Argument from authority, from the omniscient Wikipedia: http://en.wikipedia.org/wiki/Radian
"Although the radian is a unit of measure, it is a dimensionless quantity." "The radian is a unit of plane angle, equal to 180/pi (or 360/(2 pi)) degrees, or about 57.2958 degrees,.... It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level." "… the radian is now considered an SI derived unit." On Nov 23, 2009, at 1:31 PM, 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: owner-ccp...@jiscmail.ac.uk >> [mailto:owner-ccp...@jiscmail.ac.uk] On Behalf Of James Holton >> Sent: 23 November 2009 16:35 >> To: CCP4BB@jiscmail.ac.uk >> 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:ccp...@jiscmail.ac.uk] On >> Behalf Of Ian >>> Tickle >>> Sent: Sunday, November 22, 2009 10:57 AM >>> To: CCP4BB@JISCMAIL.AC.UK >>> 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 >>> >>> >>> Disclaimer >>> This communication is confidential and may contain >> privileged information >>> intended solely for the named addressee(s). It may not be >> used or disclosed >>> except for the purpose for which it has been sent. If you >> are not the >>> intended recipient you must not review, use, disclose, >> copy, distribute or >>> take any action in reliance upon it. If you have received >> this communication >>> in error, please notify Astex Therapeutics Ltd by emailing >>> i.tic...@astex-therapeutics.com and destroy all copies of >> the message and >>> any attached documents. >>> Astex Therapeutics Ltd monitors, controls and protects all >> its messaging >>> traffic in compliance with its corporate email policy. 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