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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 >>> >>> >>>