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: [log in to unmask]
>> [mailto:[log in to unmask]] On Behalf Of James Holton
>> Sent: 23 November 2009 16:35
>> To: [log in to unmask]
>> 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:[log in to unmask]] On
>>>
>> Behalf Of Ian
>>
>>> Tickle
>>> Sent: Sunday, November 22, 2009 10:57 AM
>>> To: [log in to unmask]
>>> 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
>>>
>>>
>>>
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