| Topic: |
Science > Physics |
| User: |
"Josh" |
| Date: |
06 Aug 2007 03:51:02 PM |
| Object: |
The Basics, Pi and arc-lengths |
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), or pi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to the circle's radius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and a circle's circumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi. Pi appears to come before arc-length before radian.
PROBLEM: pi = c/2r, where c is circumference. But of course,
circumference is in terms of pi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, and pi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
.
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| User: "Stephen Montgomery-Smith" |
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| Title: Re: The Basics, Pi and arc-lengths |
06 Aug 2007 04:12:54 PM |
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Josh wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), or pi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to the circle's radius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and a circle's circumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi. Pi appears to come before arc-length before radian.
PROBLEM: pi = c/2r, where c is circumference. But of course,
circumference is in terms of pi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, and pi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
Apparently the answer to this problem was not so obvious to Archimedes,
so you are in good company. Someone (actually someone rather
distinguished in a lecture) explained that Archimedes defined arclength
of a curve by drawing inscribed polygons, and circumscribed polygons,
and then taking limits as the size of each edge goes to zero. But
Archimedes apparently realized (and it seems that you can see this in
his original writings) that there was no obvious a priori reason why the
limits of the circumferences two polygons should become the same value.
This is a more modern treatment: http://en.wikipedia.org/wiki/Arc_length
Really, you have asked a good question. But try sci.math rather than
sci.physics.
Stephen
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| User: "Stephen Montgomery-Smith" |
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| Title: Re: The Basics, Pi and arc-lengths |
06 Aug 2007 04:24:38 PM |
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Stephen Montgomery-Smith wrote:
Josh wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), or pi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to the circle's radius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and a circle's circumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi. Pi appears to come before arc-length before radian.
PROBLEM: pi = c/2r, where c is circumference. But of course,
circumference is in terms of pi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, and pi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
Apparently the answer to this problem was not so obvious to Archimedes,
so you are in good company. Someone (actually someone rather
distinguished in a lecture) explained that Archimedes defined arclength
of a curve by drawing inscribed polygons, and circumscribed polygons,
and then taking limits as the size of each edge goes to zero. But
Archimedes apparently realized (and it seems that you can see this in
his original writings) that there was no obvious a priori reason why the
limits of the circumferences two polygons should become the same value.
This is a more modern treatment: http://en.wikipedia.org/wiki/Arc_length
See here as well:
http://en.wikipedia.org/wiki/Archimedes#Mathematics
Incidentally, I once heard a lecture about how the ancient Chinese used
similar methods to calculate pi, but had much more accurate answers than
Archimedes obtained.
Also, one definition of pi is to say that it is the first positive root
of the equation sin(x)=0, where sin(x) is defined by its power series.
Personally I think this approach is rather contrived - on the other hand
it is way easier to rigorously establish than the "arc-length" approach.
Stephen
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| User: "Josh" |
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| Title: Re: The Basics, Pi and arc-lengths |
06 Aug 2007 06:12:43 PM |
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On Aug 6, 2:24 pm, Stephen Montgomery-Smith
<step...@math.missouri.edu> wrote:
Stephen Montgomery-Smith wrote:
Josh wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), orpi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to thecircle'sradius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and acircle'scircumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi.Piappears to come before arc-length before radian.
PROBLEM:pi= c/2r, where c is circumference. But of course,
circumference is in terms ofpi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, andpi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
Apparently the answer to this problem was not so obvious to Archimedes,
so you are in good company. Someone (actually someone rather
distinguished in a lecture) explained that Archimedes defined arclength
of a curve by drawing inscribed polygons, and circumscribed polygons,
and then taking limits as the size of each edge goes to zero. But
Archimedes apparently realized (and it seems that you can see this in
his original writings) that there was no obvious a priori reason why the
limits of the circumferences two polygons should become the same value.
This is a more modern treatment:http://en.wikipedia.org/wiki/Arc_length
See here as well:
http://en.wikipedia.org/wiki/Archimedes#Mathematics
Incidentally, I once heard a lecture about how the ancient Chinese used
similar methods to calculatepi, but had much more accurate answers than
Archimedes obtained.
Also, one definition ofpiis to say that it is the first positive root
of the equation sin(x)=0, where sin(x) is defined by its power series.
Personally I think this approach is rather contrived - on the other hand
it is way easier to rigorously establish than the "arc-length" approach.
Stephen- Hide quoted text -
- Show quoted text -
From what I am finding, it seems the correct logical approach to the
chicken-or-egg issue is that pi must come first. If pi comes first,
all of the rest of circles, triangles, trig, etc. follow.
We are always taught that C=pi*d, but in reality, pi=C/d is more
correct. Then, by similarity of circles, you can proceed to calculate
circumferences using pi. Pi seems to be some intrinsic universal
property, a ratio between diameters and circumferences.
This step leads, unfortunately, to the big question: why is the ratio
of a circle's circumference to its diameter equal to this crazy number
3.14159... that we call pi? I think I'm probably better off leaving
that question to the ghosts of Einstein and Feynman, or perhaps to the
deity that came up with this irrational (pun intended) system.
.
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| User: "Tom Potter" |
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| Title: Re: The Basics, Pi and arc-lengths |
07 Aug 2007 08:06:14 AM |
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"Josh" <jjreicher@gmail.com> wrote in message
news:1186441963.803750.191190@e9g2000prf.googlegroups.com...
On Aug 6, 2:24 pm, Stephen Montgomery-Smith
<step...@math.missouri.edu> wrote:
Stephen Montgomery-Smith wrote:
Josh wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), orpi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to thecircle'sradius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and acircle'scircumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi.Piappears to come before arc-length before radian.
PROBLEM:pi= c/2r, where c is circumference. But of course,
circumference is in terms ofpi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, andpi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
Apparently the answer to this problem was not so obvious to Archimedes,
so you are in good company. Someone (actually someone rather
distinguished in a lecture) explained that Archimedes defined arclength
of a curve by drawing inscribed polygons, and circumscribed polygons,
and then taking limits as the size of each edge goes to zero. But
Archimedes apparently realized (and it seems that you can see this in
his original writings) that there was no obvious a priori reason why
the
limits of the circumferences two polygons should become the same value.
This is a more modern treatment:http://en.wikipedia.org/wiki/Arc_length
See here as well:
http://en.wikipedia.org/wiki/Archimedes#Mathematics
Incidentally, I once heard a lecture about how the ancient Chinese used
similar methods to calculatepi, but had much more accurate answers than
Archimedes obtained.
Also, one definition ofpiis to say that it is the first positive root
of the equation sin(x)=0, where sin(x) is defined by its power series.
Personally I think this approach is rather contrived - on the other hand
it is way easier to rigorously establish than the "arc-length" approach.
Stephen- Hide quoted text -
- Show quoted text -
From what I am finding, it seems the correct logical approach to the
chicken-or-egg issue is that pi must come first. If pi comes first,
all of the rest of circles, triangles, trig, etc. follow.
We are always taught that C=pi*d, but in reality, pi=C/d is more
correct. Then, by similarity of circles, you can proceed to calculate
circumferences using pi. Pi seems to be some intrinsic universal
property, a ratio between diameters and circumferences.
This step leads, unfortunately, to the big question: why is the ratio
of a circle's circumference to its diameter equal to this crazy number
3.14159... that we call pi? I think I'm probably better off leaving
that question to the ghosts of Einstein and Feynman, or perhaps to the
deity that came up with this irrational (pun intended) system.
"pi" is basically a constant used to express discrete reality
(events) in terms of continuous reality (space).
Auto-correlation is used to determine discrete reality
and cross-correlation is used to determine continuous reality.
Auto-correlation involves a set of events associated with
a single point or body,
cross-correlation involves a set of events associated with
two points or bodies.
Discrete reality is expressed in "N" counts of
cycles or events (Which are one point entities)
in a sample,
continuous reality is expressed in "N" counts of
cycles*x (Where x is a two point entity)
in a sample.
Note that x can be degrees, grads, mils, radians,
hertz seconds, etc. x can be any real number.
In other words,
if x is an integer you are expressing discrete reality.
if x is a real number you are expressing continuous reality.
The most fundamental two points
that can be equated to discrete reality (Cycles)
is the diameter of a cycle (Circle)
as Josh indicated.
2 pi is a Dedekind Cut
that separates discrete reality from continuous reality.
As can be seen,
2 pi is the math equivalent of physic's "c"
the so-called speed of light.
In other words, 2 pi is a constant used to
express discrete one point realities,
which are basically times,
in terms of continuous two point realities,
which are basically spaces,
just as "c" is used to express time intervals
in terms of spaces.
--
Tom Potter
*** Time Magazine Person of the Year 2006 ***
*** May 2007 Anti-Bigot Award ***
http://home.earthlink.net/~tdp
http://tdp1001.googlepages.com/home
http://no-turtles.com
http://www.frappr.com/tompotter
http://spaces.msn.com/tdp1001
http://www.flickr.com/photos/tom-potter
http://tom-potter.blogspot.com
--
Posted via a free Usenet account from http://www.teranews.com
.
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| User: "Sam Wormley" |
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| Title: Re: The Basics, Pi and arc-lengths |
11 Aug 2007 12:47:00 PM |
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Tom Potter wrote:
As can be seen, 2 pi is the math equivalent of physic's "c"
the so-called speed of light.
Potter, 2 pi does crop up often in physics expressions. And
c is not "so-called speed of light", but is in fact defined.
http://scienceworld.wolfram.com/physics/SpeedofLight.html
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| User: "Tom Potter" |
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| Title: Re: The Basics, Pi and arc-lengths |
13 Aug 2007 10:11:35 PM |
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"Sam Wormley" <swormley1@mchsi.com> wrote in message
news:oCmvi.41048$Xa3.34202@attbi_s22...
Tom Potter wrote:
As can be seen, 2 pi is the math equivalent of physic's "c"
the so-called speed of light.
Potter, 2 pi does crop up often in physics expressions. And
c is not "so-called speed of light", but is in fact defined.
http://scienceworld.wolfram.com/physics/SpeedofLight.html
Sam makes a good point when he points out,
that the so-called "speed of light" is defined
for a particular set of conditions.
As I have pointed out,
Special Relativity is basically
the addition of tangents, and
that "c", the so-called "speed of light"
is just a constant that equates
the time interval between points A and B,
to the number of King's body parts between points A and B.
In other words, "c' is not a velocity,
it is a constant used to set the units of space,
in terms of the more fundamental units of time,
and there is nothing magic about "c",
as it just a tangent that indicates that
there were no intervening interactions between
a point A and a point B.
tangent(x) = velocity / c
Thanks for bringing up this point Sam.
Your pal,
--
Tom Potter
*** Time Magazine Person of the Year 2006 ***
*** May 2007 Anti-Bigot Award ***
http://home.earthlink.net/~tdp
http://tdp1001.googlepages.com/home
http://no-turtles.com
http://www.frappr.com/tompotter
http://spaces.msn.com/tdp1001
http://www.flickr.com/photos/tom-potter
http://tom-potter.blogspot.com
--
Posted via a free Usenet account from http://www.teranews.com
.
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| User: "Tom Potter" |
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| Title: Re: The Basics, Pi and arc-lengths |
14 Aug 2007 09:49:53 AM |
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"Sam Wormley" <swormley1@mchsi.com> wrote in message
news:oCmvi.41048$Xa3.34202@attbi_s22...
Tom Potter wrote:
As can be seen, 2 pi is the math equivalent of physic's "c"
the so-called speed of light.
Potter, 2 pi does crop up often in physics expressions. And
c is not "so-called speed of light", but is in fact defined.
http://scienceworld.wolfram.com/physics/SpeedofLight.html
Sam makes a good point when he points out,
that the so-called "speed of light" is defined
for a particular set of conditions.
As I have pointed out,
Special Relativity is basically
the addition of tangents, and
that "c", the so-called "speed of light"
is just a constant that equates
the time interval between points A and B,
to the number of King's body parts between points A and B.
In other words, "c' is not a velocity,
it is a constant used to set the units of space,
in terms of the more fundamental units of time.
Thanks for bringing up this point Sam.
Your pal,
--
Tom Potter
*** Time Magazine Person of the Year 2006 ***
*** May 2007 Anti-Bigot Award ***
http://home.earthlink.net/~tdp
http://tdp1001.googlepages.com/home
http://no-turtles.com
http://www.frappr.com/tompotter
http://spaces.msn.com/tdp1001
http://www.flickr.com/photos/tom-potter
http://tom-potter.blogspot.com
--
Posted via a free Usenet account from http://www.teranews.com
.
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| User: "Stephen Montgomery-Smith" |
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| Title: Re: The Basics, Pi and arc-lengths |
06 Aug 2007 07:39:37 PM |
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Josh wrote:
On Aug 6, 2:24 pm, Stephen Montgomery-Smith
<step...@math.missouri.edu> wrote:
Stephen Montgomery-Smith wrote:
Josh wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), orpi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to thecircle'sradius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and acircle'scircumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi.Piappears to come before arc-length before radian.
PROBLEM:pi= c/2r, where c is circumference. But of course,
circumference is in terms ofpi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, andpi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
Apparently the answer to this problem was not so obvious to Archimedes,
so you are in good company. Someone (actually someone rather
distinguished in a lecture) explained that Archimedes defined arclength
of a curve by drawing inscribed polygons, and circumscribed polygons,
and then taking limits as the size of each edge goes to zero. But
Archimedes apparently realized (and it seems that you can see this in
his original writings) that there was no obvious a priori reason why the
limits of the circumferences two polygons should become the same value.
This is a more modern treatment:http://en.wikipedia.org/wiki/Arc_length
See here as well:
http://en.wikipedia.org/wiki/Archimedes#Mathematics
Incidentally, I once heard a lecture about how the ancient Chinese used
similar methods to calculatepi, but had much more accurate answers than
Archimedes obtained.
Also, one definition ofpiis to say that it is the first positive root
of the equation sin(x)=0, where sin(x) is defined by its power series.
Personally I think this approach is rather contrived - on the other hand
it is way easier to rigorously establish than the "arc-length" approach.
Stephen- Hide quoted text -
- Show quoted text -
From what I am finding, it seems the correct logical approach to the
chicken-or-egg issue is that pi must come first. If pi comes first,
all of the rest of circles, triangles, trig, etc. follow.
We are always taught that C=pi*d, but in reality, pi=C/d is more
correct. Then, by similarity of circles, you can proceed to calculate
circumferences using pi. Pi seems to be some intrinsic universal
property, a ratio between diameters and circumferences.
Exactly. Pi is a universal constant. Other similar numbers are e, the
golden ratio, and the roots of the Bessel functions. That these numbers
are what they are is somehow what makes the universe what it is. But
these numbers are also what they are because of the laws of logic. So
the only way to change these numbers is not only to change the universe,
but the very rules of logic themselves.
This step leads, unfortunately, to the big question: why is the ratio
of a circle's circumference to its diameter equal to this crazy number
3.14159... that we call pi? I think I'm probably better off leaving
that question to the ghosts of Einstein and Feynman, or perhaps to the
deity that came up with this irrational (pun intended) system.
However one can use mathematical logic to derive formulae that this
mysterious number must satisfy. Simple ones are like
pi=4(1-1/3+1/5-1/7+1/9 .... +(-1)^n/(2n+1)+...)
but if you want to calculate it to millions of decimal places, there are
some rather amazing (and for me incomprehensible) formulae, due I think
to Ramanujan.
Stephen
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| User: "PD" |
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| Title: Re: The Basics, Pi and arc-lengths |
07 Aug 2007 10:41:23 AM |
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On Aug 6, 3:51 pm, Josh <jjreic...@gmail.com> wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), or pi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to the circle's radius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and a circle's circumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi. Pi appears to come before arc-length before radian.
PROBLEM: pi = c/2r, where c is circumference. But of course,
circumference is in terms of pi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, and pi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
I think it's a pretty fair guess that people knew how to measure, or
even cared about measuring, length before angle.
I think it's a pretty fair guess that measuring the diameter of a
circle came first and was done with a straight stick, either an arm or
a rod.
It's also a fair guess that the circumference was first measured by
rolling the circle on the ground and then measuring the distance on
the ground for one revolution, with an arm or a rod.
I don't know when somebody figured out that one was a little over
three times the other.
I also don't know when somebody tried it again with a spoked wheel, so
that the rolled wheel only turned 1/4 of a revolution, or 1/6th, or
1/12th. That's probably the start of the notion of "angle"
measurement, though.
In this scenario, the sequence would have then been
diameter
circumference
pi
angle
radian
PD
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| User: "John C. Polasek" |
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| Title: Re: The Basics, Pi and arc-lengths |
07 Aug 2007 02:20:06 PM |
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On Mon, 06 Aug 2007 13:51:02 -0700, Josh <jjreicher@gmail.com> wrote:
This may be a chicken-or-the-egg question, but maybe not. I'd like to
see if I'm the only person who wonders about this. Seems the more I
study QM, the less I understand arithmetic.
Which comes first: the radian (theta), the arc-length (s), or pi?
Where r is the radius...
s=(r)*(theta)
One radian is defined as the angle 2 radii subtend while bordering an
arc-length equal to the circle's radius. So it would appear that the
radian does not define arc-length. Arc-length appears to come before
radian.
The arc-length is really only a distance measurement (tough with a
straight-edge though) and a circle's circumference is defined as
2*pi*r. Circumference, however, is really just the full arc-length. So
again...it appears that arc-length is, in a sense, defined in terms of
pi. Pi appears to come before arc-length before radian.
PROBLEM: pi = c/2r, where c is circumference. But of course,
circumference is in terms of pi. And so we are stuck. Furthermore, if
you reverse the order, you can derive arc-length from radian, and pi
from arc-length. So either way, you define the variables in terms of
each other.
Ideas?
When calculus began to deal with angles, it was inescapable to define
an angular measure like radians to make the following come true:
ds = r*dtheta
which is clearly an improvement over
ds = r*d(degrees)*pi/180
With r constant and using theta = 1 radian
s = r*theta = r*1
Just now, too late to take back the cogitation, I remember to ask the
question "What is the OP going to do with the answer?" and, "Is this
the answer to it?"
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