I agree that the official SI documentation has priority, but as I read it there is no discrepancy between it and Wikipedia. The official SI position (and that of NIST and IUPAC) is that the radian is a dimensionless unit (i.e., a unit of dimension 1).
Quoting at length from the SI brochure: "2.2.3 Units for dimensionless quantities, also called quantities of dimension one Certain quantities are defined as the ratio of two quantities of the same kind, and are thus dimensionless, or have a dimension that may be expressed by the number one. The coherent SI unit of all such dimensionless quantities, or quantities of dimension one, is the number one, since the unit must be the ratio of two identical SI units. The values of all such quantities are simply expressed as numbers, and the unit one is not explicitly shown. Examples of such quantities are refractive index, relative permeability, and friction factor. There are also some quantities that are defined as a more complex product of simpler quantities in such a way that the product is dimensionless. Examples include the 'characteristic numbers' like the Reynolds number Re = ρvl/η, where ρ is mass density, η is dynamic viscosity, v is speed, and l is length. For all these cases the unit may be considered as the number one, which is a dimensionless derived unit. Another class of dimensionless quantities are numbers that represent a count, such as a number of molecules, degeneracy (number of energy levels), and partition function in statistical thermodynamics (number of thermally accessible states). All of these counting quantities are also described as being dimensionless, or of dimension one, and are taken to have the SI unit one, although the unit of counting quantities cannot be described as a derived unit expressed in terms of the base units of the SI. For such quantities, the unit one may instead be regarded as a further base unit. In a few cases, however, a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian and the steradian. The radian and steradian have been identified by the CGPM as special names for the coherent derived unit one, to be used to express values of plane angle and solid angle, respectively, and are therefore included in Table 3." "The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities." pp 119-120, "The International System of Units (SI)". International Bureau of Weights and Measures (BIPM). http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf also see http://physics.nist.gov/cuu/Units/units.html http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf On Nov 23, 2009, at 4:03 PM, marc.schi...@epfl.ch wrote: > I would believe that the official SI documentation has precedence over > Wikipedia. In the SI brochure it is made quite clear that Radian is just > another symbol for the number one and that it may or may no be used, as is > convenient. > > Therefore, stating alpha = 15 (without anything else) is perfectly valid for > an angle. > > Marc > > > > Quoting Douglas Theobald <dtheob...@brandeis.edu>: > >> 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. 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 >>>>> >>>> >>>> >>> >>> >>> 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 >> >
