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STANDARD INTERNATIONAL Généralités 1: Partie unités et Grandeurs General Part 1: units and Quantities

ISO 80000-1:2009(E) 80000-1 Reference number © First edition 2009-11-15 ISO 2009 ISO Provided by : www.spic.ir 80000-1:2009(E) ISO ii in Switzerland Published requester. of the the country bodyin ISO's member writing be in at ISO either from permission address without andthe including photocopying microfilm, or mechanical, electronic no be or part or reproduced any of utilized this publication in may otherwise specified, byan form Unless reserved. rights All © 2009 ISO the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. given at the address Secretariat Central please the to is it inform found, aproblem event relating that unlikely the were for Every printing. care optimized parameters hasbeen taken to ensure that the is forfile use suitable by ISO bod member eation PDF-cr the relative the file; to to this General in PDF create can Info the be found used file of the software products Details of a Adobe Incorporated. is Systems Adobe trademark area. this in noliability accepts Se The Central ISO policy. licensing Adobe's not infringing of responsibility the accept therein parties file, downloading this editin the performing onthe computer andinstalled to arelicensed areembedded which unless the typefaces not beedited shall or viewed beprinted b this file policy, may with licensing Adobe's In accordance embedded typefaces. PDFcontain file This may Web www.iso.org www.iso.org Web E-mail [email protected] Fax +412274909 47 Tel. +41 2274901 11 56 • postale Case office copyright ISO

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Provided by : www.spic.ir ISO 80000 consists of the following parts, under the general title title general the under parts, thefollowing of consists 80000 ISO added. been has C Annex normative anew ⎯ normative; become B have and A Annexes ⎯ added; been have 99:2007 Guide ISO/IEC with accordance in definitions ⎯ follow; then andunits first come quantities that emphasize to changed been has thestructure ⎯ following: the are standard theprevious from changes technical major The ISO ISO and the Amendments 31-0:1992/Amd.2:2005 ISO 31-0:1992/Amd.1:1998, 1000:1992/Amd.1:1998. incorporates It also ISO 1000:1992. and ISO 31-0:1992 andreplaces cancels ISO of 80000-1 edition first This units and 25,Quantities IEC/TC ISO ISO/TC Committee by Technical inwas prepared co-operation units and with 80000-1 12, Quantities rights. patent such all or any identifying for responsible held be not shall ISO rights. patent of ISO of subject the be elements may the of 80000-1 some that possibility the to drawn is Attention a vote. casting bodies themember % of 75 least at by approval requires Standard International an as Publication voting. for bodies tothemember circulated are committees technical the by adopted Standards International Draft Standards. International toprepare is committees technical of task main The 2. Part Directives, ISO/IEC in the given the rules with accordance in drafted are Standards International standardization. electrotechnical of matters all on (IEC) Commission Electrotechnical International with the closely collaborates ISO work. inthe part take also ISO, with liaison in non-governmental, and governmental organizations, International committee. that on berepresented to theright has established been has committee atechnical which for inasubject bodyinterested member Each committees. technical ISO through out carried normally is Standards International preparing of work The bodies). member (ISO bodies standards national of federation aworldwide is Standardization) for Organization International (the ISO Foreword 80000-1:2009(E) ISO iv ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ Part 12: Solid state physics physics state Solid 12: Part numbers Characteristic 11: Part Part 10:Atomic and nuclear physics physics molecular and chemistry Physical 9: Part Part 8: Acoustics Part 7: Light Thermodynamics 5: Part PartMechanics 4: time and Space 3: Part technology and sciences natural in the used be to and symbols signs Mathematical 2: Part Part 1: General . Quantities and units and Quantities © ISO 2009 – All rights reserved 2009–© ISO rights All : Provided by : www.spic.ir IEC 80000 consists of the following parts, under the general title title general the under parts, following the of consists IEC 80000 © ISO 2009 – All rights reserved 2009–© ISO rights All ⎯ ⎯ ⎯ Part 14: Telebiometrics related to human physiology technology and science Information 13: Part Electromagnetism 6: Part

Quantities and units and Quantities : ISO 80000-1:2009(E) 80000-1:2009(E) ISO v

Provided by : www.spic.ir ISO/IEC ISO/IEC Guide SI Brochure inthe and 99:2007 in appear ISO does ISQ in used However, evolved. has series harmonized thepresent which 31, from 31. inISO system this for used thephrase was which based”, theSIis which on quantities of the“system for notation shorthand a is ISQ rests. technology and science modern of all which on equations and quantities of system expanding and evolving continually and infinite theessentially to assign to notation aconvenient simply is that ISQ however, berealized, Itshould used. would be term new the that announced been already ithad published; vacuum) is defined to be equal to 1, i.e. of dimension one; in the CGS-EMU system, the magnetic constant the systemsof quantities by which they aredefined the SI, is named the the named is the SI, of theunits of basis the as used thequantities them among the relations including quantities, of system The quantities. thecorresponding among therelations from developed are and role important an play theunits among therelations units of system inany because important, are sciences, thephysical of equations the simply are which quantities, the among relations the Nonetheless, developed. was quantities of asystem of concept the before long used and were developed interest of quantities of values express to used theunits of Most quantities. their for units defining from them prevent not will this listed, thatare examples familiar tomore quantities their relate can they that provided However, Standard. International another in or Standard International this in belisted not may using in interested are they quantities that the find may fields specialized in particular working readers some that inevitable It is presented. is them among relations andthe quantities used commonly themore of aselection instead, Standard; International inthis relations and quantities these all list to notpossible itis Thus, developed. are technology and science of fields new as evolving continually are and number in infinite essentially are them among relations the and quantities The Forelectric and magnetic units in the CGS-ESU, CGS-EMU NOTE technologists. and scientists all to familiar are and today textbooks scientific of majority the in presented are They sciences. physical the throughout use for accepted universally almost those are here used quantities the among relations and quantities The mesures et poids des générale Conférence d’unités theFrench: (from SI the Units, of System International the for basis the is one that the is Standard International in this given presentation The convention. of matter a only is touse treatment ways. Which butdifferent, consistent, in many betreated can units of systems and quantities of Systems 0.1 Quantities Introduction 80000-1:2009(E) ISO vi 2) MKS 1) CGS system, MKS similar complications. In mechanics, Newton’s law ofmotion in its general form is written they are not of dimension one. The Gaussian system is related to the CGS-ESU and CGS-EMU systems and there are (permeability ofvacuum) is defined to beequal to1,i.e. ofdi consistency with the new new the with consistency ), adopted by the General Conference on Weights and Measures, the CGPM (from the French: theFrench: (from CGPM the andMeasures, Weights on Conference General the by adopted ), = = metre-kilogram-second. metre-kilogram-second. centimetre-gram-second; ESU 2) , c

= 1/ g n , where International System of Quantities System International Quantities and units and Quantities g n isthe standard acceleration offree fall; in the ISQ,

= electrostatic units; EMU ). . In the CGS-ESU system, the electric constant series that was under preparation at the time they were were they time at the preparation under was that series

[8] , Edition 8:2006. In both cases, this was to ensure toensure was this cases, both In 8:2006. , Edition mension one,in contrast tothose quantities inthewhere ISQ , denoted “ISQ”, in all languages. This name was not not was name This languages. all in “ISQ”, , denoted = electromagnetic units. 1) andGaussian systems, there is a difference in c

c Système international ⋅ m a . Inthe old technical ε 0 (the permittivity of µ 0

Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009–© ISO rights All l,sub-letter). respectively; part, inthe number nn,running number; part (pp, pp-nn.l written are units of numbers item The sub-number). s, respectively; part, the in number running nn, number; part (pp, pp-nn.s written are quantities of numbers item The respectively. pages, right-hand and left-hand the on given also are units and quantities additional Some pages. right-hand onthe given are units) other some (and theSI of units andthe pages, left-hand the on given are the ISQ, of a subset are which them, among relations and quantities the Standard, International this 14 of to 3 In parts Arrangement of the tables 0.4 Brochure. SI the of part as presented are which methods, such recommends CIPM the and units, the realizing for methods experimental review to helpful often is it Nonetheless, SI unit. any realize to used be could physics of laws the with consistent method Any advances. science as developed be may methods new and aunit, of thevalue of realization practical the ways for different many be may There units. base the realizing from in principle follows units derived of values the Realizing importance. particular of is units base the of values the Realizing science. in quantity any of thevalue of measurements in making step the essential is This theunit. of value the with theunit as kind thesame of quantity some of thevalue thatcompare measurements tomake theunit of definition the touse is aunit of value realize the To values of the Realizing units 0.3 Brochure SI the in presented are realization, practical their for advice etmesures des poids international Comité theFrench: (from CIPM the andMeasures, for Weights Committee theInternational of committees theadvisory of responsibility the are and SI the of heart the at are realization, practical their and units, base these of definitions The respectively. and candela, mole, kelvin, ampere, second, kilogram, themetre, are units base corresponding The intensity. luminous and substance, of amount temperature, thermodynamic current, electric time, mass, length, are quantities base The units. base seven and quantities base seven are SI, there the in and Standard International Inthis the of quantities. base terms in quantities thederived defining relations the to corresponding units base the of powers of products as units derived defining then and quantities base corresponding of set small a for units base of a set defining first by developed is units of A system 0.2 Units are simply chosen by convention and no attempt is made to define them otherwise then operationally. operationally. then otherwise them define to made is attempt no and convention by chosen simply are mesures theFrench: (from BIPM the Measures, and of Weights Bureau International ). Note that in contrast to the base units, each of which has a specific definition, the base quantities quantities base the definition, aspecific has which of each units, tothebase in contrast Notethat ).

). The current definitions of the base units, and and units, base the of definitions current The ).

[8] , published by and obtainable from the from obtainable and by , published Bureau international des poids et poids des international Bureau ISO 80000-1:2009(E) 80000-1:2009(E) ISO vii

Provided by : www.spic.ir Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009 –© ISO rights All andareference anumber of means by expressed be can that amagnitude has property the where substance, or body, aphenomenon, of property quantity 3.1 The content in this clause is essentially the same as in ISO/IEC Guide 99:2007. Some notes and examples are modified. NOTE apply. definitions and terms following the document, this of purposes the For Termsanddefinitions 3 terms (VIM) 99:2007, Guide ISO/IEC applies. amendments) any (including document thereferenced of edition thelatest references, undated For applies. cited edition only the references, dated For document. this of application the for indispensable are documents referenced following The 2 Normative references 80000-1. ISO of scope the outside are properties nominal and quantities Ordinal Standard. International this of parts other to introduction an as and technology, and science of fields various the within use general for intended are ISO in 80000-1 down laid principles The SI. Units, of System International and the ISQ, ofQuantities, System the International especially systems, unit andcoherent symbols, unit and quantity units, quantities, of systems quantities, ISO concerning definitions and information general gives 80000-1 1 Scope General Part 1: units and Quantities INTERNATIONAL STANDARD INTERNATIONAL International vocabulary of metrology — Basic and general concepts and associated associated and concepts general and Basic — metrology of vocabulary International

ISO 80000-1:2009(E) ISO 1

Provided by : www.spic.ir OE6 Adapted from ISO/IEC Guide 99:2007, definit NOTE 6 ‘biological quantity’, or ‘base quantity’ and ‘derived quantity’. The concept ’quantity’ may be generically divided into, e.g. ‘physical quantity’, ‘chemical quantity’, and NOTE 5 Aquantity asdefined here is a scalar. However, avect is also considered to be a quantity. NOTE 4 for quantities are written in italics. A given symbol can indicate different quantities. 2 Symbols for quantities are given in the ISO 80000 and IEC 80000 series, NOTE 3 Areference can be a measurement unit, a measurement procedure, areference material,such. For magnitudeor acombination of a of quantity, see 3.19. NOTE 2 individual quantities in the right hand column. following table. The left hand side of the table shows specific The generic concept ‘quantity’ canbe divided into several levels of specific concepts, asshown inthe NOTE 1 80000-1:2009(E) ISO nature” (inEnglish, “quantities of the same kind”). corresponding ‘kinds of quantity’. In French, the term InEnglish, the terms for quantities in theleft half ofthe table in 3.1, Note 1,are often used for the NOTE 4 Quantities ofthe same kind withinquantities a ofgiven the samesystem ofdimension quantities are not necessarilyhave the ofthesame quantity same kind. dimension. However, NOTE 3 The division of the concept ‘quantity’ into several kinds is to some extent arbitrary. NOTE 2 Kindof quantity is often shortened to “kind”, e.g. in quantities ofthe same kind. NOTE 1 quantities comparable to mutually common aspect kind ofquantity 3.2 relative permeability,and massfraction. although they have the same dimension. Similarly for heat capacity and entropy, as well as for number of entities, The quantities moment of force and energy are,by convention, not regarded as being of the sameEXAMPLE kind, the same kind, namely, of the kind ofquantity called energy. The quantities heat, kinetic energy, and potential energy are generally considered to be quantities of EXAMPLE 2 the same kind, namely, of the kind ofquantity called length. The quantities diameter, circumference, and wavelength are generally considered to be quantities of EXAMPLE 1 HRC(150 kg) HRC(150 kg) Rockwell C hardness (150 kg load), number concentration of entity B, entity B,c amount-of-substance concentration of electric resistance, electric charge, energy, length, l E B

Q wavelength, radius, r kinetic energy, heat, Q R λ λ T C B

Rockwell C hardness of steel sample radius of circle A, electric resistance of resistor kinetic energy ofparticle (Erys; B C(Erys; number concentration of erythrocytes in blood sample i amount-of-substance concentration ofethanol in wine sample wavelength of the sodium D radiation, electric charge of the proton, heat of vaporization of sample , c i (C “nature” is only used in expressions such as “grandeurs de même

2 ion 1.1, in which there is an additional note. H 5 OH) i ) concepts under ‘quantity’. These are generic concepts for the or a tensor, the components ofwhich are quantities, r A

λ i (150 kg) T (Na; D) i

R i

i , . The symbols Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009 –© ISO rights All Inthe ISQ, the quantity dimension offorce is denoted by dim EXAMPLE 1 factor numerical any omitting quantities, base the to corresponding factors of powers of product a as quantities of asystem of quantities base the on aquantity of thedependence of expression dimension dimension ofa quantity quantity dimension 3.7 Adapted from ISO/IEC Guide 99:2007, def NOTE 3 The International System of Units (SI) (see item 3.16) is based on the ISQ. NOTE 2 This system of quantities is published in the14. ISO 80000 and IEC 80000 series NOTE 1 intensity luminous and substance, of amount temperature, thermodynamic current, electric time, mass, length, quantities: base seven the on based quantities of system ISQ Quantities of System International 3.6 Adapted from ISO/IEC Guide 99:2007, definition NOTE Inasystem ofquantities having thebase quantities length andmass, mass density isa derived quantity defined as the quotient of mass and volume (length to the power three). EXAMPLE thatsystem of quantities the base of terms in defined quantities, of a system in quantity, derived quantity 3.5 Adapted from ISO/IEC Guide 99:2007, definition NOTE 4 ‘Number of entities’ can beregarded as a base quantity in any system of quantities. NOTE 3 Base quantities are referred to as being mutually independentproduct of powers since of a the base other quantitybase quantities. cannot be expressed as a NOTE 2 The subset mentioned in the definition is termed the “set of base quantities”. NOTE 1 subset that within quantities theother of terms in be expressed can subset the in no quantity where quantities, of system agiven of subset chosen conventionally in a quantity quantity base 3.4 Adapted from ISO/IEC Guide 99:2007, def NOTE 2 empirical relations only. Ordinal quantities (see 3.26), such as Rockwell C hardness, and nominal properties (see 3.30), such asof light, colour are usually not considered to be part of a system of NOTE 1 quantities those relating equations non-contradictory of with aset together quantities of set system ofquantities 3.3 Adapted from ISO/IEC Guide 99:2007, definition 1.2, been added. NOTE 5 XML The setof base quantities in the International System of Quantities (ISQ) is given in 3.6. EXAMPLE

inition 1.6, in which Note 1 is different. inition 1.3, in which Note 1 is different. quantities because they arerelated to other quantities through 1.5, in which the example is slightly different. which1.4, in the definition isslightlydifferent. in which “kind” appears as an admitted term. Note 1 has F

= LMT − 2 . Quantities andunits ISO 80000-1:2009(E) 80000-1:2009(E) ISO , Parts to 3 , Parts 3

Provided by : www.spic.ir OE6 Adapted from ISO/IEC Guide 99:2007, definition 1.7, in which “dimension of a quantity” and“dimension” are given as admitted terms. NOTE 6 exponents are zero, see 3.8. exponents, are positive, negative, or zero. Factors with exponent zero and the exponent 1 are usually omitted. allWhen quantity powers of the dimensions of the base quantities according to the definition of the derived quantity. The dimension of a The conventional symbolic representationroman (upright) ofthe type. The conventionaldimension symbolic ofa base representationquantity is ofthe dimensiona single of upper aderived case letter quantity isthein product of NOTE 2 Apower ofafactor is the factor raised to an exponent. Each factor is the dimension of a base quantity. NOTE 1 4 The period, EXAMPLE 3 Inthe same system of quantities, dim EXAMPLE 2 80000-1:2009(E) ISO Some quantities of dimension onearederived defined unit isthenumbersymbol one,1. as the ratios oftwo quantities of the same kind. The coherent NOTE 3 The measurement units andmore information valuesnumber.a than of quantities of dimension one arenumbers, but such quantities convey NOTE 2 Clausesymbol1, see 5.This dimensionnumber isnota dimension one” reflects the convention in which the symbolic representation of the dimension for such quantities is the The term “dimensionless quantity” is commonlyfact used that and isall exponentskept here for historicalare zero in the reasons. symbolic It stems representation from the of the dimension for such quantities. The term “quantity of NOTE 1 dimension quantity its in quantities base the to corresponding thefactors of exponents the all which for quantity dimensionless quantity quantity one ofdimension 3.8 Thus, thedimension of aquantity component B, and ML OE5 Symbols representing the dimensions ofthe base quantities in the ISQ are: NOTE 5 Inagiven system ofquantities, NOTE 4 Inderiving the dimension of a quantity, no account is taken of its scalar, vector, ortensor character. NOTE 3 quantities having the same quantity dimension are not necessarily ofthe same kind. ⎯ quantities of different quantity dimensions are always ofdifferent kinds, and ⎯ quantities of the same kind have thesame quantity dimension, ⎯ Hence T =π Q 2 is denoted by dim

are zero zero are i T L ) dim ( g l or Cg − =⋅ 3 isalso the quantity dimension of mass density, − Q T 1/2 TCgl , of a particle pendulum of length . = luminous intensity intensity luminous of amount substance temperature thermodynamic current electric time mass length . Q () isdenoted by dim Base quantity where , but theneutral element for multiplication of dimensions. Q

ρ = Cg B L

() = α ML M l at a place with the local acceleration of free fall = β in which Note 5 and Examples 2 and 3 are different and − Symbol for dimension T 2 J N Θ I T M L 3 γ isthe quantity dimension of mass concentration of π g

I δ

ρ ε . N ζ J η where theexponents, named dimensional © ISO 2009 – All rights reserved 2009 –© ISO rights All g is is Provided by : www.spic.ir NOTE 5 Adapted from ISO/IEC GuideISO/IECfrom Adapted 99:2007, definition 1.9,in and in which “measurement unit” and “unit” are given as admitted terms. NOTE 5 Fora given mass”. quantity,“unit of the short term “unit” is often combined with the quantity name, such as “mass unit” or NOTE 4 © ISO 2009 – All rights reserved 2009 –© ISO rights All 1.11] 99:2007, Guide [ISO/IEC The metreper second, symbol m/s,with and thecentimetre theSI.The knot, per equal second, to symbolone nautical cm/s, are mile derivedper units of ThespeedSI. inthekilometre perhour, symbol km/h,a measurement is of unit speed outsideSI but theaccepted foruse EXAMPLE quantity aderived unitfor measurement derived unit 3.11 Adapted from ISO/IEC Guide 99:2007, definition 1.10,last sentence inNote which3 hasbeenadded. in the example in Note 2 is slightly different. The NOTE 4 For number ofentities,Compare Note thenumber 3 inone, 3.4.symbol 1, NOTE 3 Abase unit may also serve for aderived quantity ofthe same quantity dimension. NOTE 2 Ineach coherent system of units, there is only one base unit for each base quantity. NOTE 1 quantity base a for convention by adopted is unitthat measurement unit base 3.10 to 10 Measurement units of quantities of dimension one are givennumbers. special Innames, some cases, e.g. radian, steradian,these measurement and decibel, units or are NOTE 3 example,usedjoulethenevera unitof energy, (J)is but as as is called hertz (Hz) when usedforfrequencies andbecquerel when (Bq) used foractivities ofradionuclides. As another to be used with quantities of specific kind only. For example, the measurement unit ‘second to the power minus one’ (1/s) not considered to be quantities of the same kind. However, in some cases special measurement unit names are restricted botha measurementname andsymbolheatcapacity of unitof Measurement units ofquantitiessymbol ofthe sameeven qua when the quantities are not ofthe same kind. Forexample, joule per kelvin and J/K arerespectively the NOTE 2 Measurement units are designated by conventionally assigned names and symbols. NOTE 1 anumber as one first the to quantity thesecond of ratio the express to be compared can kind thesame of quantity other any which with convention, by andadopted defined quantity, scalar real unit unit measurement measurement of unit 3.9 “dimensionless quantity” is given as an admitted Adapted term. fromISO/IEC Guide 99:2007, definition 1. NOTE 5 Numbers of entities are quantities of dimension one. NOTE 4 quantum system. number. Plane angle,EXAMPLE solid angle, refractive index, relative permeability, mass fraction, friction factor,Mach coherent derived unit in theSI. The derived quantity rainfall,EXAMPLE when defined as areic volume (volume per area), has the metre as a length. IntheSI, the metre is the base unit of length. Inthe CGS systems, theEXAMPLE centimetre is the base unit of − EXAMPLE 3 and microgram per kilogram equal to 10

Number of turns in a coil, number of molecules in a given sample, degeneracy of the energy levels of a

− 9 . hour, is a measurement unit of speed outside the SI. are expressed by quotients such as millimole per mole equal ntity dimension may bedesignated by thesame name and a unit of moment of force, i.e. the newton metre (N · m). and ameasurement unit ofentropy, which aregenerally can be regarded asa base unit inany system ofunits. 8, in which Notes 1 and 3are different and in which which the definition and Note 2 are slightly different

ISO 80000-1:2009(E) 80000-1:2009(E) ISO 5

Provided by : www.spic.ir 6 Adapted from ISO/IEC Guide 99:2007, definition 1.15, in which Example 1 is different and in which “off-system unit” is given as an admitted term. NOTE Day, hour, minute are off-system measurement units of time with respect to the SI. EXAMPLE 2 SI. unit derived acoherent is quantity derived each unitfor measurement the which in quantities, of system agiven on based units, of system coherent system ofunits 3.14 1.13] 99:2007, Guide [ISO/IEC quantities of system given a for rules, given with accordance in defined andsubmultiples, multiples with their together units, andderived units base of set system ofunits 3.13 1.12] 99:2007, Guide [ISO/IEC Coherencecan bedeterminedonlywith respect units. toaparticular system ofquantitiesandgiven a setofbase NOTE 2 Apower ofabase unit is the base unit raised to an exponent. NOTE 1 80000-1:2009(E) ISO The electronvolt (≈ EXAMPLE 1 unit measurement off-system unit off-systemunit measurement 3.15 Adapted from ISO/IEC Guide 99:2007, definition 1.14, inwhich Note 2is different. NOTE 3 Foracoherent system of units, numerical value equations have thesame form,including numericalthe corresponding factors, as quantity equations. See examples of numerical value equations in 3.25. NOTE 2 Asystem of units can be coherent only with respect toa system of quantities and the adopted base units. Setof coherent SI units and relations between them. NOTE 1 EXAMPLE one than factor proportionality withnoother base units for that, unit derived coherent derived unit 3.12 OE4 The coherent derived unit for every derived quantity ofdimension one inone, agivensymbol system 1.The nameof and units issymbol the number ofthe measurement unit one are generally not indicated. NOTE 4 Aderived unit can be coherent with respect to one system of quantities but not to another. NOTE 3 of velocity when velocity is defined by the quantity equation Ifthe metre, the second, and the mole are base units, the metre EXAMPLEper second is the coherent derived unit coherent derived unit in theSI. The centimetre per second is the coherent derived unit of speed in aEXAMPLE CGS system of units but is not a coherent derivedunitssystem insuch ofquantities. a quantity equation derived unit of amount-of-substance concentration when amount-of-substance concentration is defined by the

that does not belong to a given a to notbelong does that

a given system of quantities and for a chosen set of base units, is a product of powers of powers of a product is units, base of set achosen for and quantities of system a given c

n / V . The kilometre perhour andthe knot, given as examples ofderived units in3.11, arenot 1,602 18 × 10 –19 J) is an off-system measurement unit of energy with respect to the

system of units of system v

= d r

/ d t and the mole per cubic metre is the coherent © ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir on Weights and Measures (CGPM) (CGPM) Measures and on Weights Conference General the by adopted use, their for rules with together symbols, and names their and prefixes of aseries including andsymbols, names their Quantities, of System International the on based units, of system SI International S 3.16 © ISO 2009 – All rights reserved 2009 –© ISO rights All Adapted from ISO/IEC Guide 99:2007, definition 1.17, inwhich Notes 1 and 2are different. NOTE 3 Source: IEC 80000-13:2008. SIprefixesnot be refer usedstrictly to to represent powers 1024 of10,andbits (2 should not be used for powers of2.For example, 1 kbit should NOTE 2 SIprefixes for decimal multiples ofSI base units and SIderived units are given in 6.5.4. NOTE 1 The houris a non-decimal multiple ofthe second. EXAMPLE 2 The kilometre is a decimal multiple ofthe metre. EXAMPLE 1 one than greater integer an unitby measurement given a by multiplying unitobtained measurement multiple ofaunit 3.17 Adapted from ISO/IEC Guide 99:2007, definition 1.16, inwhich Notes 1 and 5are different. NOTE 6 Forthe SIprefixes for multiples of units and submultiples of units, see 6.5.4. NOTE 5 Inquantity calculus, the quantity ‘numberunit one,symbol of 1. entities’ is often considered to be abase quantity, with thebase NOTE 4 Forafullpublished description by the and Bureau explanation International of the desInternational Poids System et Mesures ofUnits, (BIPM) see edition andavailable 8 onof theSIbrochure the BIPM website. NOTE 3 The baseunits and the coherent derived units ofthe SIform a coherent set, designated the “setofcoherentunits”. SI NOTE 2 The SIisfounded on the seven base quantitiesbase of units,the ISQ and the seenames 6.5.2. and symbols of thecorresponding NOTE 1 Prefixes forbinary multiples are: ystem of Units Units of ystem (2 (2 (2 (2 (2 (2 (2 (2 10 10 10 10 10 10 10 10 ) ) ) ) ) ) ) ) Factor Value Factor Value 5 6 7 8 1 2 3 4

10 bits), which is a kibibit (1Kibit). 1 208 925 819 614 629 174 706 176 yobi yobi 1 208925819614629 174706176 1 180 591 620 717 411 303 424 zebi zebi 1 180591620717411 303424 5 2 0 0 4 7 exbi 1 152921504606846 976 1 125 899 906 842 624 pebi pebi 1 125899906842624 1 099 511 627 776 tebi tebi 1 099511627776 1 073 741 824 gibi gibi 1 073741824 1 048 576 mebi mebi 1 048576 1 024 kibi kibi 1 024 Name Symbol Prefix Yi Yi Ei Ei Pi Pi Mi Ki Ki Gi Gi Ti Ti Zi Zi ISO 80000-1:2009(E) 80000-1:2009(E) ISO 7

Provided by : www.spic.ir OE4 Inthe case of vector ortensor NOTE 4 Aquantity value can be presented in more than one way (see Examples 1, 2 and 8). NOTE 3 The number can be complex (seeExample 5). NOTE 2 anumber and a reference material (see Example 10). ⎯ anumber and a reference to a measurement procedure (see Example 7),or ⎯ a numberproduct andameasurement of a Examples(seeunit ⎯ According tothe type ofreference, a quantity value is either NOTE 1 Amount-of-substance concentration oflutropin in a given EXAMPLE 10 8 Electric impedance ofa given EXAMPLE 5 Celsius temperature of agiven sample: EXAMPLE 4 Curvature of a given arc: EXAMPLE 3 Mass ofagiven body: EXAMPLE 2 Length of a given rod: EXAMPLE 1 aquantity of magnitude expressing together andreference number value value ofa quantity quantity value 3.19 1.18] 99:2007, Guide [ISO/IEC SI prefixes for decimal submultiples of SIbase units and SI derived units are given in 6.5.4. NOTE Forplane angle, the second is anon-decimal submultiple of the minute. EXAMPLE 2 The millimetre is a decimal submultiple of the metre. EXAMPLE 1 80000-1:2009(E) ISO Molality ofPb EXAMPLE 9 one than greater integer an by unit measurement given a dividing by unitobtained measurement submultiple ofa unit 3.18 NOTE 5 Adapted from AdaptedGuide ISO/IEC 99:2007, definition1.19which “value of a quantity” and “value” are given as admitted terms. NOTE 5 XML Mass fraction of cadmium in a given sample of copper: EXAMPLE 8 Rockwell Chardness of a given sample (150 kg load): EXAMPLE 7 Refractive index ofagiven sample of glass: EXAMPLE 6 XML Force acting on agiven particle, e.g. in Cartesian components ( EXAMPLE generally not indicated for quantities of dimension one (see Examples 6 and 8), or (− where each component is aquantity. 31,5; 43,2; 17,0) is anumerical-value vector andN (newton) isthe unit, or( sample of plasma (WHO international standard 80/552): where jis the imaginary unit: 2+ in a given sample of water: quantities, each component has aquantity value. circuit element at a given frequency, , in which Example 10 and Note 4 are different and in 1,2, 3, 4, 5, 8and 9); the measurement unit one is F x ; F

F x ; y

; F

International Units per litre) 5,0 IU/l (WHO 43,5 HRC(150 kg) 3 µg/kg or3 1,32 (7 1,76 µmol/kg − 0,152 kg or 152 g 112 m 5,34 m or 534 cm F y 5 ; z

−

31,5 N; 43,2 N; 17,0N) × 10 − 9

Provided by : www.spic.ir value of a quantity” Adapted from ISO/IEC Guide and “numerical 99:2007, value”definition are given1.20, as an admitted terms. NOTE 3 measurement units ofQ respectively. (The symbol number in the expression of a quantity value, aquantity of expression inthe number value numerical aquantity of value numerical numerical quantity value 3.20 © ISO 2009 – All rights reserved 2009 –© ISO rights All Forthe quantities in Example 1of item 3.22, [ EXAMPLE 1 units measurement other or units derived coherent units, base between relation mathematical unit equation 3.23 1.22] 99:2007, Guide [ISO/IEC EXAMPLE 1 units measurement of independent quantities, of system given in a quantities between relation mathematical quantity equation 3.22 1.21] 99:2007, Guide [ISO/IEC In quantity calculus, quantity equations are preferred to numerical value equations because quantity equations are independent of the choice of measurement units, whereas numerical value equations are not(see also 4.2 and 6.3). NOTE quantities to applied operations and rules mathematical of set calculus quantity 3.21 { Forquantities that have a measurement unit (i.e. those other than ordinal quantities), the numerical value NOTE 2 considered as a part ofthe numerical quantity Forquantities value. of dimension one, the reference is a measurement unit which is a number and this is not NOTE 1 EXAMPLE 2 J := J EXAMPLE 2 units. EXAMPLE 2 OE Adapted from ISO/IEC Guide 99:2007, definit NOTE 1km/h = EXAMPLE 3 EXAMPLE 3 the duration of the electrolysis, andF Q } of aquantity XML Foraquantity value of EXAMPLE { same quantity value can be expressed as 5 721 g in which case the numerical quantity value quantity value, which remains 3. mmol/mol. The unit mmol/mol is numerically equal to 0,001, but this number 0,001 is not part of the numerical Inan amount-of-substance fraction equal to 3 mmol/mol, the numericalEXAMPLE quantity value is 3 and the unit is m } = (5721 g)/g = Q Q T n

is frequently denoted { 1 = =

(1/2) = kg m kg It

/ ζ

5 721. See 3.19. Q F 1 where n 2 , (1/3,6) m/s. m 2 :=

Q Q /s v denotes “is by definition equal to” as given in ISO 80000-2:2009, item 2-7.3.) 2 3 2

2 and

, where Tis the kinetic energy and , where J,kg, m,andsare thesymbols for thejoule, kilogram, metre, andsecond, where

Q is the amount ofsubstance ofaunivalent component, 3 Q is the Faraday constant. , respectively, provided that these measurement units are in acoherent 1 ,

Q Q m 2 } and

= = 5,721 kg, the numerical quantity value is{

Q /[Q Q

3 other than any number serving as the reference reference the as serving number any than other denote different quantities, andwhere ], where [ ion 1.23, inwhich the Example 2 is different. Q ] denotes] themeasurement unit. v is the speed ofa specified particle of mass Q

in which Note 2 is different and in which “numerical other than other 1 ] = [Q 2 ] [Q 3

ordinal quantities quantities ordinal ] where [Q ζ m I isthe electric current and is a numericalis factor. } 1 ], [ = (5,721 kg)/kg ISO 80000-1:2009(E) 80000-1:2009(E) ISO Q 2 ] and[Q 3 = ] denote the m 571 The 5,721. . system of t is 9

Provided by : www.spic.ir OE Adapted fromISO/IEC Guide 99:2007, definition 1.25, inwhich “numerical quantity value equation” is given as an admitted term. NOTE Inthe quantity equation for kinetic energy of a particle, EXAMPLE 2 or both. 10 exist quantities those among operations algebraic no which butfor kind, thesame of quantities other with magnitude, to according established, be can relation ordering atotal which for procedure, measurement a conventional by defined quantity, ordinal quantity 3.26 Forthe quantities in the first example in item 3.22, { EXAMPLE 1 units measurement specified and equation quantity agiven on based values, quantity numerical between relation mathematical numericalequation quantity value equation value numerical 3.25 1.24] 99:2007, Guide [ISO/IEC The measurement units may belong to different systems ofunits. NOTE EXAMPLE km/m 80000-1:2009(E) ISO Adapted from ISO/IEC Guide 99:2007, 1.27, inwhich “measurement scale” is given as an admitted term. NOTE Rockwell C hardness scale. EXAMPLE 3 Time scale. EXAMPLE 2 Celsius temperature scale. EXAMPLE 1 quantities values quantity of set ordered scale measurement scale quantity-value 3.27 1.26] definition 99:2007, Guide [ISO/IEC Ordinal quantities are arranged according to ordinal quantity-value scales (see 3.28). NOTE 2 dimensions. Differences Ordinal and ratiosquantities of can ordinalenter intoquantities empirical have relations nophysical only andmeaning. have neither measurement units nor quantity NOTE 1 Subjective level of abdominal pain on a scale from zero to five. EXAMPLE 4 Earthquake strength on the Richter scale. EXAMPLE 3 Octane number for petroleum fuel. EXAMPLE 2 Rockwell Chardness. EXAMPLE 1 quantities for units twomeasurement of ratio units between factor conversion 3.24 numerical values of { T } = (1/2) XML (km/h)/(m/s) = EXAMPLE 2 h/s EXAMPLE 1 ×

that of 2 × 3 2 is anumerical value equation giving the numerical value 9of kind kind Q

1 , = = Q 1 000 and thus 1 km 3 600 and thus 1 h 2 and Q

of quantities of 3 (1/3,6) and thus 1km/h , respectively, provided that they are expressed in base units or coherent derived units = 3 600 s. =

1000 m. of a given kind of quantity of kind agiven of

of the same kind same the of = (1/3,6) m/s

Q 1 } =

ζ { T

Q used in ranking, according to magnitude, to magnitude, according ranking, in used

= 2 (1/2) } {Q T in joules. in 3 }

1 = }, { 2k and 2kg Q 2 } and{ v Q

= 3 3 m/s, then then m/s, 3 } denote the Provided by : www.spic.ir letter country codes, nor currencies are treated here. here. treated are currencies nor codes, country letter here. Neither nottreated are resistance) corrosion and hardness (e.g. tests conventional and of theresult as expressed or scales) colour-intensity quantities Ordinal treated. are body quantity-value scale quantity-value scale value ordinal ordinal quantity-value scale 3.28 © ISO 2009 – All rights reserved 2009 –© ISO rights All called generally a category, such constitute would on so and wavelengths heights, distances, Diameters, comparable. mutually are that quantities of categories into together grouped be may Quantities ofquantity Kind 4.2 Standard, International In this The ofquantity concept 4.1 4 Quantities Adapted from ISO/IEC Guide 99:2007, 1.30, inwhich Note 2 isdifferent. NOTE 3 International Standard. “Nominal property value” is not to be confused with “nominal quantity value”, which is not used in this NOTE 2 means. Anominal property has a value, which can be expressed in words, by alpha-numerical codes, or by other NOTE 1 Sequence of amino acids in a polypeptide. EXAMPLE 5 ISO two-letter country code. EXAMPLE 4 Colour of a spot test in chemistry. EXAMPLE 3 Colour of a paint sample. EXAMPLE 2 Sex ofahuman being. EXAMPLE 1 nomagnitude has property the where substance, or body, aphenomenon, of property nominal property 3.30 1.29] definition 99:2007, Guide [ISO/IEC agreement formal by defined scale quantity-value conventional reference scale 3.29 Adapted from ISO/IEC Guide 99:2007, 1.28, inwhich “ordinal value scale” is given as an admitted term. NOTE 2 Anordinal quantity-value scale may be established by measurements according to a measurement procedure. NOTE 1 Scale ofoctane numbers forpetroleum fuel EXAMPLE 2 Rockwell C hardness scale EXAMPLE 1 Mathematic operations can be performed on quantities other than ordinal quantities, as explained below. below. explained as quantities, ordinal than other onquantities beperformed can operations Mathematic kind the same of quantities called are quantities comparable , arranged according to quantity-value scales (such as the Beaufort scale, the Richter scale the the scales scale, as scale Richter (such Beaufort quantity-value to according , arranged

for

ordinal quantities quantities ordinal ─

Quantity calculus Quantity calculus quantities nominal used for the quantitative description of a phenomenon, substance or or substance a phenomenon, of description quantitative the for used

properties , such as the sex of a human being or the ISO two- ISO the or being human a of sex the as , such . ISO 80000-1:2009(E) 80000-1:2009(E) ISO . Mutuallylength 11

Provided by : www.spic.ir between quantities is called a is called quantities between equations non-contradictory of set Each quantities. derived the define to used are equations which choice of amatter also is It quantities. base be to considered are quantities which and many how choice of amatter It is quantities 12 The Faraday constant, EXAMPLE 2 The Planck constant EXAMPLE 1 constants called are quantities Such circumstances. all under constant be to considered are quantities Some constants and Universal empirical constants 4.4 a is called quantities between equation Each quantities. new define or nature of laws express that equations through related are Quantities quantities System of 4.3 80000-1:2009(E) ISO EXAMPLE 3 called are They measurement. by obtained generally are values Their others. on butdepend circumstances, some under constant be may quantities Other calculus quantity called is quantities of division and multiplication subtraction, addition, operations mathematical the Performing quantities. new in resulting algebra, of rules the to according another one by divided and multiplied are Quantities kind. same the beof also must equation an in sign anequal of side oneach quantities Hence, quantities. comparable mutually of category same the to belong they unless subtracted or be added cannot quantities more or Two called called are quantities Such independent. mutually as kinds different of quantities some toconsider convenient It is where Theory shows that where pendulums, can be expressed by onequantity equation The result of measuring at a certain station the length base quantities base by means of equations. equations. of means by fundamental physical constants physical fundamental or TCl C g C isan empirical constant that depends on the location. = = is the local acceleration offree fall, which is another empirical constant. 2 π g . In quantity calculus, the algebraic expressions should be quantities or numbers. numbers. or quantities be should expressions algebraic the calculus, In quantity .

. Other quantities, called called quantities, . Other system ofquantities . quantity equation quantity ─ andderived Base quantities quantities , h

F =

6,626 068 96(33) = 96 485,339 9(24) 96485,339 . . derived empirical constants empirical ×

10 /o [CODATA 2006]. C/mol quantities,

–34 l and the periodic time J·s [CODATA 2006]. J· s are defined or expressed in terms of base of terms in expressed or defined are . T , for each of several particle © ISO 2009 – All rights reserved 2009 –© ISO rights All universal Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009 –© ISO rights All IEC the defines 80000 ISO in given and multipliers, 80000 including equations, quantity and quantities base of choice special The International System of Quantities, ISQ 4.6 coefficients called often are . factors numerical than other multipliers Constant EXAMPLE 3 more or one include also may A multiplier EXAMPLE 2 constants. universal more or one include may A multiplier EXAMPLE 1 called then are and numerical purely be may multipliers Such chosen. quantities of the system on i.e. equations, in the occurring quantities the for chosen definitions the on depend multipliers These multipliers. constant contain sometimes quantities between Equations equations quantity in multipliers Constant 4.5 length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. intensity. luminous and substance, of amount temperature, thermodynamic current, electric time, mass, length, ISQ: the in quantities base seven are There equations. quantity by units the base of terms in defined be can where law ofmotion is In the now obsolete MKS system, length, mass, and time are the three base quantities. In that system, the Newton In the ISQ, theCoulomb law for electric charges is particle inclassical mechanicsis In the CGS system, length, mass, and time are the three base quantities. In that system, the kinetic energy of a free fall adopted by theCGPM 1901. (Inthis system, force and mass have the same dimension.) constant. where where Tm F F F T F = = = is kinetic energy, isforce, is force, force, is 2 1 4 g 1 n π 1 ε ma v 0 2

12 m r 2 q is mass, 1 and International System of Quantities of System International

m q 2 is mass and are two electric charges, r a is acceleration, and v conventional quantity values quantity conventional is speed. The same relation is also true in the ISQ. g n isa conventional quantity value, i.e. the standard acceleration of is distance and and distance is numerical factors , denoted “ISQ” in all languages. Derived quantities quantities Derived languages. all in “ISQ” , denoted . ε 0 . is a universal constant, i.e. the electric ISO 80000-1:2009(E) 80000-1:2009(E) ISO 13

Provided by : www.spic.ir The The and is expressed by a number. number. a by expressed is and one of dimension mlre toy L molar L entropy L electric T tension energy frequency 14 L LMT illuminance LT force speed EXAMPLE aquantity of dimension the ISQ, the in Hence, respectively. J, N and I,Θ M,T, L, denoted are quantities thebase of the dimensions intensity, andluminous substance of amount temperature, thermodynamic current, electric time, mass, length, quantities base seven the with In theISQ, Forhistorical reasons, a quantity ofdimension one is often called NOTE quantities base of powers of product a as be expressed can terms these of Each terms. of asum of consist may expression The equation. an of means by quantities quantity any consideration, under quantities of system In the 5 Dimensions 80000-1:2009(E) ISO form the in expressed be may relation this units, and quantities of treatments In formal The wavelength of one of the sodium spectral lines is EXAMPLE 1 the called is number That number. a and unit this of aproduct unit,as this of interms beexpressed can kind thesame of quantity other then any the called quantity reference a as chosen is kind agiven of quantity a of example aparticular If and numerical Units values 6.1 6 Units A denoted product dimensional the has zero to equal all are exponents dimensional whose A quantity the called where A, B, C, … denote the dimensions of the base quantities quantities thebase of dimensions the denote B, C,… A, where by a numerical factor factor a numerical by efcec 1 efficiency L L magnetic flux entropy 0 B 0 dimension is the numerical value of the wavelength expressed in metres. Here, Here, Quantity Dimension dim dim dim dim Q λ C

≈ 0 = 5,896 … { Q Q Q λ dimensional exponents dimensional

= } ·[ = = is the symbol for the quantity wavelength, m is the symbol for the unit of length, the metre, and 5,896 1, where the symbol 1 denotes the corresponding dimension. Such a quantity is called a a called is quantity a Such dimension. corresponding the denotes 1 symbol the where 1, A L × α Q α 10 (see 3.7) of the quantity thequantity of 3.7) (see M

] B β β –7

T C m γ γ I … ξ δ , i.e. Θ ε N ζ ξ J A 2 –2 2 2 2 2 η –1 α MT MT MT MT MT

–1 J

B –2

β –2 –2 –3 –2 –2 . C

numerical value numerical I I Θ Θ –1 2 γ …, where the …, where

–1 –1

N –1 Q is then expressed by the by expressed then is

set of exponents exponents of set of the quantity expressed in this unit. this in expressed quantity the of A , B , C , … from a chosen set, sometimes multiplied multiplied sometimes set, achosen , …from Q A can be expressed in terms of the base thebase of terms in be expressed can , dimensional product dimensional α B Q , dimensionless , β in general becomes becomes in general C , , …, respectively, and , …,respectively, γ , … is the same for each term. term. each for same the , …is . See Note 1in 3.8. © ISO 2009 – All rights reserved 2009 –© ISO rights All

α unit unit , β , (see 3.9), 3.9), (see γ quantity quantity , … are , …are × 10 –7 ,

Provided by : www.spic.ir Equations between units, such as [ as such units, between Equations preferable to indicate the numerical value explicitly as the ratio of the quantity to the unit. unit. the to the of the quantity as ratio explicitly value thenumerical toindicate preferable compound units compound called are numerical values, or combinations of dimension one of quantities (see Clause 5). 5). Clause (see quantities of one dimension of combinations or values, numerical numbers, are etc., functions, trigonometric functions, logarithmic functions, exponential of arguments The is independent of the unit. theunit. of independent is unit, the and value thenumerical of product the is which quantity, the because value numerical the first times is that unit another in expressed is aquantity If unit. a by multiplied respectively, tensor, or vector value a numerical as expressed be also can tensors and Vectors above. described as expressed be can thequantity of value where unit [ © ISO 2009 – All rights reserved 2009 –© ISO rights All exp( EXAMPLE 2 { as such values, numerical between Equations The speed, EXAMPLE 1 Q quantities, two of quotient the and product The 6.2 Mathematical operations intables. columns of headings and graphs in use for recommended particularly is notation This EXAMPLE 3 { e.g. subscript, a as unit the using and symbol quantity the around brackets) (curly braces by placing indicated be unit could a in particular expressed a quantity of value numerical unit. The particular a in expressed thequantity of value numerical andthe itself quantity the between todistinguish essential It is EXAMPLE 2 Thus, the product { theproduct Thus, OE Aquantity defined as A/Bis called ‘quotient of Aby B’ or ‘Aper B’,but not ‘A per unit B’. NOTE 1 / Q Thus, ifthe particle travels a distance where the metre, leads to a numerical value which is 10 Thus, Changing the unit for the wavelength in the previous example from the metre to the nanometre, which is 10 v Q 2 Q QQQ Q

, and the quotient [ quotient the and , = 1 1 Q 222 1 Q

Q l / is the symbol for the quantity, [ thequantity, for symbol the is 2 t λ v =⋅ 2

] of the quantity thequantity of ]

= l = ≈ {} isthe distance travelled inthe duration { {

5,896 QQ l / Q t 11

= 1 } λ } {Q (6 m)/(2 s) /nm ⎣⎦ ⎡⎤ × ⎣⎦ ⎡⎤ Q 10 E 2 } ·[ ≈ / 1 kT Q 589,6 } { –7 expressed in the unit [Q unit inthe expressed ); ln( Q

Q m Q 1 2

v = . ] [ } is the numerical value { value thenumerical is } 1 Q = , of a particle in uniform motion is given by p 3 m/s 3 ]/[ 5,896 1 Q /kPa); sin( /kPa); Q Q 2 2 ] 2 . Similarly, the quotient { quotient the . Similarly, ] isthe unit [ × Q 10 1 π Q –7 /3); cos( l 2

= ] × 6 m in thedurationm 6 Q 10 = ] is the symbol for the unit and { and unit the for symbol the is ] [ Q 9 Q nm 1 ω 1 ] [ / t Q t Q . ]. For vectors and tensors, the components are quantities that that quantities are components the andtensors, vectors ]. For + Q 9 1 = 2 the numerical value of the quantity expressed in metres. k and Q 2 589,6 nm α quantity the of ] times the first unit, the new numerical value becomes 1/ becomes value numerical new unit,the thefirst times Q ], are called called ], are Q ) 1 1 Q Q 2 2 Q } 2 } of the quantity the quantity of } , satisfy the relations relations the , satisfy = 1 }/{ t {

= Q Q 2s, the speed 1 2 } { unit equations unit } is the numerical value { value numerical the is } Q Q 2 numerical value equations value numerical called are }, 1 / Q 2 . Units such as [ Q 1 ,

Q Q v , 2 } is the symbol for the numerical numerical the for symbol the is } isequalto . , and the product [ product the , and ISO 80000-1:2009(E) 80000-1:2009(E) ISO Q Q 1 1 λ / ] [ } Q nm Q 2 } of the quantity quantity the of } . It is, however, however, . Itis, 2 ] and ] and [ Q 1 ] [ Q Q 2 –9 ] is the the ] is 1 times ]/[ Q 15 2 k . ]

Provided by : www.spic.ir 16 EXAMPLE recommended. strongly is and preferred is normally quantity equations of theuse Therefore, choice. this of independent being of advantage the have equations quantity whereas units, of choice the on depend equations) unit course of (and equations value Numerical frequently. less used are equations used; unit generally are equations value numerical and equations Quantity technology. and science in used are equations, unit and equations, value numerical equations, quantity i.e. above, introduced equation of types three The and Quantity numerical value equations equations 6.3 EXAMPLE 3 80000-1:2009(E) ISO a tochoose convenient, more inpractice and however, possible, It is equations. value numerical inthe factors numerical additional of the appearance to lead would quantity each for theunit of choice anindependent butmaking arbitrarily, bechosen might Units systems Coherent of units 6.4 context. same the in stated be clearly shall units the used, not are subscripts If equations. such in thesubscripts omit to not recommended is it chosen, units on the depend equations value numerical in factors numerical Since situation. physical same the describe they although quantities different are ofthatratio, thelogarithm as such thatratio, of function andany kind thesame of quantities two of ratio The ln called called derived called are unitsquantities derived the of units The quantities. of system the in equations the with accordance in units thebase of terms in expressed are quantities derived called are quantities base the of units The defined. is quantity base each for unit one only and one first units, of a system such establish To quantities. of system a chosen in equations thecorresponding as factors, numerical the including form, same the exactly have equations value numerical occurs in the expressions for the derived units in terms of the base units. units. thebase of terms in units derived the for expressions inthe occurs ever 1 than other factor nonumerical units, of system coherent a such 1. In symbol one, unit the acquires one dimension of quantity a particular, In respectively. units, base the for those by dimension quantity the in dimensions base for symbols the by replacing obtained is unit the of expression the i.e. question, in quantity the of dimension the to corresponds unit each of expression the units, of system coherent In a p andln where { choices, it would generally be different. The number 3,6 that occurs in this numerical value equation results from the particular units chosen; with other A simple quantity equation is speed, distance, and duration, respectively, the following numerical value equation is derived: Using, for example, kilometre perhour(symbol km/h), me as given in 6.2. coherent v { v p

= } 0 v km/h have no meaning in quantity calculus where

} l / km/h t

with respect to the system of quantities, including the equations in question. question. in equations the including quantities, of system tothe respect with p = /

3,6{ p = 0

v and ln( /(km/h). l } m /{ t } p s /

p 0 ) are different quantities. Note that in mathematics for numbers, ln( p denotes pressure. tre (symbol m)and second (symbol s) as the units for . A system of units defined in this way is is way this in defined units of A system . system of units in suchway a that the base units base © ISO 2009 – All rights reserved 2009 –© ISO rights All . Then, the units of all all of units the . Then, p / p 0 ) = ln p

− lnp 0 , but Provided by : www.spic.ir The SI comprises ISQ. the to with respect units of system a coherent force kg · m/s © ISO 2009 – All rights reserved 2009 –© ISO rights All entropy kg expressed in terms of theSI base · frequency s m/s speed m units Symbol forSIderived unit Quantity EXAMPLE 1 L substitutions: formal following the using by quantities derived ISQ corresponding the of products dimensional the from beobtained can units theSIbase of terms in units derived SI the coherent for expressions The SI derived units 6.5.3 1. in Table listed are units base SI seven The units SIbase 6.5.2 SIunits. of system thecoherent form that together ⎯ ⎯ CGPM Measures, and on Weights Conference The General 6.5.1 SI of System Units, The International 6.5 from its coherent derived unit in the SI in terms of the base units. units. thebase of in terms SI in the unit derived coherent its from obtained be can ISQ the in quantity derived a of dimension the Hence, reversible. are substitutions These I T M energy kg · m derived derived units base units, and International System of Units System International − 1 → m → A → s → kg

2 2 /(s /s 2 2 2

·K)

hroyai eprtr kli K intensity luminous kelvin substance of amount temperature thermodynamic current electric second time kilogram mass metre length Table 1 — SI base units for the ISQ base quantities the 1 ISQ —SIbaseunitsTable for base ISQ base quantity , denoted SI , denoted Θ 1 J N

→ K → 1 → cd → mol [1960] ( [1960] in all languages, was adopted by the 11th International International 11th by the adopted was languages, all in Conférence générale des poids etmesures poids des générale Conférence oa nrp kg· m efficiency 1 kg· molarm entropy s photon irradiance magnetic flux expressed in terms of theSI base kg· m electric potential Symbol forSIderived unit units Quantity mole mol mol mole candela cd cd candela mee A ampere Name Symbol SI base unit m m s s kg kg − 1 /m 2 2 2 2

/(s /(s /(s 2 2 3 ·K mol) ·A) ·A) ISO 80000-1:2009(E) 80000-1:2009(E) ISO ). The SI is SIis The ). 17

Provided by : www.spic.ir 18 EXAMPLE 2 80000-1:2009(E) ISO units. incompound andsymbols names special use to advantageous often It is 3. 2and Tables in listed are CGPM the by approved those exist; andsymbols names special units, derived theSI of some For quantities. of system the with coherent units of system toaspecific refer units whereas quantities, of toasystem refer Dimensions not. do dimensions whereas amagnitude, has unit Each one. than other factors numerical any include units nor dimensions Neither cannot. dimensions whereas equations, quantity in used be thus can and quantities of cases special are Units luiac u lx illuminance degree Celsius lux luminous flux H Celsius temperature tesla inductance henry magnetic flux density siemens magnetic flux electric conductance F electric resistance capacitance farad volt electric potential difference electric charge W power watt J energy joule pressure, stress Hz force newton N frequency hertz solid angle plane angle Using thederivedVvolt,unit 1 Using the derived unit joule, 1 J ISQ derived quantity Table 2 — SI derived units with special 2Table —SIderived units with and symbols names radian steradian weber pascal lumen ohm coulomb = = 1m 1 m pca ae Specialsymbol Special name 2 2 · kg/(s ·kg/s 3 2 ·A),the symbol forthe unit for magnetic fluxmay bewritten V ·s. , the symbol for theunit for molar entropy may bewritten J/(K · mol). S V rad sr T Wb Pa Ω C lm ºC

lm/m Wb/A A·s kg· m/s m = Ω J/s = cd · sr K s N/m m/m V· s − 2 − 1 /m 1

2 2 2 2

= = 1 2 1

Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009 –© ISO rights All m symbol unit, anSIbase also is which equaltometre, metre square by divided metre cubic is unit SIderived coherent The unit. aderived in expressed hence and quantity derived a is which area), per (volume volume areic as given is rainfall example, For unit. SI base toan beequal can cases in some unit SI derived an that benoted It should coherent with respect to the ISQ. ISQ. the to respect with coherent Table in listed SI prefixes the with formed are 4. These units SI coherent the of submultiples and multiples decimal values, numerical small or large avoid to In order SI 6.5.4 prefixes unit. base SI an as it adopted yet not has CGPM the although unit, abase as considered often is 1, one, symbol theunit case, inthis Hence, quantities. base any of other interms beexpressed itcannot because quantity abase as considered is number counting thequantity Here, atom. an in protons of e.g.number number, newton per newton equal to one, symbol N/N symbol one, to equal newton per newton is factor friction for SIunit derived the example, for unit, derived SI an generally is 1, symbol unitone, The catalytic activity dose equivalent absorbed dose becquerel activity (of aradionuclide) Table 3 — SI derived units with special names names special with units derived 3—SI Table ISQ derived quantity Factor 10 10 10 10 10 10 10 10 10 10 1 2 3 6 9 12 15 18 21 24 dc da deca h hecto k kilo M mega G giga tr T tera P peta E exa Z zetta 3 /m 2

= m.

yotta Y Name Symbol

sievert gray katal Prefix pca ae Specialsymbol Special name Table 4 — SI prefixes prefixes 4—SI Table = human health health human 1. Consider, however, the unit one for the quantity counting counting thequantity for one unit the however, 1. Consider, and symbolsadmittedreasons for ofsafeguarding Sv Bq kat Gy SI multiple units SI multiple SI derived unit Factor 10 10 10 10 10 10 10 10 10 10 − − − − − − − − − − 24 21 18 15 12 9 6 3 2 1

and and yocto y yocto zepto z atto a femto f pico p nano n micro milli m centi c deci d Name Symbol SI submultiple units SI submultiple kat = Gy Sv Bq = units and SIderived units Expressed inSI base ISO 80000-1:2009(E) 80000-1:2009(E) ISO = = J/kg s J/kg mol/s Prefix − 1

are not not are 19

Provided by : www.spic.ir cubic decimetre, symbol dm gram would be given the same status as an additional unit used with the SI as the litre, symbol l, equal to the submultiple It hasbeen proposed to adopt a new nameforthe SIbase unit ofmass, without aprefix. Atthesame time, the submultiple 20 The SIprefixes are also used together with the ISO currency codes, e.g. NOTE 2 EXAMPLE 3 4. Clause inIEC80000-13:2008, given is multiples these of derivation and origin the on information Additional 3.17. in given are multiples binary for Prefixes multiples. binary todenote used not be shall 10.They of powers exact denote SIprefixes The nm Write (nanometre) for 10 EXAMPLE 2 used. be not shall prefixes Compound EXAMPLE 1 80000-1:2009(E) ISO dimension. same the having but kinds, different of quantities between distinction facilitate to units derived for expressions in used be may andsymbols names special Such EXAMPLE 2 context. the on depending not, or used be could that andsymbols names special has one unit the however, quantities, such certain of case In the windingturns Number ina of EXAMPLE1 numerically. expressed is aquantity when such explicitly out written not generally Itis 1. symbol one, theunit is one dimension of anyquantity for unit SI coherent The one unit The 6.5.5 units. compound for symbols toform symbols unit other with combined be can that and power negative or toapositive be raised that can submultiple or multiple adecimal for symbol anew with it forming attached, directly is it which to symbol unit thesingle with tobecombined isconsidered aprefix of symbol The gram and kilogram can continue to be used, as the litre and centilitre are used today. gram (symbol g), e.g. milligram, sy Forhistorical reasons, thenames name oftheof SIbasethe decimaluni multiples and submultiples of the kilogram are formed by adding the prefixes to the submultiple NOTE 1 1 GSEK 1 MUSD = 1 kGBP = kEUR 1 1 Kibit 1 kbit 1 cm plane angle, level of apower quantity, 1 1 solid angle, Ω µ s /km − 3 1

= = = 1 000bit (10 = (10 1024 bit = = 1 1000 GBP (British pound) 1000 EUR (European euro) 1000 000 000 SEK (Swedish crown) 1000USD (US dollar) Ω − − Ω 2 α 6 /(10 m)

= = 2,3sr 0,52rad=

3 –1 3

m) =

= 10 10 = = − 10 6 2,3 6 m s 3 L , andthetonne, symbol t,equal tothemultiple megagram, symbol Mg. Ifthisis accepted, − − F 0,52 3 3 1

= Ω 12 Np /m mbol mginsteadof microkilogram ( = 12 (see Table 5) − N 9 m,not m = 25 × 1= µ m. 25 t of mass, the kilogram, contains the SIprefix “kilo”.The µ kg). © ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir used. Mass and volume fractions can also be expressed in units such as as such units in expressed be also can fractions volume and Mass used. percentage by mass or percentage by volume. Additional information, such as %( as such information, Additional volume. by percentage or mass by percentage example, about,for speak to itis meaningless numbers, are mil” and“per cent” “per units the Since © ISO 2009 – All rights reserved 2009 –© ISO rights All ‰ 1 where ‰, symbol mille), per (or mil per Also, reflection factor, EXAMPLE 4 1 % %,where symbol cent, per cases, In some Ithasbeen proposed to adopt aspecial name and symbol for theunit one and its symbol 1forgeneral use, whichcould becombinedwith prefixes. NOTE 10. of powers using by expressed be may value numerical the Instead, not. may 1, symbol its or itself, one unit but the SI prefixes, with combined be may theunitone for andsymbols names Special EXAMPLE 3 These units are given in Tables 5 and 6. and 5 Tables in given are units These CIPM Measures, and for Weights Committee the International by recognized are that units non-SI certain are There 6.5.6 Other units recommended. 10is of powers of use the Instead, used. be not shall and ambiguous and language-dependent are ppt ppband pphm, ppm, as such Abbreviations mass fraction is “the mass fraction of B is is B of fraction mass “the is fraction mass a example, for expressing, wayof preferred 7.2.The also See %. symbol theunit to be attached not therefore 0,78 is fraction amass If misleading. is it because name, a quantity in used be not shall “percentage” the term using natural logarithms. logarithms. natural using original symbol l l symbol and usedis byISO apropername is it derived of because IEC original not aperson. from b a level mass volume plane angle time The unit neper, symbol Np, neper, Theunit is symbol with the coherent but notSI, adopted by yet the as CGPM an unit. areSI Levels in defined the I has TheCGPM l andL approvedthe two symbols for due the litre between andto of l the confusion risk 1in Only fonts. t some angular velocity, attenuation coefficient, = ( 78 78 or 78 then percentage the is %, % Comité InternationalPoids des Mesures et Quantity Quantity ω

= 1 a/ photon flux, 17 rad/s α

08 pm curvature, 0,83Np/m bel neper tonne t litre second minute degree day hour minute r

= b 83 %

aeSmo Definition Name Symbol Table 5 — Units used with the SI the with used 5—Units Table = 0,83 w B

= := Np B l, L ″ ′ º d h min

0,78” or “the mass fraction of B is w is B of fraction mass “the or 0,78”

0,01, is used as a submultiple of the coherent unit one. one. unit thecoherent of asubmultiple as used is 0,01, :=

= b a

0,78? Instead, the unambiguous term “fraction” shall be shall “fraction” term unambiguous the Instead, 0,78? 0,001, is used as a submultiple of the coherent unit one. one. unit coherent the of asubmultiple as used is 0,001, ), as having to be retained for use together with the SI. SI. the with together use for retained be to having as ), k

Φ = 0,34 m

= 37 1 B:= 1 Np := 1º 1 d:= 1 h:= 1 min := 1 l:= 1 1 1 t:= ″ ′

:= := := × (1/60)º ( 10 (1/60) − 1dm 1 000kg 24 h h 24 min 60 1 (1/2) ln 10 Np≈ π

Unit /180) rad ln e 6 60 s s − 3 ′ 1

µ g/g

= 10 1,151 293 − 6 or ml/m or B ISO 80000-1:2009(E) 80000-1:2009(E) ISO

m = 78 %”. Furthermore, Furthermore, %”. 78 / m ) or %( 3

= 10 − 9 V . / V ) shall shall ) SQ he 21

Provided by : www.spic.ir shall be used. used. be shall unit the for symbol international the by followed numerals Arabic quantities, physical of values express To deprecated. is units such all of use the units, atomic the for Except units. UScustomary and units Imperial units, CGS units, atomic e.g. exist, units other Many 22 (1 var var, symbol thevar, as such OIML, or IEC ISO, by adopted are purposes special for units Some 80000. IEC and 80000 ISO of parts other the in pages) (right-hand pages unit the on column Remarks the in appropriate, where given, are SI. They the with together temporarily used be to CIPM the by recognized are that units non-SI certain also are There 80000-1:2009(E) ISO symbol number, theMach as such numbers, characteristic for Symbols signs. modifying other or with subscripts sometimes alphabet, Greek or Latin the from letters single generally are quantities for Symbols General 7.1.1 Symbols for quantities 7.1 7 Printing rules Symbols for quantities are given in ISO 80000 (parts 3 to 5 and 7 to 12) and IEC 80000 (parts 6, 13 and 14). 14). and 13 6, (parts 80000 IEC and 12) 7to and to5 3 (parts 80000 ISO in given are quantities for Symbols 80000-2. ISO in given are quantities tensor and vector for Notations asentence. of end the at e.g. punctuation, normal for except stop afull by not followed is symbol quantity The text. the of rest the in used type the of irrespective type, (sloping) italic in written always are symbols quantity The product. in a factors as theyoccur if symbols other from beseparated symbols two-letter thatsuch recommended is It capital. is always which of initial the theLatinalphabet, from letters with two written however, a length mass energy The dalton was earlier called Thedalton := Table 6 — Units used with the SI, whose values in SI units are obtained experimentally experimentally are obtained SI units in values SI,whose the with used 6—Units Table Quantity Quantity 1 V · A) for reactive power. power. reactive for ·A) 1V srnmclui ua astronomical unit dalton electronvolt a unified atomic mass unit unified atomic Da aeSmo Definition Name Symbol , symbol u. symbol , eV a

Unit conventional value approximately equal to 1 ua= Sun and the Earth the mean value ofthe distance between the 1 eV in vacuum passing through a potential difference of 1 V [CODATA 2006] 12 1/12 of the mass of an atom of the nuclide [CODATA 2006] 1 Da = kinetic energy acquired by an electron in C atrestand in its ground state = 1,495 978 706 91(6) 1,602 176487(40) 1,660 538 782(83) © ISO 2009 – All rights reserved 2009 –© ISO rights All

× × 10 10 × 10 − − 19 27 11 J kg m

Ma , are, , are, Provided by : www.spic.ir OE Foralist of common subscripts, see IEC 60027-1. NOTE The following principles for the printing of subscripts apply. apply. subscripts of the printing for principles following The subscripts. of use by made be can a distinction interest, of are values different or applications different quantity, one for when, or symbol letter same the have quantities different context, a given in When, Subscripts 7.1.2 printed. tobe are quantities for symbols which in type italic of thefont about implied or made is No recommendation © ISO 2009 – All rights reserved 2009 –© ISO rights All EXAMPLE 1 ambiguity. any avoid to inserted are parentheses unless line same the on sign division a or sign by amultiplication followed not be shall (/) asolidus acombination, Insuch quotients. or products themselves are both or denominator or numerator the where cases to extended be can procedures These when sans The serif solidus fonts are "/"used. can easily The horizontal be bar isconfused often preferable with tothe denote italic upper-case division. "I" or the italic lower-case "l", in particular NOTE 2 ways: thefollowing of one in indicated is another by quantity one of Division Insome fields, e.g. vector algebra, distinction is made between NOTE 1 ways: thefollowing of in one indicated is combination this quantities, more or two of a product in combined are quantities for symbols When Combination ofsymbols for quantities 7.1.3 EXAMPLE ⎯ ⎯ Do not write notwrite Do Other subscripts, such as those representing words or fixed numbers, are printed in roman (upright) type. type. (upright) in roman printed are numbers, fixed or words representing those as such subscripts, Other is number, arunning as such variable, mathematical a or quantity aphysical represents that Asubscript F Σ c C Italic subscripts Roman subscripts I g λ i ik x n printed in italic (sloping) type. type. (sloping) italic in printed ab p (

b ab a ab

( c a c / n , , ω

a a = n

/ ( = b ab

, , bc

(x (p (i / a a ab a i λ n c , k : running number) · : x : pressure) : running number)

: wavelength)

b = − = : running numbers) b − 1

-component) ( ab without a space between between a space without 1 , a , a

/ c a

b − × · ) 1

/ b b

−

= 1

(bc ), not a

a and and µ C S T g c 3 m n 1/2 r g (r: relative)

(g: gas) (m: (m: molar) (1/2: half) b − 1 (n: normal) (3: third) , as , as ab − 1 could be misinterpreted as ( as misinterpreted be could a · b and

b . ISO 80000-1:2009(E) 80000-1:2009(E) ISO ab ) − 1 . 23

Provided by : www.spic.ir operations. operations. mathematical other certain of theuse from arise that ambiguities remove to used be also may Parentheses 24 EXAMPLE 3 as such operators dyadic for signs most of sides both on bespaces shall There EXAMPLE 2 example for operations, monadic over and division and multiplication over priority has (powering) Exponentiation expressions. in compound subtraction and addition over priority have anddivision Multiplication ambiguity. avoid to required when employed are parentheses provided subtraction, and addition involve denominator the and numerator the where cases in used be can solidus The 80000-1:2009(E) ISO EXAMPLE 1 quantities. the for expressions of difference or thesum as written be shall expression the or value, numerical complete the after symbol unit common the placing values, numerical the combine to used be shall parentheses either then quantities, of adifference or asum is be expressed to thequantity If symbol. unit the and value numerical the between no space be shall there case which in angle, plane for second and minute degree, units the for are rule this to exceptions only The temperature. a Celsius expressing when aspace by preceded be shall Celsius the degree for ºC symbol the rule, this with accordance in that, noted be also should ‰.It mil, per and % cent, per units tothe applies also rule this that be noted It should symbol. unit the and value numerical the between space a leaving aquantity, for expression in the value numerical the after placed be shall unit the of symbol The quantities for Expressions 7.1.4 Write velocity is equal to distance per duration or EXAMPLE 4 equations. and expressions in be used shall quantities for abbreviations, or words never but Symbols, 80000-2. ISO in given are andtechnology insciences use for recommended andsymbols signs mathematical Other solidus), and relations, such as as such relations, and solidus), l t U

bc dbc cd bad ab a = =

Note that in this example, the ambiguity could also beremoved by altering the order of operations. ln (a a (a a / = / 12 m 23,6ºC, not

230 + +

= + + x

b b =

a b b +

·c /

) (c )/(

y (b ×

−

+ c (1+ + = 7 m = 7m ·

+ + d

(ln d c

d d =

), (not a = ), parentheses are required ), parentheses are required

5 %)V= a

a x t

(12 +

) = + ( 23,6º C 23,6º + ( b

b y

/ − ·c

, not ln( not , c / 7) m 7)

) b 230 · ) +

+ c d

) , parentheses are notrequired d = , parentheses are notrequired × 5 m, not 12 m, not 5 1,05 V 1,05 = x ,

< + , u

y ) , but not after monadic operators operators monadic after not , but ≈ 242V, not − 7 m 7 U

= 230 V 230 + v 5% =

l / t , but not velocity − a 2 + is equal to equal is and and − . = + distance/duration orv , − , − © ISO 2009 – All rights reserved 2009 –© ISO rights All ± ( a , 2 × ), not( ), and · (but not for the the not for ·(but and − a ) 2 .

l per t . Provided by : www.spic.ir not be used in the place of symbols. symbols. of place the in used be not shall subscripts, with or italics in presented example, for terms, abbreviated multiletter or quantities Descriptive terms or names of quantities shall not be arranged in the form of an equation. Names of © ISO 2009 – All rights reserved 2009 –© ISO rights All and tera. The latter form may also be written without a space, i.e. Nm, provided that special care is taken when the symbol for one of the units is the same as the symbol for a prefix. This is the case for m, metre and milli, and for T, tesla NOTE ways: thefollowing of one in indicated be shall units more or two of multiplication by formed unit A compound Combination ofsymbols for units 7.2.2 printed. be to are units for symbols which in type roman of font the about implied or made is No recommendation “the Write water content is 170 kg/m EXAMPLE 3 symbols. mathematical and symbols unit than else nothing contain shall units for Expressions EXAMPLE 2 permitted. not is consideration under measurement of context or quantity the of nature special the about information giving of means a as symbol a unit to attachment Any used. be shall no other, and this, then exists, unit a for symbol aninternational When 7.2.5. also See language. to language from differ which names, unit for applicable not is initial a capital with symbols unit writing for rule The EXAMPLE 1 e.g. position, inexponent signs containing symbols unit for made is An exception ofaperson. name aproper from aunitderived for thesymbol in acapital is letter initial thatthe except case, lower are letters These alphabet. Greek or theLatin from letters more oneor of consist units for symbols Most asentence. of end at the e.g. punctuation, normal for except stop full a by followed not is and plural the in unaltered remain shall symbol unit The text. the of rest the in used type the of irrespective type, (upright) in roman written always are units for Symbols General 7.2.1 Namesandsymbols for units 7.2 Write EXAMPLE 2 µ Ω shannon mol mole Sh second s volt V micrometre m w N · m, N m Nm N ·m, P U ohm mech B max

= 0,76=

= 500V,notU 750 not W, 76 %, neither 0,76 ( ρ P =

= V 500 V m 750 W

and not max mech

m / m

density ) nor76 %( = volume mass m 3 ”, not“170 kg H / m

) 2 O/m 3 ”. ° C. ISO 80000-1:2009(E) 80000-1:2009(E) ISO 25

Provided by : www.spic.ir In complicated cases, negative powers or a horizontal bar may be used. be used. may bar horizontal a or powers negative cases, In complicated ambiguity. any toavoid inserted are parentheses unless line thesame on sign adivision or sign multiplication a by be followed not shall (/) solidus A division. and multiplication over priority has Exponentiation 26 second to the power minus one, metre per second squared EXAMPLE 2 thepower of name The EXAMPLE 1 aspace. by separated names, thetwo of theconcatenation is units two of theproduct of the name language, English In the Englishcompound units names of 7.2.4 printed. tobe are prefixes for which symbols type in roman of thefont about implied or made is No recommendation attached. it is which to unit the for the symbol and prefix the for symbol the between aspace without text, the of rest in the used type the of irrespective type, (upright) in roman be printed shall prefixes for Symbols Prefixes 7.2.3 ways: thefollowing of one in indicated be shall another by unit one dividing by formed unit A compound mN means millinewton, not metre newton. EXAMPLE 80000-1:2009(E) ISO printed. tobe are numbers for symbols which in type roman of font the about implied or made is No recommendation text. the of rest the in used thetype of irrespective type, (upright) roman in printed be shall Numbers General 7.3.1 7.3 Numbers newton tesla EXAMPLE 2 name unit the only is it SIunits, For Celsius temperature Alfvén number EXAMPLE 1 initial. a capital with spelled is name theperson’s name, aperson’s containing names in quantity However, used. is initial a capital when asentence of the beginning in except French, and English in initial case lower a with spelled are units of names and quantities of Names Spelling ofnames ofquantitiesand andEnglish languages French ofunits inthe 7.2.5 metre per second, joule per kilogram kelvin, not joule per kilogram per kelvin EXAMPLE 3 parentheses). (without “per” thenone more contain never shall name compound A names. two the between “per” word the byinserting formed is units two of quotient the of name The two and three may be expressed by “squared” and “cubed”, respectively. respectively. “cubed”, and “squared” by expressed be may three two and m s , m/s, m ·s m m/s, , newton metre newton metre a of a unit is the name of that unit followed by “to the power power the by “to that unit followed of the name aunitis of degree Celsius degree − 1 , m s m , , symbol °C, that contains a capital letter. letter. a capital contains °C, that , symbol − 1

© ISO 2009 – All rights reserved 2009 –© ISO rights All n ”. However, the powers thepowers However, ”. Provided by : www.spic.ir also be a space on both sides. sides. both on aspace be also comma or by any other means. means. other by any or comma a or point a by not and space small a by be separated shall groups the used, is three of groups into separation of the cross or the dot (see also 7.1.3). The multiplication cross ( cross multiplication The 7.1.3). also (see thedot or thecross of © ISO 2009 – All rights reserved 2009 –© ISO rights All EXAMPLE 1 ( across is numbers of multiplication for sign The division and Multiplication 7.3.3 The General Conference on Weights and Measures ( NOTE 2 Inaccordance with theISO/IEC Directives, Part2,2004, RulesStandards for thestructure and drafting ofInternational NOTE 1 EXAMPLE 0,567 zero. a by preceded be shall sign decimal 1,the than less is number the of value) (absolute themagnitude If 8 a within document. consistently used be should sign decimal same The line. on the apoint or comma a either is sign decimal The Decimal 7.3.2 sign ACelsius temperature from EXAMPLE 3 as such arelation, denoting Example signs in For given 7.1.3. also examples 4. See the in shown as symbol, or thesign of sides both on aspace be shall there symbols, and signs operations, Example (see bya space the number from for beseparated However, not 3). shall and operator monadic a is sign”, of “change or sign” “same toindicate used aquantity), (or anumber before sign minus or A plus 1935. e.g. aspace, without written be always year shall The 80000-1. ISO e.g. numbers, reference as used numbers ordinal for used be not should three of groups into separation The Inthe number “1 234 EXAMPLE 2 left. the towards digit most theright- from be shall counting the sign), decimal no thus (and part decimal no is there where case In the 1234,567 8rather than 1 234,5678 0,567 8rather than 0,5678 EXAMPLE 1 where thedecimal comma is always used. documents written in theFrench language (and anumber of It is customary to use the decimal point in most documents written in the English language, and the decimal comma in In practice, the choice between these alternatives depends on customary use in the language concerned. meeting in 2003 passed unanimously the following resolution: EXAMPLE 2 preferred for the multiplication of letter symbols. symbols. letter of themultiplication for preferred is and units incompound and scalars of aproduct indicate to used be also may It cases. comparable and vectors of product scalar a indicate to used be shall (·) dot half-high The products. Cartesian and products vector in and Examples in 2), and shown 1 (as values numerical and numbers of the multiplication from the decimal sign towards the left left the towards sign thedecimal from counting three, of groups into separated be may these digits, with many numbers of reading the facilitate To XML 5+ EXAMPLE 4 “The decimal marker shall be either apoint onthe line ora comma on the line.” , thedecimalcommais signInternationalinon theline a Standards. l A =2,5 × =80 mm × 2 5 10 3

m − 25 mm 3 ”, the right-most digit is that underlined. n

± 1,6 and − the right. No group shall contain more than three digits. Where such such Where digits. thanthree more contain shall group No right. the 7 C to + °C to D

< 2 mm × 5 ) or a half-high dot (·). There shall be a space on both sides sides both on a space be shall There (·). dot or a half-high ) °C. other European languages), except insome technical areas > 5mm Conférence Générale des Poids et Mesures × ) or half-high dot (·) shall be used to indicate indicate to used be shall (·) dot half-high or ) ISO 80000-1:2009(E) 80000-1:2009(E) ISO = , < and > , there shall shall , there ) at its) at 27

Provided by : www.spic.ir besignificant. to considered are sign adecimal after digits All is 10. is 28 the of magnitude maximum 401 401 between avalue torepresent assumed generally 008 is 401 and 007,5 the case, Inthis 008,5. Annex (see number the digit last example, the in for 1 to Thus, equal B). range arounding with rounded is digit last the that so interpreted generally is it information, further any without given is number a When uncertainty and Error 7.3.4 ambiguity. any avoid to inserted are parentheses unless line same the on sign division a or sign by amultiplication followed notbe shall (/) asolidus acombination, In such quotients. or products themselves bothare or denominator thenumerator, where tocases extended be can provisions These 10 EXAMPLE 4 ways: thefollowing of one in indicated is another by number one of Division EXAMPLE 3 4 711.32 80000-1:2009(E) ISO “Numbers expressed as digits” refers tonumbers such as “12”, as opposed to“twelve”. 3) 4 e.g. omitted, be may sign themultiplication cases, In some sets. of products and Cartesian products scalar products, vector of examples contains ISO numbers. for symbols ISO also multiplication of 80000-2 anoverview 2-9.5,gives Item 80000-2:2009, that distinction in the following way: following the in distinction that indicate to recommended is It magnitude. of order the indicate to used just are or significant are zeros three theright-most if notknown is but it digits, significant three 401 contains 401 thenumber 000. Here, Consider digit(s). thelast of limits Digits of a number are called called are number a of Digits 90. 40,100 80 and 40,100 between avalue sometimes or 40,100 between 40,100 75 and a value 85 torepresent 40,100 assumed number generally the is 8 Similarly, 1. and 0 between is error the 401 401 and and 009,0 008,0 between value a 401 represents number 008 the case, 401 Inthis 008. becomes e.g.401 008,91 digits), thelast off cutting by simply (i.e. by truncation replaced Negative exponents should be avoided when the numbers are expressed with digits with expressed are numbers the when avoided be should exponents Negative the half-high dot may be used as the multiplication sign between numbers expressed with digits with expressed numbers between sign multiplication the as used be may dot the half-high sign between numbers expressed withdigits expressed numbers between sign multiplication the as beused should dot half-high not the and cross the sign, decimal the as used is thepoint If 401,0 401,0 401 401,000 401,000 401,00 401,00 b a

× 10 × a × 10

× 10 3 /

b 10 3

four four 3

− 3 3

is acceptable 3 a

b –1 ×

0.351 2 4711,32 ·0,351 2 4 711,32 error significant digits significant a · three significant digits digits significant three six significant digits five significant digits digits significant five

b in the number 401 008 is 0,5. However, in some applications rounding is is rounding 401 applications number inthe some in However, 0,5. 008 is –1 significant significant

− 3 shouldavoided be 3) if the corresponding number is considered to lie within the error error the within lie to considered is number corresponding the if . If the comma is used as the decimal sign, both the cross and cross boththe sign, thedecimal as used is thecomma . If digits digits c − 5 d , 6 × 0,351 2 ab , 7( a

b ), 3 ln 2. 3ln ), © ISO 2009 – All rights reserved 2009 –© ISO rights All 3) , except when the base when the base , except 3) . Provided by : www.spic.ir The number of atoms of a nuclide in a molecule is shown in the right subscript position, e.g. e.g. position, subscript right the in shown is a molecule in a nuclide of atoms of number The uncertainty may be expressed as exemplified by: by: exemplified as expressed be may uncertainty associated the and value anumerical normal, is quantity corresponding the for distribution assumed © ISO 2009 – All rights reserved 2009 –© ISO rights All Ifthe number of atoms is equal to1, it is not indicated, e.g. H NOTE e.g. position, superscript left in the shown is anuclide of number) (mass number nucleon The positions. and meanings the following have shall molecule or anuclide specifying superscripts and subscripts Attached 80000-9. ISO in given is elements chemical for symbols of list A complete EXAMPLE 1 asentence. of end at the e.g. punctuation, normal for except stop a full by followed be not shall symbol The case. lower is any, if letter, afollowing and acapital is letter initial The theLatinalphabet. from two oneor letters of consist symbols The thetext. of rest inthe used thetype of irrespective type, (upright) in roman beprinted shall elements chemical for Symbols elements and Chemical nuclides 7.4 Uncertainties areoftenexpressed in thefollowing manner: (23,478 2 NOTE EXAMPLE In the expression where Numerical values of quantities are often given with an associated an associated with given often are quantities of values Numerical 23,478 2thus encompassing all values between and including those limits. to 23,481 4 and lower limit equal to 23,475 0) having an extent of 0,006 4(2 Note that in the context ofengineering tolerances, 23,478 2 with expanded uncertainty”. has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused values. According to ISO/IEC Guide 98-3:2008, 7.2.2 note, “The a mathematicalfrom of view. point 23,478 2 b a l l 14 14 l is a length expressed in the unit metre, m; m; metre, unit the in expressed alength is

denotes a standard uncertainty (see ISO/IEC Guide 99) expressed in terms of the least significant significant least the of terms ISO/IEC in (see expressed 99) uncertainty Guide astandard denotes value; numerical the is = = N N a 23,478 2(32) m m 2(32) 23,478 2 ( 32 represents a standard uncertainty equal to 0,003 2. 0,003 to equal uncertainty astandard represents 32 value; numerical the is 2 23,478 l digit(s) in digit(s)

b is the length expressed in the unit metre, m; m; metre, unit in the expressed length the is ) m

H He C Ca a .

± 0,003 2 means 23,481 4 or 23,475 0, but not all values between these two ± 0,003 2 expresses the limits of azone (i.e. upper limit equal ± format should be avoided whenever possible because it 2 O. O. standard uncertainty standard × 0,003 2) symmetricallydispersed around ± 0,003 2) m. This is, however, wrong ISO 80000-1:2009(E) 80000-1:2009(E) ISO . Provided that the the that Provided . 29

Provided by : www.spic.ir A state of ionization or an excited state is shown in the right superscript position. position. superscript right in the shown is state excited an or ionization of A state 30 7.5 Greek alphabet EXAMPLE 2 80000-1:2009(E) ISO e.g. position, subscript left in the shown is anuclide of number) (atomic number proton The tt finzto: Na State of ionization: Nuclear excited state: 64 Gd 110 phi ypsilon tau sigma rho pi omicron xi nu mu lambda kappa iota theta eta zeta epsilon delta gamma beta alpha omega psi chi Ag + , PO ae Romantype Name m

3 4 − or (PO Φ Υ Τ Σ Ρ Π Ο µ Ξ Ν Μ Λ Κ Ι Θ Η Ζ Ε ∆ Γ Β Α ω ψ Ω Ψ Χ Table 7 — Greek letters letters 7—Greek Table 4 ) 3 −

ζ δ β ϕ υ τ σ ρ π ο ξ ν ϰ ϑ η ε γ α χ λ ι , , , , , ,

є ϱ ϖ κ  φ θ

Ζ Ζ Ε ∆ Γ Β Α Ω ω ω ψ Ω Ψ Χ Φ Υ Τ Σ Ρ Π µ Ο Ξ Ν Μ Λ Κ Ι Θ Η

Italic type ζ ζ δ δ β ε ϕ π ϑ ρ γ γ α χ χ υ τ σ ο ξ ν ϰ η λ λ ι , , , , , ,

є ϱ ϖ κ κ  φ φ θ 

© ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir © ISO 2009 – All rights reserved 2009 –© ISO rights All EXAMPLE 1 A.2.2 coefficient A.2.1 A.2 Coefficients, factors Names ofterms are greatly language-dependent and these recommendations apply mainly toEnglish. NOTE 2 Most oftheexamples in this annexrecommendations. aredrawn from NOTE 1 Note. inA.6.5, “molar” theterm see Also possible. wherever tension” “electric name the touse recommended is It English. than other languages many in voltage to corresponds tension” “electric name The voltage. as such rule, general this to exceptions few a are there However, unit. any corresponding of name the reflect not shall name aquantity expressed, are they which in theunit of independent always themselves are quantities Since examined. critically be should principles these from deviations terms, existing reviewing when Furthermore, quantities. new naming when followed be should presented principles the However, languages. scientific thevarious in incorporated been have which deviations frequent relatively the eliminate to rules strict impose to annex this of intention the not It is guidance. some need also may quantity aphysical of naming the symbol, an appropriate of the choice in as Just quantities. derived or related other indicate to quantities physical of names to added are etc., concentration, molar, density, specific, like terms Similarly, etc. constant, ratio, parameter, factor, coefficient, like terms with incombination formed commonly is name a exists, aquantity for name nospecial If A.1 General the multiplicative relation relation the multiplicative EXAMPLE 2 Sometimes, the term “modulus” isused instead of the term “coefficient”. NOTE diffusion coefficient: Hall coefficient: A modulus of elasticity: linear expansion coefficient: The term “coefficient” should be used when the two quantities quantities two the when used be should “coefficient” term The aquantity conditions, certain under If, or or a

factor H

. D E

Terms in names for physical quantities physical quantities namesfor Terms in A

= α

k l d · B . The quantity quantity . The σ E J

H = / = l

–

= = E

A ε α

l grad d (B T ×

A J n (normative) )

is proportional to another quantity quantity toanother proportional is Annex A k that occurs as a multiplier in this equation is often called a called often is equation in this amultiplier as occurs that

existing practice andare notintended toconstitute new

A and B have different dimensions. dimensions. different have B , this can be expressed by by can be expressed , this ISO 80000-1:2009(E) 80000-1:2009(E) ISO 31

Provided by : www.spic.ir transport phenomena, are called called are phenomena, transport called sometimes are quantities Such quantities. 32 EXAMPLE 1 A.3.3 EXAMPLE A.3.2 EXAMPLE A.3.1 Parameters, numbers, ratios A.3 EXAMPLE A.2.3 80000-1:2009(E) ISO EXAMPLE aquantity, of theratio of logarithm The Levels A.4 EXAMPLE 3 recommended. not is usage this of Extension ratio. of instead used sometimes is “index” term The EXAMPLE 2 theterm Sometimes, Reynolds number: friction factor: quality factor: coupling factor: level of apower quantity: heat capacityratio: Grüneisen parameter: mobility ratio: Prandtl number: Pr thermal diffusion ratio: refractive index: n packing fraction: amount-of-substance fraction of B: Quotients of dimension one of two quantities are often called called often are twoquantities of one dimension of Quotients of description the in occurring those as such quantities, of one dimension of combinations Some new constitute to considered often are equations in such as occur which quantities of Combinations quantities whenthetwo be used should “factor” term The

b Q µ

“fraction”

γ k

L P

is used for ratios smaller than one. than smaller ratios for used is characteristic numbers characteristic Re F γ L |X Pr

mn x

= | =

= = =

µF =

V Q η QR ρ

k /( v c

n , and a reference value of that quantity, quantity, that of value areference and , p (L /

η /

λ m c

V L ρ x L b k n f γ n

B T ) )

P = = 1/2 = =

= =

∆ C µ c

ln( parameters D r 0 n – p / / / B T A c / µ C /

P + /

n D V

0 ) and carry the term “number” in their names. names. their in “number” term the andcarry . A and ratios B have the same dimension. dimension. the same have . Q 0 , is called a called , is © ISO 2009 – All rights reserved 2009 –© ISO rights All level . Provided by : www.spic.ir quotient of that quantity by the area. Also, the noun “area” is occasionally used. used. occasionally is “area” noun the Also, the area. by quantity that of quotient andA.6.4. A.6.3 See also thevolume. by quantity that of quotient mass. recommended. not is usage this of Extension “constant”. term the including names given sometimes also are calculations mathematical constant constant © ISO 2009 – All rights reserved 2009 –© ISO rights All EXAMPLE 1 A.6.3 EXAMPLE A.6.2 EXAMPLE A.6.1 general Terms with application A.6 EXAMPLE A.5.3 decay constant for a particular nuclide: EXAMPLE matter of constant A.5.2 EXAMPLE A.5.1 Constants A.5 term “constant”. “constant”. term “constant”. “constant”. OE The adjective “mass” or the adjective “massic” is sometimes used instead of the adjective “specific”. NOTE Madelung constant for a particular lattice: gravitational constant: Planck constant: areic massareicor surface mass density: mass: volumic or density mass capacity: heat specific equilibrium constant for a chemical reaction (which varies with temperature): number density orvolumic number: areic electric charge or surface electric density: energy density orvolumic energy: e electric charge density orvolumic electric charge: specific entropy: volume: specific specific activity: The term “surface … density” or the adjective “areic” is added to the name of a quantity to indicate the the indicate to aquantity of name tothe added is “areic” adjective the or …density” “surface term The the indicate to aquantity of name addedtothe is “volumic” adjective the or noun“density” The by quantity that of quotient the indicate to aquantity of name the to added is “specific” adjective The from thatresult or circumstances particular under value thesame keep that quantities Other a called is substance, aparticular for circumstances all under value same the has that A quantity a called is circumstances all under value same the have to considered is that quantity A or

a fundamental physical constant physical fundamental a v s . Again, provided no special name exists, the name of such a quantity includes the term term the includes quantity a such of name the exists, name no special provided Again, . h

G c

c s v a

ρ = =

= = A

S C/

V A / / / m m m m α

. Unless a special name exists, its name explicitly includes the the includes explicitly name its exists, name aspecial Unless . ρ A ρ

ρ ρ ρ ρ n e

A A = = = =

= =

E N/ Q m

Q m / / / V V V V / /

A A

K p

ISO 80000-1:2009(E) 80000-1:2009(E) ISO universal 33

Provided by : www.spic.ir amount of substance. substance. of amount to indicate the quotient of that quantity by the total volume. volume. total the by quantity that of quotient the to indicate 34 EXAMPLE 2 quantities. similar between distinguish to solely aquantity of name the to added also is “linear” term The EXAMPLE 1 A.6.4 EXAMPLE 2 80000-1:2009(E) ISO function. distribution spectral a denote to English in used is concentration” “spectral term The EXAMPLE A.6.6 EXAMPLE A.6.5 A.6.2. also See area. surface the by aquantity such of quotient the indicate to a current or a flux expressing quantity a of name the to added is “areic” adjective the or “density” noun The quantity to indicate the quotient of that quantity by the length. Also, the noun “line” is occasionally used. used. occasionally is “line” noun the Also, length. the by quantity that of quotient the indicate to quantity OE However, theterm“molar” corresponding unit (in this case, mole). See A.1. NOTE magnetic flux density or areic magnetic flux: electric current density or areic electric current: mass concentration ofB: linear attenuation coefficient: linear expansion coefficient: mean linear range: molecular concentration of B: amount-of-substance concentration ofB: molar volume: cubic expansion coefficient: linear mass density orlineic mass: density ofheat flow rateor areic heat flow rate: mass attenuation coefficient: molar mass: molar internal energy: mean mass range: The adjective “molar” is added to the name of a quantity to indicate the quotient of that quantity by the the by thatquantity of quotient the indicate to aquantity of name the to added is “molar” adjective The density” … “linear term The The term “concentration” is added to the name of a quantity, especially for a substance in a mixture, amixture, in a substance for especially aquantity, of name the to added is “concentration” term The

V m

R R ρ

U m ρ

α α µ µ V C l

U V M violates the principle that the name of the quantity shall not reflect the name of a l

m m or the adjective “linear” or the adjective “lineic” is added to the name of a of name the to added is “lineic” adjective the or “linear” adjective the or

= =

V U c / n / B / n

B R µ µ α α ρ R

l m

ρ V l

= = J q =

= =

= – =

l R V J µ R − / ρ 1 − / l − i ρ

d 1

c C ρ / 1 n

d l d B B B /d

= /d = = T /d

x m N T B

q J B B B

/ / = = = V V

Φ Φ A / / A

© ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir XML EXAMPLE 2 EXAMPLE 2 use. use. number. © ISO 2009 – All rights reserved 2009 –© ISO rights All EXAMPLE 1 number. rounded as is the A: selected multiple even Rule The B.3 EXAMPLE 1 B.2 EXAMPLE 1 B.1 number EXAMPLE 2 ie ubr rounded number given number rounding range: 10 rounded number given number rounding range: 10 integral multiples: 12,1; 12,2; 12,3; 12,4; etc. rounding range: 0,1 ie ubr rounded number given number rounding range: 0,1 rounded number given number rounding range: 0,1 rounding range: 10 1 1 235,0 1 225,0 1 240 220 1 1 227,5 1 1 225,1 1 230 223,3 1 230 220 12,35 12,25 12,4 12,2 12,275 12,3 12,251 12,3 12,223 12,2 integral multiples: 1210; 1 220; 1230; 1 240; etc. Rounding means replacing the magnitude of a given number by another number called the the called number by another number agiven of themagnitude replacing means Rounding If there are two successive integral multiples equally near the given number, two different rules are in in are rules twodifferent number, given the near equally multiples integral successive two are there If therounded as accepted is thenthat number, given the nearest multiple one integral only is there If , selected from the sequence of integral multiples of a chosen rounding range. range. rounding achosen of multiples integral of thesequence from , selected

Rounding of numbers ofnumbers Rounding (normative) Annex B

ISO 80000-1:2009(E) 80000-1:2009(E) ISO rounded rounded 35

Provided by : www.spic.ir respected, it is advisable to round only in one direction. inone direction. only to round itis advisable respected, be haveto limits other or requirements safety where incases instance, For account. into be taken to have step. one only in out be carried always shall therounding Therefore, 36 B.4 computers. in used sometimes is B Rule minimized. are errors rounding the way that a in such measurements of series example, for treating, when advantage special of and preferable generally is A Rule EXAMPLE 1 rounded asselected the number. multiple is greaterRule inmagnitude B:The 80000-1:2009(E) ISO B.6 B.5 12,254 should be rounded to 12,3 and not first to 12,25 and then to 12,2. EXAMPLE XML EXAMPLE 2 ie ubr rounded number given number rounding range: 0,1 ie ubr rounded number given number rounding range: 10 − 12,35 12,25 12,4 12,3 − 1 1 235,0 1 225,0 1 240 230 − − 22 12,25 2, 1 225,0 12,35 3, 1 235,0 The rounding range should always be indicated. indicated. be always should range rounding The number therounded of theselection for criteria nospecial if only used be should above given rules The toerrors. lead may above given therules of application the by stage thanone inmore Rounding

− − − − 12,3 1 230 12,4 1 240 © ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir decimal logarithm. logarithm. decimal the on based is bel The considered. is phase, the not and amplitude, the only where applications in common logarithms of the same base shall be used. used. be baseshall thesame of logarithms only application, of field Inagiven quantities. different thus are and values different have but other each to proportional are bases with different defined Quantities itself. theargument does as consideration under situation physical the about information same the gives argument an of base specified toany logarithm The c) other other c) system. Earlier such quantities have been called called been have quantities such Earlier system. © ISO 2009 – All rights reserved 2009 –© ISO rights All A quantities Logarithmicroot-power C.2 a) follows: as classified are quantities logarithmic logarithm, the of theargument of source on the Depending bespecified. must logarithm the of the base complete, be to definition a For functions. logarithmic of means by defined quantities are quantities Logarithmic C.1 General Nonetheless the bel, symbol B, and its submultiple decibel, symbol dB, is is dB, symbol decibel, submultiple its B, and symbol bel, the Nonetheless logarithms. natural and notation complex from naturally result angle, thephase for rad, with radian, together amplitude, Np,for neper, calculations, In theoretical unit. SI an as CGPM by adopted yet not is it but SI, with the coherent theunit becomes Np, symbol neper, theunit logarithms, natural ISQ. With Quantities, of System International the in used are logarithms why natural is That used. is logarithm natural the if, only and if, simpler become also operations and relations mathematical other Many logarithm. natural the with, only and with, done usefully is ratios complex-quantity of logarithms of taking The acoustics. and in telecommunications example for quantities, root-power sinusoidal for used frequently is notation Complex cycle. one is interval time appropriate the quantity, aperiodic For used. is specified be to interval time appropriate an over value square root-mean the quantities, root-power non-sinusoidal For logarithm. the of argument the is value square root-mean the quantities, root-power time-varying sinusoidal For position vector vector position on depending aquantity i.e. meaning, another has quantity field because deprecated now is name this b) b) root-power quantity root-power theory quantities theory logarithmic quantities logarithmic logarithmic ratios logarithmic quantities logarithmic r . ; that are defined by the logarithm of the ratio of two quantities of the same kind; kind; thesame of quantities two of the ratio of logarithm the by defined are that is a quantity, the square of which is proportional to power when it acts on a linear linear a on acts it when power to proportional is which of square the quantity, a is , in which the argument is given explicitly as a number, e.g. e.g. number, a as explicitly given is argument the which in ,

Logarithmic quantities and their units units andtheir quantities Logarithmic . (normative) Annex C field quantities field

in connection with logarithmic quantities, but but quantities, logarithmic with connection in ― for historical reasons reasons historical for ISO 80000-1:2009(E) 80000-1:2009(E) ISO logarithmic information- logarithmic ― very very 37

Provided by : www.spic.ir factor 1/2 is included in the definition of the logarithmic power quantity. quantity. power logarithmic the of definition the in included 1/2is factor the because same the is quantity logarithmic the of value numerical the quantity, root-power corresponding a of square the to proportional is quantity power a When context. this in quantities power as labelled also are 38 Hart, hartley, respectively. nat;and information, unit of natural Sh; shannon, are: quantities thecorresponding of units The and 10. e, 2, are thelogarithm of bases three The used. are bases different with three logarithms theory, In information Logarithmicinformation-theory quantities C.4 a called is power to proportional is that quantity A Logarithmicpower quantities C.3 80000-1:2009(E) ISO

power quantity power . In many cases, energy-related quantities quantities energy-related cases, many In . © ISO 2009 – All rights reserved 2009 –© ISO rights All Provided by : www.spic.ir responsible for the for responsible is JCGM The and OIML). IUPAP IUPAC, ISO, ILAC, IFCC, IEC, (BIPM, Organizations International Vocabulary of Measurements of Vocabulary and units and units of the CIPM. theCIPM. of © ISO 2009 – All rights reserved 2009 –© ISO rights All the is IUPAC The IUPACD.7 the ISO is D.6 ISO the is ILAC The ILACD.5 the IFCCis The IFCC D.4 the is IEC The IEC D.3 the is CGPM The CGPM D.2 The D.1 JCGM The Secretariat of the JCGM is administered by the BIPM. BIPM. the by administered is JCGM the of Secretariat The states. The CIPM is the the is CIPM The states. of CGPM. The BIPM is the is BIPM The CGPM. of

JCGM is the is JCGM . . International organizations in the field of quantities and ofquantities units in thefield organizations International ― IEC/TC25 ― ISO/TC 12 International Organization for Standardization for Organization International ― CIPM International Electrotechnical Commission Electrotechnical International International Laboratory Accreditation Cooperation Accreditation Laboratory International International Federation of Clinical Chemistry and Laboratory Medicine Laboratory and Chemistry ofClinical Federation International International Union of Pure and Applied Chemistry Applied of and Union Pure International Joint Committee for Guides in Metrology in Guides for Committee Joint General Conference on Weights and Measures and onWeights Conference General Guide to the Expression of Uncertainty in Metrology in ofUncertainty totheExpression Guide International Committee for Weights Measures and Weights for Committee International ― BIPM

International Bureau of Weights and Measures and ofWeights Bureau International , the VIM. VIM. , the (informative) Annex D

. ISO/TC . ISO/TC 12 is . IEC/TC . IEC/TC 25 is . It is an International Joint Committee of eight eight of Committee Joint anInternational is . It

. . , consisting of delegates of all member member all of delegates of , consisting Technical Committee Technical Technical Committee Technical and operates under the supervision thesupervision under operates and , the GUM, and the andthe GUM, , the and operates under the authority authority the under operates and ISO 80000-1:2009(E) 80000-1:2009(E) ISO . 12, 12, 25, 25, International Quantities Quantities 39

Provided by : www.spic.ir the 40 the is OIML The D.9 OIML the is IUPAP The IUPAPD.8 80000-1:2009(E) ISO

the BIML, the BIML, the General Conference on Legal Metrology Legal on Conference General ― CGML International Bureau of Legal Metrology Legal of Bureau International International Organization of Legal Metrology Legal of Organization International International Union of Pure and Applied Physics Applied of Union and Pure International ― CIML ― BIML ; the CIML, the the CIML, ; the .

International Committee of Legal Metrology ofLegal Committee International . The bodies of this organization are the CGML, CGML, the are organization this of bodies . The . © ISO 2009 – All rights reserved 2009 –© ISO rights All ; and Provided by : www.spic.ir [6] IEC [6] 60027-2, [3] ISO/IEC Directives, Part 2, 2004, 2004, 2, Part Directives, ISO/IEC [3] [7] IEC [7] 80000-13:2008, © ISO 2009 – All rights reserved 2009 –© ISO rights All ISO [1] 80000-2:2009, [4] ISO/IEC Guide 98-3:2008, 98-3:2008, Guide ISO/IEC [4]

[9] [9] 2006, Constants: Physical Fundamental the of Values Recommended CODATA [2] ISO ISO [2] 80000-9, [8] BIPM, [8] [5] IEC [5] 60027-1, natural sciences andtechnology sciences natural in measurement (GUM:1995) in measurement electronics http://physics.nist.gov/cuu/Constants/index.html The International System of Units (SI) Units of System International The

Letter symbols to be used in electrical technology — Part 1: General Part — technology electrical in used tobe symbols Letter Quantities and units — Part 9: Physical chemistry and molecular physics molecular and chemistry 9: Physical Part — units and Quantities Letter symbols to be used in electrical technology — Part Part — technology electrical in and used be to symbols Telecommunications Letter 2: Quantities and units — Part in the —Part used units tobe and symbols and Quantities signs Mathematical 2: Quantities and units — Part 13: Information science and technology and science Information 13: — Part units and Quantities

Uncertainty of measurement — Part 3: Guide to the expression of uncertainty uncertainty of expression the to 3:Guide Part — ofmeasurement Uncertainty

Rules for the structure and drafting of International Standards International of drafting and structure the for Rules Bibliography Bibliography , 8 th

edition (2006), http://www.bipm.org/en/si/ (2006), edition

ISO 80000-1:2009(E) 80000-1:2009(E) ISO