So... how do you measure or report a solid angle without invoking the
steradian? sterdegrees?
Ian Tickle wrote:
James, I think you misunderstood, no-one is suggesting that we can do
without the degree (minute, second, grad, ...), since these conversion
units have considerable practical value. Only the radian (and
steradian) are technically redundant, and as Marc suggested we would
probably be better off without them!
Cheers
-- Ian
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of James Holton
Sent: 23 November 2009 16:35
To: [email protected]
Subject: Re: [ccp4bb] units of the B factor
Just because something is dimensionless does not mean it is
unit-less.
The radian and the degree are very good examples of this.
Remember, the
word "unit" means "one", and it is the quantity of something that we
give the value "1.0". Things can only be measured relative
to something
else, and so without defining for the relevant "unit", be it
a long-hand
description or a convenient abbreviation, a number by itself is not
useful. It may have "meaning" in the metaphysical sense, but its not
going to help me solve my structure.
A world without units is all well and good for theoreticians
who never
have to measure anything, but for those of us who do need to
know if the
angle is 1 degree or 1 radian, units are absolutely required.
-James Holton
MAD Scientist
Artem Evdokimov wrote:
The angle value and the associated basic trigonometric
functions (sin, cos,
tan) are derived from a ratio of two lengths* and therefore are
dimensionless.
It's trivial but important to mention that there is no
absolute requirement
of units of any kind whatsoever with respect to angles or
to the three basic
trigonometric functions. All the commonly used units come
from (arbitrary)
scaling constants that in turn are derived purely from convenience -
specific calculations are conveniently carried out using
specific units (be
they radians, points, seconds, grads, brads, or papaya
seeds) however the
units themselves are there only for our convenience (unlike
the absolutely
required units of mass, length, time etc.).
Artem
* angle - the ratio of the arc length to radius of the arc
necessary to
bring the two rays forming the angle together; trig
functions - the ratio of
the appropriate sides of a right triangle
-----Original Message-----
From: CCP4 bulletin board [mailto:[email protected]] On
Behalf Of Ian
Tickle
Sent: Sunday, November 22, 2009 10:57 AM
To: [email protected]
Subject: Re: [ccp4bb] units of the B factor
Back to the original problem: what are the units of B and
<u_x^2>? I haven't been able to work that out. The first
wack is to say the B occurs in the term
Exp( -B (Sin(theta)/lambda)^2)
and we've learned that the unit of Sin(theta)/lamda is 1/Angstrom
and the argument of Exp, like Sin, must be radian. This means
that the units of B must be A^2 radian. Since B = 8 Pi^2 <u_x^2>
the units of 8 Pi^2 <u_x^2> must also be A^2 radian, but the
units of <u_x^2> are determined by the units of 8 Pi^2. I
can't figure out the units of that without understanding the
defining equation, which is in the OPDXr somewhere. I suspect
there are additional, hidden, units in that definition. The
basic definition would start with the deviation of scattering
points from the Miller planes and those deviations are probably
defined in cycle or radian and later converted to Angstrom so
there are conversion factors present from the beginning.
I'm sure that if the MS sits down with the OPDXr and follows
all these units through he will uncover the units of B, 8 Pi^2,
and <u_x^2> and the mystery will be solved. If he doesn't do
it, I'll have to sit down with the book myself, and that will
make my head hurt.
Hi Dale
A nice entertaining read for a Sunday afternoon, but I think you can
only get so far with this argument and then it breaks down,
as evidenced
by the fact that eventually you got stuck! I think the
problem arises
in your assertion that the argument of 'exp' must be in units of
radians. IMO it can also be in units of radians^2 (or
radians^n where n
is any unitless number, integer or real, including zero for that
matter!) - and this seems to be precisely what happens
here. Having a
function whose argument can apparently have any one of an infinite
number of units is somewhat of an embarrassment! - of
course that must
mean that the argument actually has no units. So in
essence I'm saying
that quantities in radians have to be treated as unitless,
contrary to
your earlier assertions.
So the 'units' (accepting for the moment that the radian is a valid
unit) of B are actually A^2 radian^2, and so the 'units' of
8pi^2 (it
comes from 2(2pi)^2) are radian^2 as expected. However
since I think
I've demonstrated that the radian is not a valid unit, then
the units of
B are indeed A^2!
Cheers
-- Ian
