how does the equation cos(x)= (exp(ix) + exp(-ix))/2
and the sine equivalent fit into this? Clearly exponentials are not restricted to angles ... indicating that x (and by implication angles) have no dimensions. Marc Schiltz's previously cited Taylor expansion demonstrates this even better: sin(x) = x/1! - x^3/3! + x^5/5! ..... etc to infinity If you assume for a moment that x does have a dimension, lets call it [X], then the equation is dimensionally unbalanced [?] = [X]^1 - [X]^3 + [X]^5 ...... etc and is therefore invalid. It only makes sense if x, and its sine, are dimensionless Pete On 23 Nov 2009, at 16:42, marc.schi...@epfl.ch wrote: > Dale Tronrud wrote: >> While it is true that angles are defined by ratios which result in >> their values being independent of the units those lengths were measured, >> common sense says that a number is an insufficient description of an >> angle. If I tell you I measured an angle and its value is "1.5" you >> cannot perform any useful calculation with that knowledge. > > > I disagree: you can, for instance, put this number x = 1.5 (without units) > into the series expansion for sin X : > > x - x^3/(3!) + x^5/(5!) - x^7/(7!) + ... > > and compute the value of sin(1.5) to any desired degree of accuracy > (four terms will be enough to get an accuracy of 0.0001). Note that > the x in the series expansion is just a real number (no dimension, no > unit). > > > > Yes it's >> true that the confusion does not arise from a mix up of feet and meters. >> I would have concluded my angle was 1.5 in either case. >> >> The confusion arises because there are differing conventions for >> describing that "unitless" angle. I could be describing my angle as >> 1.5 radians, 1.5 degrees, or 1.5 cycles (or 1.5 of the mysterious >> "grad" on my calculator). > > > > These are just symbols for dimensionless factors : > > 1 rad = 1 > 1 degree = pi/180 > 1 grad = pi/200 > > Thus : > > 1.5 rad = 1.5 > 1.5 degree = 0.0268 > 1.5 grad = 0.0236 > > and all these numbers (which have no units !!!) can be put into the > series expansions for trigonometric functions. > > In my opinion, it is actually best not to use the symbol rad. As we can > see from this discussion, it mostly creates confusion. > > > > For me to communicate my result to you >> I would need to also tell you the convention I'm using, and you will >> have to perform a conversion to transform my value to your favorite >> convention. If it looks like a unit, and it quacks like a unit, I >> think I'm free to call it a unit. >> >> I think you will agree that if we fail to pass the convention >> along with it value our space probe will crash on Mars just as hard >> as if we had confused feet and meters. >> >> The result of a Sin or Cos calculation can be treated as "unitless" >> only because there is 100% agreement on how these results should be >> represented. Everyone agrees that the Sin of a right angle is 1. > > > This is not a simple matter of agreement (or convention), it is > contained in the very definition of the sine function. > > >> If I went off the deep end I could declare that the Sin of a right >> angle is 12 and I could construct an entirely self-consistent description >> of physics using that convention. > > > I challenge you to draw a right triangle on paper where the length of > one of the sides measures 12 times the length of the hypotenuse. > > Of course, you can say that your "crazy Tronrud Sin" is defined > differently, but then we are really speaking about something else. You > can define whatever crazy quantity you want. But the need for a function > which describes the ratio of the length of a side of a right triangle > to the length of its hypotenuse will inevitably arise at some point in > physics and mathematics. And the "crazy Tronrud Sin" will not do this > job. So the proper sine and cosine functions will eventually have to > be invented. > > > > > In that case I would have to be >> very careful to keep track of when I was working with traditional >> Sin's and when with "crazy Tronrud Sin's". When switching between >> conventions I would have to careful to use the conversion factor of >> 12 "crazy Tronrud Sin's"/"traditional Sin" and I'd do best if I >> put a mark next to each value indicating which convention was used >> for that particular value. Sounds like units to me. >> >> Of course no one would create "crazy Tronrud Sin's" because the >> pain created by the confusion of multiple conventions is not compensated >> by any gain. When it comes to angles, however, that ship has sailed. >> While mathematicians have very good reasons for preferring the radian >> convention you are never going to convince a physicist to change from >> Angstrom/cycle to Angstrom/radian when measuring wavelengths. You >> will also fail to convince a crystallographer to measure fractional >> coordinates in radians. We are going to have to live in a world that >> has some angular quantities reported in radians and others in cycles. >> That means we will have to keep track of which is being used and apply >> the factor of 2 Pi radian/cycle or 1/(2 Pi) cycle/radian when switching >> between. >> >> I agree with Ian that the 8 Pi^2 factor in the conversion of >> <u_x^2> to B looks suspiciously like 2 (2 Pi)^2 and it is likely >> a conversion of cycle^2 to radian^2. I can even imagine that the >> derivation of effect of distortions of the lattice points that lead >> to these parameters would start with a description of these distortions >> in cycles, but I also have enough experience with this sort of problem >> to know that you can only be certain of these "units" after going >> back to the root definition and tracking the algebra forward. >> >> In my opinion the Mad Scientist is right. B and <u_x^2> represent >> the same quantity reported with different units (or conventions if >> you will) and the answer will be something like B in A^2 radian^2 >> and <u_x^2> in A^2 cycle^2. It would be much clearer it someone >> figured out exactly what those units are and we started properly >> stating the units of each. I'm sorry that I don't have the time >> myself for this project. >> >> Dale Tronrud >> >> P.S. As for your distinction between the "convenience" units used to >> measure angles and the "absolutely required" units of length and mass: >> all units are part of the coordinate systems that we humans impose on >> the universe. Length and mass are no more fundamental than angles. >> Feet and meters are units chosen for our convenience and one converts >> between them using an arbitrary scaling constant. In fact the whole >> distinction between length and mass is simply a matter of convenience. >> In the classic text on general relativity "Gravitation" by Miser, >> Thorne and Wheeler they have a table in the back of "Some Useful >> Numbers in Conventional and Geometrized Units" where it lists the >> mass of the Sun as 147600 cm and and the distance between the Earth >> and Sun as 499 sec. Those people in general relativity are great >> at manipulating coordinate systems! >> >>> -----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 >> > > -- > Marc SCHILTZ http://lcr.epfl.ch
