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. The Company accepts no liability or responsibility for any onward transmission or use of emails and attachments having left the Astex Therapeutics domain. Unless expressly stated, opinions in this message are those of the individual sender and not of Astex Therapeutics Ltd. The recipient should check this email and any attachments for the presence of computer viruses. Astex Therapeutics Ltd accepts no liability for damage caused by any virus transmitted by this email. E-mail is susceptible to data corruption, interception, unauthorized amendment, and tampering, Astex Therapeutics Ltd only send and receive e-mails on the basis that the Company is not liable for any such alteration or any consequences thereof. Astex Therapeutics Ltd., Registered in England at 436 Cambridge Science Park, Cambridge CB4 0QA under number 3751674
