system. The radian is the only
measure of plane angle that is coherent. Thus it is the only possible
choice of plane angle unit for SI.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
It's only love that gets you through
-- Sade
Jukka K. Korpela
>> Why would a quantity need a dimensionless unit?
>
> Because it's the only way to measure a plane angle as a coherent unit,
> and SI is a system of coherent units.
This depends on the definition of the quantity. If we defined the angle
differently, then rotation would be the coherent unit. Instead of going in
circles, why don't you explain why it would be impossible (or just a bad
idea) to define the quantity angle in a more natural way, relating to
rotation?
> If you don't know what a
> coherent unit system is, it's in the part you clipped out while
> replying.
I'm not clipping out anything; I simply quote the texts I comment on. And
you haven't told us what a coherent system is; you just kept repeating the
phrase. I know what a coherent system is, but not from your postings.
> I didn't even use the word _natural_ in that sentence, so what
> supposed inconsistency are you complaining about?
In your first message in this thread, you wrote:
"The radian is the only natural unit of angular measure."
>> Dimensional analysis, trigonometry, calculues, and coherence are
>> rather far from the natural meaning of "natural".
>
> Well, they are precisely the things that make you realize that if you
> use anything other than radians, then you're going to get yourself
> into trouble.
Trouble, maybe, in some sense, though you haven't said half a word about the
specific trouble that would arise. But getting into trouble is different
from doing the impossible.
> You seem to be getting into a self-set semantic trap about the meaning
> of the word _natural_.
It was the word you chose to use, and it is natural to interpret it in its
natural meaning. You might just as well say that you didn't mean "natural"
at all but something like "based on considerations of dimensional analysis,
trigonometry, calculus, and coherence" - which are rather unnatural things
to the natural man (including the peoples that used angles for millennia
before the radian was invented).
> You don't deny that the official SI unit of
> plane angle is the radian, right?
You don't deny that 2 + 2 equals 4, right?
Your strawman argument, formulated as a question, is just ridiculous. The
current SI definitions are implied here; we are discussing whether things
could or should be different.
> Well, do you know why it was chosen?
I have some idea of it. I have noticed that you, despite apparently
believing that the reasons were not just good but compelling (i.e., no other
choice was possible), have not even tried to give an explanation. Use one
tenth of your words to actual arguments and examples, and you'll be much
more convincing.
> SI is a coherent unit system. The radian is the only
> measure of plane angle that is coherent. Thus it is the only possible
> choice of plane angle unit for SI.
You failed to specify why the angle is defined the way it is. That's really
the heart of the matter here. Of course, if you define the angle as arc
length divided by radius length, you of course get a dimensionless quantity
such that the full angle is two times pi. Such a definition is by no means
the only possibility. If you define the angle as the amount of rotations
needed to produce it, then you also get a dimensionless unit (at least in
the most natural approach), but a different one, making the full angle equal
one.
--
Yucca, http://www.cs.tut.fi/~jkorpela/
Epiphany
> Jukka K. Korpela wrote:
>> Erik Max Francis wrote:
>>> The radian is the only natural unit of angular measure.
>>
>> Epiphany's point was that revolution is a fundamental and natural
>> concept. For example, the sum of the angles of a triangle is 0.5
>> revolutions. Mesopotamian cultures used a revolution as the basis of
>> angle measurements and divided it into parts using their number
>> system. Defining a revolution as the unit of angular measure would
>> surely be the most natural approach, and naturally if we did that, we
>> would use decimal submultiples and multiples thereof.
Yes, that is exactly my point! I fully *understand* the utility of the
radian while asking why do even fully metric nations use the degree of
arc and not the radian?
I know that one can obtain the radian by considering the circle with
unit radius. This seems basic and inescapable to me. But then I
realized there is another kind of unit circle, a circle with unit
*circumference*. And I further realized that a decimal measure based on
that unit would avoid the non-integer nature of the radian circumference.
So then I wanted to point out, speculate, and suggest that if there is
such global resistance to going with radians, maybe we would like
decimal revolutions better. A revolution may need a 2*pi conversion
factor when calculating some quantities, but the revolution still has a
naturalness since it derives from the unit circumference.
> If you want a coherent unit system, then radians are the only possible
> choice, because they're the only unit of a plane angle that's actually
> dimensionless. The power of SI is not only that it is metric, but that
> it is coherent -- that is, the conversion factor between all fundamental
> and derived units (and supplementary units, like the radian) is 1.
Then why don't there seem to be more references to radians when I
listen, view, or read media from countries beyond the U.S.? Am I just
missing them all? Here in the U.S. I can get rulers with
centimeter-millimeter markings at my local supermarket, but that same
supermarket only sells protractors with degrees. My planetarium program
doesn't seem to have any option for entering locations in radians should
I want to do so. Why do we seem so wedded to degrees when logic
supports the radian? Is the radian actually inconvenient in some way?
\ | /
-Epiphany-
/ | \
Jukka K. Korpela
> Is the radian actually inconvenient in some way? \ | /
Certainly it is. Consider common angles like 90°, 45°, 60°. In addition to
being common as such, they constitute a frame of reference, so that when you
see something like 88.5°, you immediately recognize it as just a little
smaller than the right angle.
In radians, angles like 90°, 45°, 60° are not just irrational, they're
transcendental. :-)
In rotations/revolutions, they would be 0.25 (= 1/4), 0.125 (= 1/8), and
0.1666... (= 1/6).
--
Yucca, http://www.cs.tut.fi/~jkorpela/
Terry Wells
Whilst fully concurring with your observations I do feel that I have to
point out the following:
Quote: "they constitute a frame of reference,"
Quote: "you immediately recognize it as just a little smaller than....."
These are exactly the same arguments that are put forward by the proponents
of the Imperial/English system of measurements!
"I know what six inches looks like"
"I can recognize a piece of 2 x 4 when I see one"
"I know if my pint is short"
Perhaps you have to learn to think in radians just as we like to tell the
Imperial proponents to learn to think in mm and m etc.
--
Terry Wells
Erik Max Francis
> On 21-Feb-11 00:00, Erik Max Francis wrote:
>> Jukka K. Korpela wrote:
>>> Erik Max Francis wrote:
>>>> The radian is the only natural unit of angular measure.
>>>
>>> Epiphany's point was that revolution is a fundamental and natural
>>> concept. For example, the sum of the angles of a triangle is 0.5
>>> revolutions. Mesopotamian cultures used a revolution as the basis of
>>> angle measurements and divided it into parts using their number
>>> system. Defining a revolution as the unit of angular measure would
>>> surely be the most natural approach, and naturally if we did that, we
>>> would use decimal submultiples and multiples thereof.
>
> Yes, that is exactly my point! I fully *understand* the utility of the
> radian while asking why do even fully metric nations use the degree of
> arc and not the radian?
Fully metric nations don't necessarily use solely SI units. How often
do you see the outside temperature reported in kelvins? Or the time it
takes to do something in kiloseconds?
It's pretty simple: It's a combination of convenience and and inertia.
Some SI units, despite being ideally suited to engineering and physics
work, are a little hard to grasp or are counterintuitive. And then
there's specialized units that get used in particular disciplines, where
they're unlikely to switch despite perfectly good SI counterparts, such
as electron-volts in physics or parsecs in astronomy. Astronomers don't
even measure right ascension in degrees, for cryin' out loud, they
measure it in _hours_.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
In principle I am against principles.
-- Tristan Tzara
Erik Max Francis
> Erik Max Francis wrote:
>
>>> Why would a quantity need a dimensionless unit?
>>
>> Because it's the only way to measure a plane angle as a coherent unit,
>> and SI is a system of coherent units.
>
> This depends on the definition of the quantity. If we defined the angle
> differently, then rotation would be the coherent unit. Instead of going
> in circles, why don't you explain why it would be impossible (or just a
> bad idea) to define the quantity angle in a more natural way, relating
> to rotation?
If a plane angle (or some compound unit of angle, such as an angular
speed) appears in an equation outside of a trigonometric function, then
you must use radians or you will get the wrong answer. Period.
Reviewing past discussions, I see I've had exactly this discussion with
you in the past. There are plenty of examples, whether you're talking
about the velocity of a simple harmonic oscillator (an angular velocity
appears because of the chain rule in calculus), or the change in angular
momentum given a certain torque applied through a certain angular
displacement, or even something as simple as the subtended length of arc
by the central angle of a circle. Use radians (or rad/s in the angular
velocity case), or you will get the wrong answer. That's really all
there is to it. That's what coherence _means_. It means the conversion
factor between different units to create compound units is always 1 --
that is, there are never arbitrary conversion factors.
If you use another unit system, you will have to be smart enough to know
that you need to substitute in appropriate conversion factors in the
right place. This defeats the purpose of using a coherent unit system,
which is the entire purpose for SI.
The same arguments all apply as well to solid angles and the steradian.
>> If you don't know what a
>> coherent unit system is, it's in the part you clipped out while
>> replying.
>
> I'm not clipping out anything; I simply quote the texts I comment on.
> And you haven't told us what a coherent system is; you just kept
> repeating the phrase. I know what a coherent system is, but not from
> your postings.
If you know what it is, what are you complaining about me not defining
it for, then?
>> Well, do you know why it was chosen?
>
> I have some idea of it. I have noticed that you, despite apparently
> believing that the reasons were not just good but compelling (i.e., no
> other choice was possible), have not even tried to give an explanation.
> Use one tenth of your words to actual arguments and examples, and you'll
> be much more convincing.
I gave a really, really good argument: That SI is a coherent unit
system, and the radian is the only coherent unit of plane angle. You
claim to know what these terms mean, so maybe you'd have to explain why
you don't find them compelling. Or even blatantly obvious.
>> SI is a coherent unit system. The radian is the only
>> measure of plane angle that is coherent. Thus it is the only possible
>> choice of plane angle unit for SI.
>
> You failed to specify why the angle is defined the way it is.
Because it is the only definition of plane angle that results in a
coherent unit. For the umpteenth time.
If you want a coherent unit for a revolution/cycle, then you already
have one in SI. It's the hertz-second (Hz s). It's not terribly
useful, because it doesn't measure angular anything, but there it is.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
Erik Max Francis
> ... the change in angular
> momentum given a certain torque applied through a certain angular
For those paying attention, simple thinko here: I meant "change in
energy," not "change in angular momentum."
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
This is how the world ends / Not with a bang but with a whimper
-- T.S. Elliot
Julian Bradfield
> There are plenty of examples, whether you're talking
> about the velocity of a simple harmonic oscillator (an angular velocity
> appears because of the chain rule in calculus), or the change in angular
> momentum given a certain torque applied through a certain angular
> displacement, or even something as simple as the subtended length of arc
> by the central angle of a circle. Use radians (or rad/s in the angular
> velocity case), or you will get the wrong answer. That's really all
> there is to it. That's what coherence _means_. It means the conversion
> factor between different units to create compound units is always 1 --
> that is, there are never arbitrary conversion factors.
OK, so I know intuitively what coherent means, but I'm not sure I know
what it really means. Take the arc length subtended by a central
angle. If our unit of angle is degrees, then the equation is
l = (180/π) θ
What makes this constant different from the 1/2 in E = (1/2)mv² ,
which means that we don't define the Joule as "the kinetic energy of a
mass of one kilogramme travelling at one metre per second".
It's kind of obvious, but I don't quite see how to make it precise.
The BIPM says "the equation between quantities has exactly the
same form as the equation between the numerical values of quantities",
but I don't see what "an equation between quantities" is.
Erik Max Francis
> On 2011-02-22, Erik Max Francis <m...@alcyone.com> wrote:
>> There are plenty of examples, whether you're talking
>> about the velocity of a simple harmonic oscillator (an angular velocity
>> appears because of the chain rule in calculus), or the change in angular
>> momentum given a certain torque applied through a certain angular
>> displacement, or even something as simple as the subtended length of arc
>> by the central angle of a circle. Use radians (or rad/s in the angular
>> velocity case), or you will get the wrong answer. That's really all
>> there is to it. That's what coherence _means_. It means the conversion
>> factor between different units to create compound units is always 1 --
>> that is, there are never arbitrary conversion factors.
>
> OK, so I know intuitively what coherent means, but I'm not sure I know
> what it really means. Take the arc length subtended by a central
> angle. If our unit of angle is degrees, then the equation is
> l = (180/π) θ
I think you mean pi/180 (and for a unit circle), but yes that's the
idea. If you use an incoherent unit for the plane angle, like the
degree, then you have to manually insert a conversion factor.
Essentially the whole point of coherent unit systems is to avoid this
tedium.
Plane angles are a special case, because the only coherent choice for
the unit is the radian, since it is effectively dimensionless (it's
defined such that it's a length divided by a length). Any other choice
and you will run into these arbitrary unit conversion factors when
writing equations.
> What makes this constant different from the 1/2 in E = (1/2)mv² ,
> which means that we don't define the Joule as "the kinetic energy of a
> mass of one kilogramme travelling at one metre per second".
> It's kind of obvious, but I don't quite see how to make it precise.
It's pretty easy. It means that when you write the equation, those
numerical constants don't change depending on the system of units you
use, provided it's coherent. So in your example, for a unit circle, the
numerical constant pi/180 will change if you were to use another angular
unit, say a grad/gradian/gon/whatever they're called these days (they
never caught on, anyway). If you use a radian, it is 1.
The difference in the factor of 1/2 for Newtonian kinetic energy is that
it _won't_ change depending on units, again as long as you use a
coherent unit system like SI. The 1/2 factor pops out of the derivation
of the kinetic energy when trying to compute work via dW = F dx. It
doesn't have anything to do with the choice of units.
> The BIPM says "the equation between quantities has exactly the
> same form as the equation between the numerical values of quantities",
> but I don't see what "an equation between quantities" is.
An equation between quantities is just an equation that has united
values in it, rather than a completely dimensionless equation (say, from
mathematics). Both the equations you gave are examples. Exceptions
would be totally dimensionless equations, like 1 + 1 = 2.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
Things are as they are because they were as they were.
-- Thomas Gold
Julian Bradfield
> It's pretty easy. It means that when you write the equation, those
Actually, I think the key point I'm missing is that the radian is now
a derived unit rather than a supplementary unit, and coherent really
means that derived units have 1 in their definitions. When I learnt SI
the radian was still a supplementary unit.
Epiphany
> Epiphany wrote:
>> radian while asking why do even fully metric nations use the degree of
>> arc and not the radian?
>
> Fully metric nations don't necessarily use solely SI units. How often do
> you see the outside temperature reported in kelvins? Or the time it
> takes to do something in kiloseconds?
>
> It's pretty simple: It's a combination of convenience and and inertia.
> Some SI units, despite being ideally suited to engineering and physics
> work, are a little hard to grasp or are counterintuitive. And then
> there's specialized units that get used in particular disciplines, where
> they're unlikely to switch despite perfectly good SI counterparts, such
> as electron-volts in physics or parsecs in astronomy. Astronomers don't
> even measure right ascension in degrees, for cryin' out loud, they
> measure it in _hours_.
If the reason is inertia, I can understand that. Change can take time,
particularly when something has become traditional. If the reason is
*convenience*, that makes me wonder whether we are using the best
definitions for our units. Or are we *always* going to be stuck with
useful units that just aren't related by powers of ten? To me, it is
the ability to simply move the decimal point that pushes SI ahead of
other, earlier systems of measurement. If we aren't going to escape
those useful non-decimal conveniences, then I just despair of having a
sufficiently helpful measuring system. Of course, if the goal is to
regularize as best we can, as much as possible, then it is certainly
useful to adopt SI even with those helpful units outside of the powers
of ten multiples.
I brought up the issue of radians versus degrees in the first place
because it seemed to me like it wasn't just a matter of degrees being
*tradition*. If it were just tradition, some would be plowing ahead
with promoting and using radians for all plane angle requirements with
their writings. Yet this doesn't seem to be the case. So I wondered if
there was a reason for the reluctance beyond tradition. Just like it's
not decimally convenient to define two standard lengths as having a 2*pi
ratio, I wondered whether this factor held up notice of the radian,
however *natural* it is.
Can anyone point me toward those already advocating and immersed in
radians?
\ | /
-Epiphany-
/ | \
Erik Max Francis
> On 22-Feb-11 02:52, Erik Max Francis wrote:
>> Epiphany wrote:
>>> I fully *understand* the utility of the
>>> radian while asking why do even fully metric nations use the degree of
>>> arc and not the radian?
>>
>> Fully metric nations don't necessarily use solely SI units. How often do
>> you see the outside temperature reported in kelvins? Or the time it
>> takes to do something in kiloseconds?
>>
>> It's pretty simple: It's a combination of convenience and and inertia.
>> Some SI units, despite being ideally suited to engineering and physics
>> work, are a little hard to grasp or are counterintuitive. And then
>> there's specialized units that get used in particular disciplines, where
>> they're unlikely to switch despite perfectly good SI counterparts, such
>> as electron-volts in physics or parsecs in astronomy. Astronomers don't
>> even measure right ascension in degrees, for cryin' out loud, they
>> measure it in _hours_.
>
> If the reason is inertia, I can understand that. Change can take time,
> particularly when something has become traditional. If the reason is
> *convenience*, that makes me wonder whether we are using the best
> definitions for our units. Or are we *always* going to be stuck with
> useful units that just aren't related by powers of ten?
How likely is it we're going to move away from using hours, minutes, and
seconds to tell time? The proper SI way to do this is to measure
multiples of seconds; i.e., dekaseconds, hectoseconds, kilseconds, etc.
Back when the metric system was original proposed, it was suggested
that the day be divided up into submultiples; i.e., decidays, centidays,
millidays, etc. What's the likelihood anybody's going to really do
either of those for everyday use, no matter how much advocating you do?
Zero.
> I brought up the issue of radians versus degrees in the first place
> because it seemed to me like it wasn't just a matter of degrees being
> *tradition*.
I don't see why you conclude that the use of degrees is anything _but_
tradition.
> If it were just tradition, some would be plowing ahead
> with promoting and using radians for all plane angle requirements with
> their writings. Yet this doesn't seem to be the case.
There's lots and lots (and _lots_) of technical work that exclusively
uses radians. You _have_ to use it if you want units to come out right
whenever plane angles (or their derived units, like angular velocity)
appear outside of trigonometric functions in any equation.
> Can anyone point me toward those already advocating and immersed in
> radians?
Depends on what you mean. Do you mean people advocating replacing the
degree with the radian in everyday usage? I doubt there are such people.
But do you mean people suggesting that radian is superior for
mathematical and technical uses? Then the answer is every undergraduate
mathematics and physics class in the world.
--
Erik Max Francis && m...@alcyone.com && http://www.alcyone.com/max/
San Jose, CA, USA && 37 18 N 121 57 W && AIM/Y!M/Skype erikmaxfrancis
Education is a state-controlled manufactory of echoes.
-- Norman Douglas
Dr J R Stockton
.com>, Sun, 20 Feb 2011 21:00:04, Erik Max Francis <m...@alcyone.com>
posted:
>
>If you want a coherent unit system, then radians are the only possible
>choice, because they're the only unit of a plane angle that's actually
>dimensionless. The power of SI is not only that it is metric, but that
>it is coherent -- that is, the conversion factor between all
>fundamental and derived units (and supplementary units, like the
>radian) is 1.
>
>Same argument goes with steradians for solid angles.
>
>If you want to think in cycles/revolutions, then you can use the unit
>hertz (defined as s^-1), which is what it's designed for.
>
>You may well be right that it may not be obvious why the radian is the
>best choice of unit for plane angle. But once you understand the
>reasons (dimensional analysis, trigonometry, calculus, etc.), it's a
>bit hard to deny. And, at any rate, since this newsgroup is about the
>metric system, it's hard to avoid acknowledging that the only possible
>choice for the plane angle is the radian.
The metre and the foot have the same dimensions, but different sizes.
Likewise the ton and the tonne, the second and the fortnight.
AIUI, it is generally acknowledged that the integers are dimensionless.
Therefore, as the real numbers, the complex numbers, and pi (and e ~
2.718) can be expressed with simple arithmetic on the integers, they are
dimensionless.
Correspondingly, the radian, the right angle, and the full turn all have
the same dimensions but of different sizes.
The most natural unit of angle is the full turn, but it is
inconveniently large. The Babylonians had the right idea, though there
is an inevitable degree of contention about how many and which prime
numbers the turn should be divisible by in integers.
Almost everyone knows how big a degree is and uses degrees; a minority
know what a radian is, and a minority of those make actual use of them.
But the radian is useful for the sort of work that applies the
differential and integral calculus to sinusoidal functions, such as
calculus teachers, electrical engineers, physicists, and the sort of
people who built the SI.
The radian is the most convenient into of angle for constructing a
general measurement system, but it would be gar from that for
constructing a building and furniture to use when working on, for
example, the measurement system.
--
(c) John Stockton, near London. *@merlyn.demon.co.uk/?.?.Stockton@physics.org
Web <http://www.merlyn.demon.co.uk/> - FAQish topics, acronyms, and links.
Michael Angelo Ravera
Radians count the number of radii that the outside of the angle
subtends. It is incredibly convenient, if you know the length of the
radius and the length of the arc that is subtended. It becomes a
direct calculation.
On the other hand, sometimes revolutions or rotations are easier to
count.
On the other hand, sometimes the fraction of the incline is easier to
calculate. Rise divided by run. You can expess that in percent, if you
like. This measurement is unitless.
In the first two cases, you can say that the angle is so many
revolutions or that it is so many radians. (or even use the old
Babalonian system, if you like)
Christoph Päper
> Epiphany wrote:
>
> How likely is it we're going to move away from using hours, minutes, and
> seconds to tell time? The proper SI way to do this is to measure
> multiples of seconds; i.e., dekaseconds, hectoseconds, kilseconds, etc.
> Back when the metric system was original proposed, it was suggested
> that the day be divided up into submultiples; i.e., decidays, centidays,
> millidays, etc. What's the likelihood anybody's going to really do
> either of those for everyday use, no matter how much advocating you do?
We probably would do that today if the second had been redefined as
one 100000th of a day, i.e. 10 microdays, or, vice versa, if the day
had been defined as 100000 (new) seconds, i.e. 100 ks or 0.1 Ms.
Instead, the relation of 1:86400 (= 24·60·60) was kept, probably in
part due to the seconds pendulum which coincidentally is about 1 m
long, just like Earth gravity coincidentally is little less than 10 m/
s^2. In the early time of development of the system of measurement we
now call “metric”, gravity (in form of said pendulum) was just as
likely to become the defining property as was water (1 kg ^= 1 l,
Celsius scale).
> There's lots and lots (and _lots_) of technical work that exclusively
> uses radians. You _have_ to use it if you want units to come out right
> whenever plane angles (or their derived units, like angular velocity)
> appear outside of trigonometric functions in any equation.
Trigonometric functions, unlike calculators, are agnostic to units,
i.e. sin(45°) = sin(45 deg) = sin(π/4 rad) = sin(50 gon) = sin(50
grad) = sin(2.7 karcmin) = sin(0.162 Marcsec) = sin(1/8 turn) =
sin(0.125 revolution) = sin(1 m, 1 m) = sin(1 slope) = sin(100%) =
sin(1000‰) … It would have made just or, at least, almost as much
sense to define sine and other functions to expect turns or
revolutions when presented with a unitless number, instead we settled
on radians and that will never change.
In CSS, for instance, ‘rad’, alongside ‘deg’, has long been defined as
one possible unit for angles, but since the language does not support
constants like π (and hardly supports vulgar fractions), this unit is
rather useless, because almost no angles in designs will be provided
in numerically nice decimal ‘rad’ values. A premultiplied unit like
‘pirad’ would have made more sense here, but CSS3 will likely
introduce the new unit ‘turn’ instead, where “1turn” is equal to
“360deg”, “400grad” or *“2pirad”. <http://dev.w3.org/csswg/css3-values/
#ltanglegt>
Andreas Prilop
> “400grad” <http://dev.w3.org/csswg/css3-values/#ltanglegt>
The name "grad(e)" is as archaic as "Centigrade".
The correct ISO name is "gon".
The archaic name "grad(e)" is really bad because the degree
is called "Grad" in German.