Mixing
Why radian is dimensionless?
$begingroup$
Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r
Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?
unit-of-measure
edited May 21 '14 at 11:09
evil999man
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
$endgroup$
add a comment |
$begingroup$
Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r
Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?
unit-of-measure
edited May 21 '14 at 11:09
evil999man
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
$endgroup$
3
$begingroup$
Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
1
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
3
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46
add a comment |
$begingroup$
Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r
Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?
unit-of-measure
edited May 21 '14 at 11:09
evil999man
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
$endgroup$
Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r
Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?
unit-of-measure
unit-of-measure
edited May 21 '14 at 11:09
evil999man
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
edited May 21 '14 at 11:09
evil999man
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
edited May 21 '14 at 11:09
evil999man
4,93311333
edited May 21 '14 at 11:09
evil999man
4,93311333
edited May 21 '14 at 11:09
evil999man
4,93311333
4,93311333
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
asked May 21 '14 at 11:05
Viacheslav KondratiukViacheslav Kondratiuk
2621213
2621213
3
$begingroup$
Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
1
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
3
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46
add a comment |
3
$begingroup$
Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
1
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
3
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46
3
3
$begingroup$
Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
$begingroup$
Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
1
1
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
3
3
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
$endgroup$
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
add a comment |
$begingroup$
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = Rtheta$. This is obviously no longer homogeneous in dimensions if $theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $theta$, and that $s=R$ for $theta=1,mathrm{rad}$, we conclude that the true form of that relation must be $$s = Rfrac{theta}{1,mathrm{rad}}.$$ (When asuuming that $mathrm{1,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$exp(x) = sum_{k=0}^inftyfrac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $sin(theta/mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $omega$ would be $mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
answered Dec 28 '15 at 10:07
polwelpolwel
26124
$endgroup$
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
add a comment |
$begingroup$
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=alephtheta,$$ and that for the area of a sector
$$a=frac{alephtheta r^2}2.$$
The universal constant $aleph$ is in per-angle-unit, and $aleph=1 text{ rad}^{-1}=0.0174533cdotstext{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=Asin(alephomega t)$$ where $omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=dot e=Aalephomega cos(alephomega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$Pi=3.141593cdotstext{ rad}=180text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$alephPi=pi.$$
Hence the famous Euler formula,
$$e^{ialephPi}=-1.$$
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
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A quantity is dimensionless if it has same magnitudes in different units.
$$1 text{rad}=dfrac{1m}{1 m}=dfrac{1 nm}{1 nm}=dfrac{1text{light year}}{1text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
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add a comment |
$begingroup$
As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
answered May 21 '14 at 11:57
PocketsPockets
505210
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add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
$endgroup$
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
add a comment |
$begingroup$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
$endgroup$
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
add a comment |
$begingroup$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
$endgroup$
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out. Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
answered May 21 '14 at 11:14
Graham KempGraham Kemp
87.2k43579
87.2k43579
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
add a comment |
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
$begingroup$
and what about degrees? are they also dimensionless?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 11:22
1
1
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
$begingroup$
$1$ degree is $frac 1{360}$ of a full rotation. It is a dimensionless number multiplied by an arbitrarily chosen constant. (360 would seem to have been chosen because it has 24 divisors, making it readily divisible.) The preferred form of a dimensionless unit is 1 of something divided by 1 of something else.
$endgroup$
– Graham Kemp
May 21 '14 at 11:31
add a comment |
$begingroup$
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = Rtheta$. This is obviously no longer homogeneous in dimensions if $theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $theta$, and that $s=R$ for $theta=1,mathrm{rad}$, we conclude that the true form of that relation must be $$s = Rfrac{theta}{1,mathrm{rad}}.$$ (When asuuming that $mathrm{1,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$exp(x) = sum_{k=0}^inftyfrac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $sin(theta/mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $omega$ would be $mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
answered Dec 28 '15 at 10:07
polwelpolwel
26124
$endgroup$
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
add a comment |
$begingroup$
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = Rtheta$. This is obviously no longer homogeneous in dimensions if $theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $theta$, and that $s=R$ for $theta=1,mathrm{rad}$, we conclude that the true form of that relation must be $$s = Rfrac{theta}{1,mathrm{rad}}.$$ (When asuuming that $mathrm{1,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$exp(x) = sum_{k=0}^inftyfrac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $sin(theta/mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $omega$ would be $mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
answered Dec 28 '15 at 10:07
polwelpolwel
26124
$endgroup$
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
add a comment |
$begingroup$
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = Rtheta$. This is obviously no longer homogeneous in dimensions if $theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $theta$, and that $s=R$ for $theta=1,mathrm{rad}$, we conclude that the true form of that relation must be $$s = Rfrac{theta}{1,mathrm{rad}}.$$ (When asuuming that $mathrm{1,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$exp(x) = sum_{k=0}^inftyfrac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $sin(theta/mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $omega$ would be $mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
answered Dec 28 '15 at 10:07
polwelpolwel
26124
$endgroup$
The fact that an angle is dimensionless is mostly a matter of convention. Indeed you can associate a dimension with an angle, and still remain consistent. Quite a few well known formulas need to be changed in that case though.
For example, you quote the arc length as $s = Rtheta$. This is obviously no longer homogeneous in dimensions if $theta$ is not dimensionless. However, if we remember that actually $s$ must be proportional to the angle $theta$, and that $s=R$ for $theta=1,mathrm{rad}$, we conclude that the true form of that relation must be $$s = Rfrac{theta}{1,mathrm{rad}}.$$ (When asuuming that $mathrm{1,rad = 1}$ is dimensionless, it yields the original expression.)
Another thing is that we can no longer feed an angle directly to analytic functions like sin, cos, exp, etc. Indeed, these are conveniently defined via a power series, e.g. $$exp(x) = sum_{k=0}^inftyfrac{x^k}{k!}.$$
If $x$ has a dimension, this expression makes not whole lot of sense. To be correct, we would have to write $sin(theta/mathrm{rad})$ when talking about the sine of an angle.
As of my understanding, these are some of the reasons why one decided that an angle should best be left dimensionless. Especially since angles are relevant in mathematics, where -unlike in physics- one typically does not care about dimensionality of quantities.
What are the advantages of assigning a dimension to angles? Mostly, the additional dimensionality carries a lot more information. Consider frequency $f$ and angular velocity $omega$. In the SI system both have the same dimension, namely 1/time. If angle had its own dimension, the unit of $omega$ would be $mathrm{rad/s}$! We could differentiate these quantities based on their units! Depending on how you introduce the new dimension, torque and work might also no longer share the same unit.
If you are interested, here is a very readable article that explains a possible way of introducing an additional dimension for angles.
answered Dec 28 '15 at 10:07
polwelpolwel
26124
answered Dec 28 '15 at 10:07
polwelpolwel
26124
answered Dec 28 '15 at 10:07
polwelpolwel
26124
answered Dec 28 '15 at 10:07
polwelpolwel
26124
26124
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
add a comment |
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
1
1
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
$begingroup$
Distinguishing between frequency and angular frequency becomes important when we take Fourier transforms (suppose, for instance, that we have to fuse together material from sources that use different conventions for their Fourier transform). Then $f$ is in cycles per second, and $omega$ is in radians per second. If we made both cycles and radians "dimensionless", i.e. let ourselves just drop those parts of the units, then we'll be wrong by factors of $2pi$ every now and then when we add up quantities that appear to be in the same unit after the dimensionless cleanse.
$endgroup$
– Evgeni Sergeev
Jul 26 '17 at 14:57
add a comment |
$begingroup$
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=alephtheta,$$ and that for the area of a sector
$$a=frac{alephtheta r^2}2.$$
The universal constant $aleph$ is in per-angle-unit, and $aleph=1 text{ rad}^{-1}=0.0174533cdotstext{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=Asin(alephomega t)$$ where $omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=dot e=Aalephomega cos(alephomega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$Pi=3.141593cdotstext{ rad}=180text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$alephPi=pi.$$
Hence the famous Euler formula,
$$e^{ialephPi}=-1.$$
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=alephtheta,$$ and that for the area of a sector
$$a=frac{alephtheta r^2}2.$$
The universal constant $aleph$ is in per-angle-unit, and $aleph=1 text{ rad}^{-1}=0.0174533cdotstext{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=Asin(alephomega t)$$ where $omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=dot e=Aalephomega cos(alephomega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$Pi=3.141593cdotstext{ rad}=180text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$alephPi=pi.$$
Hence the famous Euler formula,
$$e^{ialephPi}=-1.$$
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=alephtheta,$$ and that for the area of a sector
$$a=frac{alephtheta r^2}2.$$
The universal constant $aleph$ is in per-angle-unit, and $aleph=1 text{ rad}^{-1}=0.0174533cdotstext{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=Asin(alephomega t)$$ where $omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=dot e=Aalephomega cos(alephomega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$Pi=3.141593cdotstext{ rad}=180text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$alephPi=pi.$$
Hence the famous Euler formula,
$$e^{ialephPi}=-1.$$
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
$endgroup$
Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=alephtheta,$$ and that for the area of a sector
$$a=frac{alephtheta r^2}2.$$
The universal constant $aleph$ is in per-angle-unit, and $aleph=1 text{ rad}^{-1}=0.0174533cdotstext{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=Asin(alephomega t)$$ where $omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=dot e=Aalephomega cos(alephomega t)$$ is in $ms^{-1}$.
Another universal constant worth to know:
$$Pi=3.141593cdotstext{ rad}=180text{ deg}.$$
denotes the aperture angle of a half-circle.
It fufills
$$alephPi=pi.$$
Hence the famous Euler formula,
$$e^{ialephPi}=-1.$$
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
edited Dec 5 '18 at 9:32
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
answered Dec 28 '15 at 10:29
Yves DaoustYves Daoust
131k676229
131k676229
add a comment |
add a comment |
$begingroup$
A quantity is dimensionless if it has same magnitudes in different units.
$$1 text{rad}=dfrac{1m}{1 m}=dfrac{1 nm}{1 nm}=dfrac{1text{light year}}{1text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
$endgroup$
add a comment |
$begingroup$
A quantity is dimensionless if it has same magnitudes in different units.
$$1 text{rad}=dfrac{1m}{1 m}=dfrac{1 nm}{1 nm}=dfrac{1text{light year}}{1text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
$endgroup$
add a comment |
$begingroup$
A quantity is dimensionless if it has same magnitudes in different units.
$$1 text{rad}=dfrac{1m}{1 m}=dfrac{1 nm}{1 nm}=dfrac{1text{light year}}{1text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
$endgroup$
A quantity is dimensionless if it has same magnitudes in different units.
$$1 text{rad}=dfrac{1m}{1 m}=dfrac{1 nm}{1 nm}=dfrac{1text{light year}}{1text{light year}}=1 $$ As you noted The same units get cancelled.
However length is not so.
$$1m=100cm $$
$1ne 100$ for obvious reasons.
Degrees are just defined to be dimensionless. They don't change with size when you zoom in or out. However, radian definition of angle provides better insight.
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
answered May 21 '14 at 11:14
evil999manevil999man
4,93311333
4,93311333
add a comment |
add a comment |
$begingroup$
As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
answered May 21 '14 at 11:57
PocketsPockets
505210
$endgroup$
add a comment |
$begingroup$
As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
answered May 21 '14 at 11:57
PocketsPockets
505210
$endgroup$
add a comment |
$begingroup$
As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
answered May 21 '14 at 11:57
PocketsPockets
505210
$endgroup$
As you point out, the radian measurement of an angle is the ratio of the length of an arc the angle intercepts to the length of the radius of said arc. As both arc length and radius are measured with units of length, these units of length cancel when determining how many radians an angle is. This is why radians are dimensionless - there is no "unit" that describes what a radian measures, because it is a ratio of two different quantities with the same unit of measurement.
A measurement in degrees, however, is simply a different ratio; rather, it is the ratio of the arc to 1/360th of the circumference of the circle corresponding to the arc.
answered May 21 '14 at 11:57
PocketsPockets
505210
edited May 21 '14 at 13:58
answered May 21 '14 at 11:57
PocketsPockets
505210
answered May 21 '14 at 11:57
PocketsPockets
505210
answered May 21 '14 at 11:57
PocketsPockets
505210
505210
add a comment |
add a comment |
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Good question. Some people believe that an advantage of radians over degrees is that the former is dimensionless. Actually, both are. Also, the "why meter is not dimensionless?" question is relevant.
$endgroup$
– leonbloy
May 21 '14 at 11:47
1
$begingroup$
A radian is dimensionless because it describes a certain arc of a circle, regardless of whether that arc is the size of your thumb or the size of the known universe.
$endgroup$
– MJD
May 21 '14 at 13:07
$begingroup$
@MJD It makes sense. But I can tell the same for degree, isn't it?
$endgroup$
– Viacheslav Kondratiuk
May 21 '14 at 13:43
3
$begingroup$
Degrees are also dimensionless.
$endgroup$
– MJD
May 21 '14 at 13:46