International system of units

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Guide for the Use of the International System of Units (SI) m kg s SI mol cd K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. TaylorGuide for the Use of the International System of Units (SI) 1 Introduction 1.1 Purpose of Guide The International System of Units was established in 1960 by the 11th General Conference on Weights and Measures (CGPM— see Preface). Universally abbreviated SI (from the French Le Système International d’Unités), it is the modern metric system of measurement used throughout the world. This Guide has been prepared by the National Institute of Standards and Technology (NIST) to assist members of the NIST staff, as well as others who may have need of such assistance, in the use of the SI in their work, including the reporting of results of measurements. 1.2 Outline of Guide The Preface gives the principal Federal Government actions taken since 1988 regarding the SI and introduces the international body— the CGPM—that is responsible for the SI. A check list immediately follows the Preface to help NIST authors review the conformity of their manuscripts with proper SI usage and the basic principles concerning quantities and units. A detailed Contents, the aim of which is to simplify the use of the Guide, follows the check list. This introductory chapter gives the purpose of the Guide and its outline, while Chapter 2 summarizes and clarifies the NIST policy on the use of the SI in NIST publications. Chapter 3 notes the existence of a number of publications on the SI and gives the two organizational units at NIST to which questions concerning the SI may be directed and from which additional information about the SI may be obtained. Chapter 4 discusses the fundamental aspects of the SI, including the two current classes of SI units: base, and derived; those derived units that have special names and symbols, including the degree Celsius; and the SI prefixes that are used to form decimal multiples and submultiples of units. Chapter 5 discusses units that are outside the SI and indicates those that may be used with it and those that may not. It also gives (see Sec. 5.4) precise definitions of the terms “SI units” and “acceptable units” as used in this Guide. Chapter 6 gives the rules and style conventions for printing and using units, especially unit symbols and SI prefix symbols. Chapters 7 and 8, which some readers may view as the most important parts of this Guide, provide, respectively, the rules and style conventions for expressing the values of quantities, and clarifying comments on some often troublesome quantities and their units. Chapter 9 gives the rules and style conventions for spelling unit names. Chapter 10 further elaborates on printing and using symbols and numbers in scientific and technical documents and is intended to assist NIST authors prepare manuscripts that are consistent with accepted typesetting practice. Appendix A gives the definitions of the SI base units, while Appendix B gives conversion factors for converting values of quantities expressed in units that are mainly unacceptable for use with the SI to values expressed mainly in SI units. Appendix B also includes a simplified discussion of rounding numbers and rounding converted numerical values of quantities. Appendix C discusses in some detail most of the references included in Appendix D—Bibliography, which concludes the Guide. 1 Guide for the Use of the International System of Units (SI) 2 NIST Policy on the Use of the SI In accordance with various Federal Acts, the Code of Federal Regulations, and Executive Order 1 12770 (see Preface), it is NIST policy that the SI shall be used in all NIST publications. When the field of application or the special needs of users of NIST publications require the use of other units, the values of quantities shall first be expressed in acceptable units, where it is to be understood that acceptable units include the SI units and those units recognized for use with the SI; the corresponding values expressed in the other units shall then follow in parentheses. (For precise definitions of the terms “SI units” and “acceptable units” as used in this Guide, see Sec. 5.4.) Exceptions to this policy require the prior approval of the NIST Director. The following three sections—2.1 Essential data, 2.1.1 Tables and graphs, and 2.2 Descriptive information—elaborate upon this policy. 2.1 Essential data Essential data express or interpret quantitative results. All such data shall be given in acceptable units. In those cases where — the sole use of acceptable units would compromise good communication, or — units other than acceptable units have been specified as a contractual requirement, values of quantities shall be given in acceptable units followed, in parentheses, by the values of the same quantities given in the other units. Exceptions may sometimes be necessary for commercial devices, technical standards, or quantities having special legal significance; examples include commercial weights and measures devices and the related laws and regulations. However, even in such cases, values of quantities expressed in acceptable units should be used when possible with the same values expressed in other units following in parentheses. 2.1.1 Tables and graphs In tables, values of quantities expressed in acceptable units and the corresponding values expressed in other units may be shown in parallel columns, with the acceptable-unit column preceding the other-unit column. In graphs, axes labeled in other units shall be given secondary status. This may preferably be done by placing scale marks on and labeling the left-hand ordinate and bottom abscissa in acceptable units, and placing scale marks on and labeling the right-hand ordinate and top abscissa in other units. Alternatively, lighter-weight scale marks and smaller type may be employed to indicate other units using the same ordinate and abscissa as is used for the acceptable units. 2.2 Descriptive information Descriptive information characterizes arrangements, environments, the generalized dimensions of objects, apparatus, or materials, and other attributes that do not enter directly into calculations or results. When necessary for effective communication, such information may be expressed using customary terms that are widely used and recognized. Examples include common drill sizes and traditional tools used in the United States, U.S. standard fastener sizes, commercial pipe sizes, and other common terms used in the trades, the professions, the marketplace, sports, and various social activities. When such descriptive information is given, values in acceptable units are not required. For example, it is permissible to refer to a “36-inch pipeline” or a “half-inch drill” without first giving the value in an acceptable unit. 1 The NIST policy on the use of the SI is set forth in the NIST Administration Manual, Chapter 4, Communications, Subchapter 4.09, NIST Technical Communications Program, Appendix D—Use of Metric Units. 2 Guide for the Use of the International System of Units (SI) 3 Other Sources of Information on the SI 3.1 Publications Appendix C briefly describes a number of publications that deal with the SI and related topics; citations for these publications are given in Appendix D—Bibliography. Additional information about the SI is also available from the two NIST organizational units indicated in Secs. 3.2 and 3.3. 3.2 Fundamental Constants Data Center Questions concerning the more fundamental aspects of the SI and subtle aspects of proper SI usage may be directed to: Fundamental Constants Data Center Physics Laboratory National Institute of Standards and Technology 100 Bureau Drive, Stop 8420 Gaithersburg, MD 20899-8420 Telephone: (301) 975-3200 Fax: (301)-990-1350 SI Units Web Page: http://physics.nist.gov/cuu/Units/index.html 3.3 Metric Program Questions concerning Federal Government use of the SI and Federal Government policy on the use of the SI by U.S. industry and the public may be directed to: Metric Program Technology Services National Institute of Standards and Technology 100 Bureau Drive, Stop 2600 Gaithersburg, MD 20899-2600 Telephone: (301) 975-4004 Fax: (301) 975-8091 Email: TheSInist.gov http://nist.gov/metric 4 The Two Classes of SI Units and the SI Prefixes Since the 1995 edition of this Guide, the 20th CGPM, which met October 9 - 12, 1995, decided to eliminate the class of supplementary units as a separate unit class in the SI. The SI now consists of only two classes of units: base units and derived units. The radian and steradian, which were the two supplementary units, are now subsumed into the class of SI derived units. Thus the SI units are currently divided into base 2 units and derived units, which together form what is called “the coherent system of SI units.” The SI also includes the prefixes to form decimal multiples and submultiples of SI units. 2 According to Ref. 4: ISO 31-0, a system of units is coherent with respect to a system of quantities and equations if the system of units is chosen in such a way that the equations between numerical values have exactly the same form (including the numerical factors) as the corresponding equations between the quantities (see Secs. 7.11 and 7.14). In such a coherent system, of which the SI is an example, no numerical factor other than the number 1 ever occurs in the expressions for the derived units in terms of the base units. 3 Guide for the Use of the International System of Units (SI) 4.1 SI base units Table 1 gives the seven base quantities, assumed to be mutually independent, on which the SI is founded, and the names and symbols of their respective units, called “SI base units.” Definitions of the SI base units are given in Appendix A. The kelvin and its symbol K are also used to express the value of a temperature interval or a temperature difference (see Sec. 8.5). Table 1. SI base units SI base unit Base quantity Name Symbol length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd 4.2 SI derived units Derived units are expressed algebraically in terms of base units or other derived units. The symbols for derived units are obtained by means of the mathematical operations of multiplication and division. For example, the derived unit for the derived quantity molar mass (mass divided by amount of substance) is the kilogram per mole, symbol kg/mol. Additional examples of derived units expressed in terms of SI base units are given in Table 2. (The rules and style conventions for printing and using SI unit symbols are given in Secs. 6.1.1 to 6.1.8.) Table 2. Examples of SI coherent derived units expressed in terms of SI base units SI coherent derived unit Derived quantity Name Symbol 2 area square meter m 3 volume cubic meter m speed, velocity meter per second m/s 2 acceleration meter per second squared m/s −1 wavenumber reciprocal meter m 3 density, mass density kilogram per cubic meter kg/m 3/ specific volume cubic meter per kilogram m kg 2 current density ampere per square meter A/m magnetic field strength ampere per meter A/m 2 luminance candela per square meter cd/m amount-of-substance concentration 3 amount concentration , concentration mole per cubic meter mol/m 4.2.1 SI coherent derived units with special names and symbols Certain SI coherent derived units have special names and symbols; these are given in Table 3. Consistent with the discussion in Sec. 4, the radian and steradian, which are the two former supplementary units, are included in Table 3. The last four units in Table 3 were introduced into the SI for reasons of safeguarding human health. 4 Guide for the Use of the International System of Units (SI) Table 3. The 22 SI coherent derived units with special names and symbols. (a) SI coherent derived unit Special name Special Expression in Expression in symbol terms of other terms of SI base SI units units (b) (b) plane angle radian rad 1 m/m (b) (c) (b) 2 2 solid angle steradian sr 1 m /m (d) −1 frequency hertz Hz s −2 force newton N m · kg · s 2−1−2 pressure, stress pascal Pa N/m m · kg · s 2 −2 energy, work, joule J N · m m· kg · s amount of heat 2−3 power, radiant flux watt W J/s m · kg · s electric charge, coulomb C s · A amount of electricity (e) 2−3 –1 electric potential difference , volt V W/A m · kg · s · A electromotive force −2−1 4 2 capacitance farad F C/V m · kg · s · A 2−3−2 electric resistance ohm Ω V/A m · kg · s · A −2−1 3 2 electric conductance siemens S A/V m · kg · s · A 2−2−1 magnetic flux weber Wb V · s m · kg · s · A 2−2−1 magnetic flux density tesla T Wb/m kg · s · A 2−2−2 inductance henry H Wb/A m · kg · s · A (f) Celsius temperature degree Celsius ºC K (c) luminous flux lumen lm cd · sr Cd 2−2 illuminance lux lx lm/m m · cd (d)−1 activity referred to becquerel Bq s (g) a radionuclide 2−2 absorbed dose, gray Gy J/kg m · s specific energy (imparted), kerma (h) 2−2 dose equivalent, sievert Sv J/kg m · s ambient dose equivalent, directional dose equivalent, personal dose equivalent −1 catalytic activity katal kat s · mol (a) The SI prefixes may be used with any of the special names and symbols, but when this is done the resulting unit will no longer be coherent. (See Sec. 6.2.8.) (b) The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities. (See Sec 7.10.) (c) In photometry the name steradian and the symbol sr are usually retained in expressions for units. (d) The hertz is used only for periodic phenomena, and the becquerel is used only for stochastic processes in activity referred to a radionuclide. (e) Electric potential difference is also called “voltage” in the United States. (f) The degree Celsius is the special name for the kelvin used to express Celsius temperatures. The degree Celsius and the kelvin are equal in size, so that the numerical value of a temperature difference or temperature interval is the same when expressed in either degrees Celsius or in kelvins. (See Secs. 4.2.1.1 and 8.5.) (g) Activity referred to a radionuclide is sometimes incorrectly called radioactivity. (h) See Refs. 1, 2, on the use of the sievert. 4.2.1.1 Degree Celsius In addition to the quantity thermodynamic temperature (symbol T), expressed in the unit kelvin, use is also made of the quantity Celsius temperature (symbol t) defined by the equation t = T − T 0, 5 Guide for the Use of the International System of Units (SI) where T = 273.15 K by definition. To express Celsius temperature, the unit degree Celsius, symbol ºC, 0 which is equal in magnitude to the unit kelvin, is used; in this case, “degree Celsius” is a special name used in place of “kelvin.” An interval or difference of Celsius temperature, however, can be expressed in the unit kelvin as well as in the unit degree Celsius (see Sec. 8.5). (Note that the thermodynamic temperature T is 0 exactly 0.01 K below the thermodynamic temperature of the triple point of water (see Sec. A.6).) 4.2.2 Use of SI derived units with special names and symbols Examples of SI derived units that can be expressed with the aid of SI derived units having special names and symbols are given in Table 4. Table 4. Examples of SI coherent derived units expressed with the aid of SI derived units having special names and symbols. SI coherent derived unit Derived quantity Name Symbol Expression in terms of SI base units −1−1 dynamic viscosity pascal second Pa · s m · kg · s 2−2 moment of force newton meter N · m m · kg · s –2 surface tension newton per meter N/m kg · s −1−1−1 angular velocity radian per second rad/s m · m · s = s 2−1−2−2 angular acceleration radian per second squared rad/s m · m · s = s 2−3 heat flux density, watt per square meter W/m kg · s irradiance 2−2−1 heat capacity, entropy joule per kelvin J/K m · kg · s · K 2−2−1 specific heat capacity, joule per kilogram kelvin J/(kg · K) m · s · K specific entropy 2−2 specific energy joule per kilogram J/kg m · s −3−1 thermal conductivity watt per meter kelvin W(m · K) m · kg · s · K 3−1−2 energy density joule per cubic meter J/m m · kg · s −3−1 electric field strength volt per meter V/m m · kg · s · A 3−3 electric charge density coulomb per cubic meter C/m m · s · A 2−2 surface charge density coulomb per square meter C/m m · s · A 2−2 electric flux density, coulomb per square meter C/m m · s · A electric displacement −3−1 4 2 permittivity farad per meter F/m m · kg · s · A −2−2 permeability henry per meter H/m m · kg · s · A 2−2−1 molar energy joule per mole J/mol m · kg · s · mol 2−2−1−1 molar entropy, joule per mole kelvin J/(mol · K) m · kg · s · K · mol molar heat capacity −1 exposure (x and γ rays) coulomb per kilogram C/kg kg · s · A 2−3 absorbed dose rate gray per second Gy/s m · s 4−2−3 2−3 radiant intensity watt per steradian W/sr m · m · kg · s = m · kg · s 2 2−2−3−3 radiance watt per square meter W/(m · sr) m · m · kg · s = kg · s steradian 3−3−1 catalytic activity katal per cubic meter kat/m m · s · mol concentration The advantages of using the special names and symbols of SI derived units are apparent in Table 4. Consider, for example, the quantity molar entropy: the unit J/(mol · K) is obviously more easily understood 2−2−1−1 than its SI base-unit equivalent, m · kg · s · K · mol . Nevertheless, it should always be recognized that the special names and symbols exist for convenience; either the form in which special names or symbols are used for certain combinations of units or the form in which they are not used is correct. For example, because of the descriptive value implicit in the compound-unit form, communication is sometimes facilitated if magnetic flux (see Table 3) is expressed in terms of the volt second (V · s) instead of the 2−2−1 weber (Wb) or the combination of SI base units, m · kg · s · A . 6 Guide for the Use of the International System of Units (SI) Tables 3 and 4 also show that the values of several different quantities are expressed in the same SI unit. For example, the joule per kelvin (J/K) is the SI unit for heat capacity as well as for entropy. Thus the name of the unit is not sufficient to define the quantity measured. A derived unit can often be expressed in several different ways through the use of base units and derived units with special names. In practice, with certain quantities, preference is given to using certain units with special names, or combinations of units, to facilitate the distinction between quantities whose values have identical expressions in terms of SI base units. For example, the SI unit of frequency is −1 specified as the hertz (Hz) rather than the reciprocal second (s ), and the SI unit of moment of force is specified as the newton meter (N · m) rather than the joule (J). Similarly, in the field of ionizing radiation, the SI unit of activity is designated as the becquerel (Bq) −1 rather than the reciprocal second (s ), and the SI units of absorbed dose and dose equivalent are designated as the gray (Gy) and the sievert (Sv), respectively, rather than the joule per kilogram (J/kg). 4.3 Decimal multiples and submultiples of SI units: SI prefixes Table 5 gives the SI prefixes that are used to form decimal multiples and submultiples of units. They allow very large or very small numerical values (see Sec. 7.1) to be avoided. A prefix name attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit. For example, 3 one kilometer, 1 km, is equal to one thousand meters, 1000 m or 10 m. When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent. (See footnote 2 for a brief discussion of coherence.) The rules and style conventions for printing and using SI prefixes are given in Secs. 6.2.1 to 6.2.8. The special rule for forming decimal multiples and submultiples of the unit of mass is given in Sec. 6.2.7. Table 5. SI prefixes Factor Prefix Name Symbol Factor Prefix Name Symbol 24 3 8 −1 10 = (10 ) yotta Y 10 deci d 21 3 7 −2 10 = (10 ) zetta Z 10 centi c 18 3 6 −3 3−1 10 = (10 ) exa E 10 = (10 ) milli m 15 3 5 −6 3−2 10 = (10 ) peta P 10 = (10 ) micro µ 12 3 4 −9 3−3 10 = (10 ) tera T 10 = (10 ) nano n 9 3 3 −12 3−4 10 = (10 ) giga G 10 = (10 ) pico p 6 3 2 −15 3−5 10 = (10 ) mega M 10 = (10 ) femto f 3 3 1 −18 3−6 10 = (10 ) kilo k 10 = (10 ) atto a 2 −21 3−7 10 hecto h 10 = (10 ) zepto z 1 −24 3−8 10 deka da 10 = (10 ) yocto y Note: Alternative definitions of the SI prefixes and their symbols are not permitted. For example, it is unacceptable to use kilo (k) to 10 20 30 represent 2 = 1024, mega (M) to represent 2 = 1 048 576, or giga (G) to represent 2 = 1 073 741 824. See the note to Ref. 5 on page 74 for the prefixes for binary powers adopted by the IEC. 5 Units Outside the SI Units that are outside the SI, that is, non-SI units, may be divided into three categories: — those units that are accepted for use with the SI by the CIPM and hence this Guide; — those units that are not accepted for use with the SI by the CIPM, but are temporarily accepted for use with the SI by this Guide; and — those units that are not accepted for use with the SI by either the CIPM or this Guide and in the view of this Guide must strictly be avoided. 5.1 Units accepted for use with the SI The following four sections discuss in detail the units this Guide accepts for use with the SI. 7 Guide for the Use of the International System of Units (SI) 5.1.1 Hour, degree, liter, and the like Certain units that are not part of the SI are essential and used so widely that they are accepted by the CIPM, and thus by this Guide, for use with the SI 2, 3. These units are given in Table 6. The combination of units of this table with SI units to form derived units should be restricted to special cases in order not to lose the advantages of the coherence of SI units. (The use of SI prefixes with the units of Table 6 is discussed in Sec. 6.2.8.) Additionally, this Guide recognizes that situations on occasion will require the use of time-related units other than those given in Table 6; such as using intervals of time be expressed in weeks, months, or years. In such cases, if a standardized symbol for the unit is not available, the name of the unit should be written out in full. (See Sec. 8.1 for a suggestion regarding the symbol for year and Chapter 9 for the rules and style conventions for spelling unit names.) Table 6. Non-SI units accepted for use with the SI by the CIPM and this Guide Name Symbol Value in SI units minute min 1 min = 60 s hour time h 1 h = 60 min = 3600 s day d 1 d = 24 h = 86 400 s degree º 1º = (π/180) rad (a) plane angle minute ' 1' = (1/60)º = (π/10 800) rad second " 1" = (1/60)' = (π/648 000) rad (h) 2 4 2 hectare ha 1ha = 1 hm = 10 m (b) 3−3 3 liter L , l 1 L = 1 dm = 10 m (c) 3 metric ton 1 t = 10 kg T (d, f ) see footnote (g) regarding the numerical neper Np (e, f ) value of logarythmic ratio quantities such B bel (e, f ) as the neper, the bel, and the decibel dB decibel (a) See also Sec. 7.2. (b) The alternative symbol for the liter, L, was adopted by the CGPM in order to avoid the risk of confusion between the letter l and the number 1 (see Ref. 1 or 2). Thus, although both l and L are internationally accepted symbols for the liter, to avoid this risk the symbol to be used in the United States is L (see Refs. 2 and 6). The script letter l is not an approved symbol for the liter. (c) This is the name to be used for this unit in the United States (see Refs. 2 and 6); it is also used in some other English-speaking countries. However, this unit is called “tonne” in Ref. 1 and is the name used in many countries. (d) The statement LA = n Np (where n is a number) is interpreted to mean that ln(A / A ) = n. Thus when L = 1 Np, A / A = e. The 2 1 A 2 1 symbol A is used here to denote the amplitude of a sinusoidal signal, and L is then called the Napierian logarithmic amplitude A ratio, or the Napierian amplitude level difference. (e) The statement L = m dB = (m / 10) B (where m is a number) is interpreted to mean that lg(X / X ) = m/10. Thus when L = 1 B, X 0 X 1/10 X / X = 10, and when L = 1 dB, X / X = 10 . If X denotes a mean square signal or power-like quantity, L is called a power 0 X 0 X level referred to X . (See Sec. 8.7.) 0 (f) In using these units it is important that the nature of the quantity be specified, and that any reference value used be specified. These units are not SI units, but they have been accepted by the CIPM for use with the SI. For additional information on the neper and bel, see Ref. 5: IEC 60027-3, and Sec. 8.7 of this Guide. (g) The numerical values of the neper, bel, and decibel (and hence the relation of the bel and the decibel to the neper) are rarely required. They depend on the way in which the logarithmic quantities are defined. (h) This unit and its symbol are used to express agrarian area. 5.1.2 Electronvolt, astronomical unit, and unified atomic mass unit The CIPM, and thus this Guide, accepts for use with the SI the units given in Table 7 1, 2. These units are used in specialized fields; their values in SI units must be obtained from experiment and, therefore, are not known exactly. (The use of SI prefixes with the units of Table 7 is discussed in Sec. 6.2.8.) 8 Guide for the Use of the International System of Units (SI) Table 7. Non-SI Units accepted for use with the SI by the CIPM and this Guide, whose values in SI units are obtained experimentally Name Symbol Definition and Value in SI units electronvolt eV (a) astronomical unit ua (b) unified atomic mass unit u (c) dalton Da (d) (a) The electronvolt is the kinetic energy acquired by an electron in passing through a potential difference of 1 V in vacuum, −19 1.602 176 487(40) × 10 J. This value of 1 eV is the 2006 CODATA recommended value with the standard uncertainty in the last two digits given in parenthesis 19, 20. (b) The astronomical unit is approximately equal to the mean Earth-Sun distance. It is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with a mean motion of 0.017 202 098 95 radians per day (known as the Gaussian constant). The value and standard uncertainty of the astronomical unit, ua, is 1.495 978 706 11 91(6) × 10 m. This is cited from the IERS Conventions 2003 (D.D. McCarthy and G. Petit eds., IERS Technical Note 32, Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie, 2004, 12). The value of the astronomical unit in meters comes from the JPL ephemerides DE403 (Standish E.M., Report of the IAU WGAS Sub-Group on Numerical Standards, Highlights of Astronomy, Appenzeller ed., Dordrecht: Kluwer Academic Publishers, 1995, 180-184). (c) The unified atomic mass unit is equal to 1/12 times the mass of a free carbon 12 atom, at rest and in its ground state, −27 1.660 538 782(83) × 10 kg. This value of 1 u is the 2006 CODATA recommended value with the standard uncertainty in the last two digits given in parenthesis 19, 20. (d) The dalton (Da) and the unified atomic mass unit (u) are alternative names (and symbols) for the same unit, equal to 1/12 times the mass of a free carbon 12 atom, at rest and in its ground state. The dalton is often combined with SI prefixes, for example to express the masses of large molecules in kilodaltons, kDa, or megadaltons, MDa. Note: The abbreviation, AMU is not an acceptable unit symbol for the unified atomic mass unit. The only allowed name is “unified atomic mass unit” and the only allowed symbol is u. 5.1.3 Units from International Standards There are a few highly specialized units that are given by the International Organization for Standardization (ISO) or the International Electrotechnical Commission (IEC) and which in the view of this Guide are also acceptable for use with the SI. They include the octave, phon, and sone, and units used in 3 information technology, including the baud (Bd), bit (bit), erlang (E), hartley (Hart), and shannon (Sh) . It is the position of this Guide that the only such additional units NIST authors may use with the SI are those given in either the International Standards on quantities and units of ISO (Ref. 4) or of IEC (Ref. 5). 5.1.4 Natural and atomic units In some cases, particularly in basic science, the values of quantities are expressed in terms of fundamental constants of nature. The two most important of these unit systems are the natural unit (n.u.) system used in high energy or particle physics, and the atomic unit (a.u.) system used in atomic physics and quantum chemistry. The use of these units with the SI is not formally accepted by the CIPM, but the CIPM recognizes their existence and importance. Therefore, this Guide formally accepts their use when it is necessary for effective communication. In such cases, the specific unit system used must be identified. Examples of physical quantities used as units are given in Table 8. Table 8. Examples of physical quantities sometimes used as units Kind of quantity Physical quantity used as a unit Symbol speed speed of light in vacuum (n.u.) c action Planck constant divided by 2π (n.u.) ħ mass electron rest mass (n.u. and a.u.) m e electric charge elementary charge (a.u.) e length Bohr radius (a.u.) a 0 energy Hartree energy (a.u.) E h time ratio of action to energy (a.u.) ħ/ E h 3 The symbol in parentheses following the name of the unit is its internationally accepted unit symbol, but the octave, phon, and sone have no such unit symbols. For additional information on the neper and bel, see Ref. 5: IEC 60027-3, and Sec. 8.7 of this Guide. 9 Guide for the Use of the International System of Units (SI) 5.2 Other Non-SI units accepted for use with the SI Because of established practice in certain fields or countries, in 1978 the CIPM considered that it was permissible for the following units given in Table 9, nautical mile, knot, angstrom, are, barn, bar, and millimeter of mercury to continue to be used with the SI. However, these units must not be introduced in fields where they are not presently used. Further, this Guide strongly discourages the continued use of these units by NIST authors except when absolutely necessary. If these units are used by NIST authors the values of relevant quantities shall be given in terms of SI units first followed by these non-SI units in parentheses. The curie, roentgen, rad, and rem have been added to the NIST SP 330 2 and Table 9 of this Guide, since they are in wide use in the United States, especially in regulatory documents dealing with health and safety. Nevertheless, this Guide strongly discourages the continued use of the curie, roentgen, rad, and rem and recommends that SI units should be used by NIST authors only if necessary. If these units are used by NIST authors the values of relevant quantities shall be given in terms of SI units first followed by these outdated non-SI units in parentheses. Table 9. Other non-SI units accepted for use with the SI either by the CIPM and this Guide (indicated by ), or by this Guide (indicated by ) Name Symbol Value in SI units nautical mile 1 nautical mile = 1852 m knot 1 nautical mile per hour = (1852/3600) m/s −10 ångström Å 1 Å = 0.1 nm = 10 m 2−28 2 barn b 1 b = 100 fm = 10 m 5 bar bar 1 bar = 0.1 MPa = 100 kPa = 1000 hPa = 10 Pa millimeter of mercury mmHg 1 mmHg ≈ 133.322 Pa 10 curie Ci 1 Ci = 3.7 x 10 Bq −4 roentgen R 1 R = 2.58 x 10 C/kg (a) −2 rad rad 1 rad = 1 cGy = 10 Gy −2 rem rem 1 rem = 1 cSv = 10 Sv (a) When there is risk of confusion with the symbol for the radian, rd may be used as the symbol for rad. 5.3 Units not accepted for use with the SI The following two sections briefly discuss units not accepted for use with the SI. 5.3.1 CGS units Table 10 gives examples of centimeter-gram-second (CGS) units having special names. These units are not accepted for use with the SI by this Guide. Further, no other units of the various CGS systems of units, which includes the CGS Electrostatic (ESU), CGS Electromagnetic (EMU), and CGS Gaussian systems, are accepted for use with the SI by this Guide except such units as the centimeter, gram, and second that are also defined in the SI. 10 Guide for the Use of the International System of Units (SI) Table 10. Examples of CGS units with special names (not accepted for use with the SI by this Guide) Name Symbol Value in SI units −7 erg erg 1 erg = 10 J −5 dyne dyn 1 dyn = 10 N (a) 2 poise P 1 P = 1 dyn · s/cm = 0.1 Pa · s (b) 2−4 2 stokes St 1 St = 1 cm /s = 10 m /s (c)−4 gauss Gs, G 1 Gs corresponds to 10 T (c) oersted Oe 1 Oe corresponds to (1000/4π) A/m (c)−8 maxwell Mx 1 Mx corresponds to 10 Wb 2 4 2 stilb sb 1 sb = 1 cd/cm = 10 cd/m 4 phot ph 1 ph = 10 lx (d)−2−2−2 gal Gal 1 Gal = 1 cm s = 10 m s (a) The poise (P) is the CGS unit for viscosity (also called dynamic viscosity). The SI unit is the pascal second (Pa · s). 2 (b) The stokes (St) is the CGS unit for kinematic viscosity. The SI unit is the meter squared per second (m /s). (c) This unit is part of the so-called electromagnetic three-dimensional CGS system and cannot strictly speaking be compared to the corresponding SI unit, which has four dimensions when only mechanical and electric quantities are considered. (d) The gal is employed in geodesy and geophysics to express acceleration due to gravity. 5.3.2 Other unacceptable units There are many units besides CGS units that are outside the SI and not accepted for use with it, including, of course, all of the U.S. customary (that is, inch-pound) units. In the view of this Guide such units must strictly be avoided and SI units, their multiples or submultiples, or those units accepted or temporarily accepted for use with the SI (including their appropriate multiples and submultiples), must be used instead. This restriction also applies to the use of unaccepted special names for SI units or special names for multiples or submultiples of SI units, such as mho for siemens (S) and micron for micrometer (μm). Table 11 gives a few examples of some of these other unacceptable units. Table 11. Examples of other unacceptable units Name Symbol Value in SI units −15 fermi fermi 1 fermi = 1 fm = 10 m −4 photometric carat metric carat 1 metric carat = 200 mg = 2 × 10 kg torr Torr 1 Torr = (101 325/760) Pa standard atmosphere atm 1 atm = 101 325 Pa kilogram-force kgf 1 kgf = 9.806 65 N −6 micron μ 1 m = 1 µm = 10 m calorie (various) cal (thermochemical) 1 cal = 4.184 J th th −13 x unit xu 1 xu ≈ 0.1002 pm = 1.002 × 10 m 3 stere st 1 st = 1 m −9 gamma γ 1 γ = 1 nT = 10 T −9 gamma (mass) γ 1 γ = 1 µg = 10 kg −6−9 3 lambda (volume) λ 1 λ = 1 µL = 10 L = 10 m 5.4 The terms “SI units” and “acceptable units” Consistent with accepted practice 1, 2, this Guide uses either the term “SI units” or “units of the SI” to mean the SI base units and SI coherent derived units, and multiples and submultiples of these units formed by using the SI prefixes. The term “acceptable units,” which is introduced in this Guide for convenience, is used to mean the SI units plus (a) those non-SI units accepted for use with the SI (see Tables 6 - 9); and (b) appropriate multiples and submultiples of such accepted non-SI units. 11 Guide for the Use of the International System of Units (SI) 6 Rules and Style Conventions for Printing and Using Units 6.1 Rules and style conventions for unit symbols The following eight sections give rules and style conventions related to the symbols for units. 6.1.1 Typeface Unit symbols are printed in roman (upright) type regardless of the type used in the surrounding text. (See also Sec. 10.2 and Secs. 10.2.1 to 10.2.4.) 6.1.2 Capitalization Unit symbols are printed in lower-case letters except that: (a) the symbol or the first letter of the symbol is an upper-case letter when the name of the unit is derived from the name of a person; and (b) the recommended symbol for the liter in the United States is L. (See Table 6, footnote (b).) Examples: m (meter) s (second) V (volt) Pa (pascal) lm (lumen) Wb (weber) 6.1.3 Plurals Unit symbols are unaltered in the plural. Example: l = 75 cm but not: l = 75 cms Note: l is the quantity symbol for length. (The rules and style conventions for expressing the values of quantities are discussed in detail in Chapter 7.) 6.1.4 Punctuation Unit symbols are not followed by a period unless at the end of a sentence. Example: “Its length is 75 cm.” or “It is 75 cm long.” but not: “It is 75 cm. long.” 6.1.5 Unit symbols obtained by multiplication Symbols for units formed from other units by multiplication are indicated by means of either a half- high (that is, centered) dot or a space. However, this Guide, as does Ref. 6, prefers the half-high dot because it is less likely to lead to confusion. Example: N · m or N m Notes: −1 1. A half-high dot or space is usually imperative. For example, m · s is the symbol for the meter per 3−1 second while ms−1 is the symbol for the reciprocal millisecond (10 s — see Sec. 6.2.3). 2. Reference 4: ISO 31-0 suggests that if a space is used to indicate units formed by multiplication, the space may be omitted if it does not cause confusion. This possibility is reflected in the common practice of using the symbol kWh rather than kW · h or kW h for the kilowatt hour. Nevertheless, this Guide takes the position that a half-high dot or a space should always be used to 12 Guide for the Use of the International System of Units (SI) avoid possible confusion; for this same reason, only one of these two allowed forms should be used in any given manuscript. 6.1.6 Unit symbols obtained by division Symbols for units formed from other units by division are indicated by means of a solidus (oblique stroke, / ) , a horizontal line, or negative exponents. −1 m Example: m/s, , or m · s s However, to avoid ambiguity, the solidus must not be repeated on the same line unless parentheses are used. 2−2 Examples: m/s or m · s but not: m/s/s 3 −3−1 3 m · kg/(s· A) or m · kg · s · A but not: m · kg/s /A Negative exponents should be used in complicated cases. 6.1.7 Unacceptability of unit symbols and unit names together Unit symbols and unit names are not used together. (See also Secs. 9.5 and 9.8.) −1 Example: C/kg, C · kg , or coulomb per kilogram but not: coulomb/kg; coulomb per kg; −1 C/kilogram; coulomb · kg ; C per kg; coulomb/kilogram 6.1.8 Unacceptability of abbreviations for units Because acceptable units generally have internationally recognized symbols and names, it is not permissible to use abbreviations for their unit symbols or names, such as sec (for either s or second), sq. 2 3 mm (for either mm or square millimeter), cc (for either cm or cubic centimeter), mins (for either min or minutes), hrs (for either h or hours), lit (for either L or liter), amps (for either A or amperes), AMU (for either u or unified atomic mass unit), or mps (for either m/s or meter per second). Although the values of quantities are normally expressed using symbols for numbers and symbols for units (see Sec. 7.6), if for some reason the name of a unit is more appropriate than the unit symbol (see Sec. 7.6, note 3), the name of the unit should be spelled out in full. 6.2 Rules and style conventions for SI prefixes The following eight sections give rules and style conventions related to the SI prefixes. 6.2.1 Typeface and spacing Prefix names and symbols are printed in roman (upright) type regardless of the type used in the surrounding text, and are attached to unit symbols without a space between the prefix name or symbol and the unit name or symbol. This last rule also applies to prefixes attached to unit names. Examples: mL (milliliter) pm (picometer) GΩ (gigaohm) THz (terahertz) 13 Guide for the Use of the International System of Units (SI) 6.2.2 Capitalization The prefix symbols Y (yotta), Z (zetta), E (exa), P (peta), T (tera), G (giga), and M (mega) are printed in upper-case letters while all other prefix symbols are printed in lower-case letters (see Table 5). Prefix names are normally printed in lowercase letters. 6.2.3 Inseparability of prefix and unit The grouping formed by a prefix symbol attached to a unit symbol constitutes a new inseparable symbol (forming a multiple or submultiple of the unit concerned) which can be raised to a positive or negative power and which can be combined with other unit symbols to form compound unit symbols. 3 3−2 3−6 3 Examples: 2.3 cm = 2.3 (cm) = 2.3 (10 m) = 2.3 × 10 m − 1−1−2−1 2 −1 1 cm = 1 (cm) = 1 (10 m) = 10 m −1−1−6−1 6−1 9−1 5000 µs = 5000 (µs) = 5000 (10 s) = 5000 × 10 s = 5 × 10 s −2 2 1 V/cm = (1 V)/(10 m) = 10 V/m Prefix names are also inseparable from the unit names to which they are attached. Thus, for example, millimeter, micropascal, and meganewton are single words. 6.2.4 Unacceptability of compound prefixes Compound prefix names or symbols, that is, prefix names or symbols formed by the juxtaposition of two or more prefix names or symbols, are not permitted. Example: nm (nanometer) but not: mµm (millimicrometer) 6.2.5 Use of multiple prefixes In a derived unit formed by division, the use of a prefix symbol (or a prefix name) in both the numerator and the denominator can cause confusion. Thus, for example, 10 kV/mm is acceptable, but 10 MV/m is often considered preferable because it contains only one prefix symbol and it is in the numerator. In a derived unit formed by multiplication, the use of more than one prefix symbol (or more than one prefix name) can also cause confusion. Thus, for example, 10 MV · ms is acceptable, but 10 kV · s is often considered preferable. Note: Such considerations usually do not apply if the derived unit involves the kilogram. For example, 0.13 mmol/g is not considered preferable to 0.13 mol/kg. 6.2.6 Unacceptability of stand-alone prefixes Prefix symbols cannot stand alone and thus cannot be attached to the number 1, the symbol for the unit one. In a similar vein, prefix names cannot be attached to the name of the unit one, that is, to the word “one.” (See Sec. 7.10 for a discussion of the unit one.) 6 3 Example: the number density of Pb atoms is 5 ×10 /m but not: the number density of Pb atoms 3 is 5 M/m 6.2.7 Prefixes and the kilogram For historical reasons, the name “kilogram” for the SI base unit of mass contains the name “kilo,” 3 the SI prefix for 10 . Thus, because compound prefixes are unacceptable (see Sec. 6.2.4), symbols for decimal multiples and submultiples of the unit of mass are formed by attaching SI prefix symbols to g, the 14 Guide for the Use of the International System of Units (SI) unit symbol for gram, and the names of such multiples and submultiples are formed by attaching SI prefix names to the name “gram.” −6−6 Example: 10 kg = 1 mg (1 milligram) but not: 10 kg = 1 µkg (1 microkilogram) 6.2.8 Prefixes with the degree Celsius and units accepted for use with the SI Prefix symbols may be used with the unit symbol ºC and prefix names may be used with the unit name “degree Celsius.” For example, 12 mºC (12 millidegrees Celsius) is acceptable. However, to avoid confusion, prefix symbols (and prefix names) are not used with the time-related unit symbols (names) min (minute), h (hour), d (day); nor with the angle-related symbols (names) º (degree), ' (minute), and " (second) (see Table 6). Prefix symbols (and prefix names) may be used with the unit symbols (names) L (liter), t (metric ton), eV (electronvolt), u (unified atomic mass unit), Da (dalton) (see Tables 6 and 7). However, although submultiples of the liter such as mL (milliliter) and dL (deciliter) are in common use, multiples of the liter such as kL (kiloliter) and ML (megaliter) are not. Similarly, although multiples of the metric ton such as kt (kilometric ton) are commonly used, submultiples such as mt (millimetric ton), which is equal to the kilogram (kg), are not. Examples of the use of prefix symbols with eV and u are 80 MeV (80 megaelectronvolts) and 15 nu (15 nanounified atomic mass units). 7 Rules and Style Conventions for Expressing Values of Quantities 7.1 Value and numerical value of a quantity The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit. More formally, the value of quantity A can be written as A = AA, where A is the numerical value of A when the value of A is expressed in the unit A. The numerical value can therefore be written as A = A / A, which is a convenient form for use in figures and tables. Thus, to eliminate the possibility of misunderstanding, an axis of a graph or the heading of a column of a table can be labeled “t/ºC” instead of “t (ºC)” or “Temperature (ºC).” Similarly, an axis or column heading can be labeled “E/(V/m)” instead of “E (V/m)” or “Electric field strength (V/m).” Examples: 1. In the SI, the value of the velocity of light in vacuum is c = 299 792 458 m/s exactly. The number 299 792 458 is the numerical value of c when c is expressed in the unit m/s, and equals c/(m/s). 3 2. The ordinate of a graph is labeled T/(10 K), where T is thermodynamic temperature and K is the unit symbol for kelvin, and has scale marks at 0, 1, 2, 3, 4, and 5. If the ordinate value of a point 3 on a curve in the graph is estimated to be 3.2, the corresponding temperature is T / (10 K) = 3.2 or T = 3200 K. Notice the lack of ambiguity in this form of labeling compared with “Temperature 3 (10 K).” 3. An expression such as ln(p/MPa), where p is the quantity symbol for pressure and MPa is the unit symbol for megapascal, is perfectly acceptable, because p/MPa is the numerical value of p when p is expressed in the unit MPa and is simply a number. Notes: 1. For the conventions concerning the grouping of digits, see Sec. 10.5.3. 15 Guide for the Use of the International System of Units (SI) 2. An alternative way of writing c/(m/s) is c , meaning the numerical value of c when c is m/s expressed in the unit m/s. 7.2 Space between numerical value and unit symbol In the expression for the value of a quantity, the unit symbol is placed after the numerical value and a space is left between the numerical value and the unit symbol. The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle: º, ', and ", respectively (see Table 6), in which case no space is left between the numerical value and the unit symbol. Example: α = 30º22'8" Note: α is a quantity symbol for plane angle. This rule means that: (a) The symbol ºC for the degree Celsius is preceded by a space when one expresses the values of Celsius temperatures. Example: t = 30.2 ºC but not: t = 30.2ºC or t = 30.2º C (b) Even when the value of a quantity is used as an adjective, a space is left between the numerical value and the unit symbol. (This rule recognizes that unit symbols are not like ordinary words or abbreviations but are mathematical entities, and that the value of a quantity should be expressed in a way that is as independent of language as possible—sees Secs. 7.6 and 7.10.3.) Examples: a 1 m end gauge but not: a 1-m end gauge a 10 kΩ resistor but not: a 10-kΩ resistor However, if there is any ambiguity, the words should be rearranged accordingly. For example, the statement “the samples were placed in 22 mL vials” should be replaced with the statement “the samples were placed in vials of volume 22 mL.” Note: When unit names are spelled out, the normal rules of English apply. Thus, for example, “a roll of 35-millimeter film” is acceptable (see Sec. 7.6, note 3). 7.3 Number of units per value of a quantity The value of a quantity is expressed using no more than one unit. Example: l = 10.234 m but not: l = 10 m 23 cm 4 mm Note: Expressing the values of time intervals and of plane angles are exceptions to this rule. However, it is preferable to divide the degree decimally. Thus one should write 22.20º rather than 22º12', except in fields such as cartography and astronomy. 7.4 Unacceptability of attaching information to units When one gives the value of a quantity, it is incorrect to attach letters or other symbols to the unit in order to provide information about the quantity or its conditions of measurement. Instead, the letters or other symbols should be attached to the quantity. Example: V = 1000 V but not: V = 1000 V max max 16 Guide for the Use of the International System of Units (SI) Note: V is a quantity symbol for potential difference. 7.5 Unacceptability of mixing information with units When one gives the value of a quantity, any information concerning the quantity or its conditions of measurement must be presented in such a way as not to be associated with the unit. This means that quantities must be defined so that they can be expressed solely in acceptable units (including the unit one — see Sec. 7.10). Examples: the Pb content is 5 ng/L but not: 5 ng Pb/L or 5 ng of lead/L 10 3 10 the sensitivity for NO molecules is 5 × 10 /cm but not: the sensitivity is 5 × 10 3 3 3 NO molecules/cm 10 10 the neutron emission rate is 5 × 10 /s but not: the emission rate is 5 × 10 n/s 18 3 18 3 the number density of O atoms is 3 × 10 /cm but not: the density is 3 × 10 O atoms/cm 2 2 the resistance per square is 100 Ω but not: the resistance is 100 Ω/square 7.6 Symbols for numbers and units versus spelled-out names of numbers and units This Guide takes the position that the key elements of a scientific or technical paper, particularly the results of measurements and the values of quantities that influence the measurements, should be presented in a way that is as independent of language as possible. This will allow the paper to be understood by as broad an audience as possible, including readers with limited knowledge of English. Thus, to promote the comprehension of quantitative information in general and its broad understandability in particular, values of quantities should be expressed in acceptable units using — the Arabic symbols for numbers, that is, the Arabic numerals, not the spelled-out names of the Arabic numerals; and — the symbols for the units, not the spelled-out names of the units. Examples: the length of the laser is 5 m but not: the length of the laser is five meters the sample was annealed at a but not: the sample was annealed at a temperature temperature of 955 K for 12 h of 955 kelvins for 12 hours Notes: 1. If the intended audience for a publication is unlikely to be familiar with a particular unit symbol, it should be defined when first used. 2. Because the use of the spelled-out name of an Arabic numeral with a unit symbol can cause confusion, such combinations must strictly be avoided. For example, one should never write “the length of the laser is five m.” 3. Occasionally, a value is used in a descriptive or literary manner and it is fitting to use the spelled- out name of the unit rather than its symbol. Thus, this Guide considers acceptable statements such 17 Guide for the Use of the International System of Units (SI) as “the reading lamp was designed to take two 60-watt light bulbs,” or “the rocket journeyed uneventfully across 380 000 kilometers of space,” or “they bought a roll of 35-millimeter film for their camera.” 4. The United States Government Printing Office Style Manual (Ref. 3, pp. 181-189) gives the rule that symbols for numbers are always to be used when one expresses (a) the value of a quantity in terms of a unit of measurement, (b) time (including dates), and (c) an amount of money. This publication should be consulted for the rules governing the choice between the use of symbols for numbers and the spelled-out names of numbers when numbers are dealt with in general. 7.7 Clarity in writing values of quantities The value of a quantity is expressed as the product of a number and a unit (see Sec. 7.1). Thus, to avoid possible confusion, this Guide takes the position that values of quantities must be written so that it is completely clear to which unit symbols the numerical values of the quantities belong. Also to avoid possible confusion, this Guide strongly recommends that the word “to” be used to indicate a range of values for a quantity instead of a range dash (that is, a long hyphen) because the dash could be misinterpreted as a minus sign. (The first of these recommendations once again recognizes that unit symbols are not like ordinary words or abbreviations but are mathematical entities—see Sec. 7.2.) Examples: 51 mm × 51 mm × 25 mm but not: 51 × 51 × 25 mm 225 nm to 2400 nm or (225 to 2400) nm but not: 225 to 2400 nm 0 ºC to 100 ºC or (0 to 100) ºC but not: 0 ºC − 100 ºC 0 V to 5 V or (0 to 5) V but not: 0 − 5 V (8.2, 9.0, 9.5, 9.8, 10.0) GHz but not: 8.2, 9.0, 9.5, 9.8, 10.0 GHz 63.2 m ± 0.1 m or (63.2 ± 0.1) m but not: 63.2 ± 0.1 m or 63.2 m ± 0.1 129 s − 3 s = 126 s or (129 − 3) s = 126 s but not: 129 − 3 s = 126 s Note: For the conventions concerning the use of the multiplication sign, see Sec. 10.5.4. 7.8 Unacceptability of stand-alone unit symbols Symbols for units are never used without numerical values or quantity symbols (they are not abbreviations). 6 mm in 1 km but not: there are many mm in a km Examples: there are 10 3 it is sold by the cubic meter but not: it is sold by the m t/ºC, E/(V/m), p/MPa, and the like are perfectly acceptable (see Sec. 7.1). 7.9 Choosing SI prefixes The selection of the appropriate decimal multiple or submultiple of a unit for expressing the value of a quantity, and thus the choice of SI prefix, is governed by several factors. These include: — the need to indicate which digits of a numerical value are significant, — the need to have numerical values that are easily understood, and 18 Guide for the Use of the International System of Units (SI) — the practice in a particular field of science or technology. A digit is significant if it is required to express the numerical value of a quantity. In the expression l = 1200 m, it is not possible to tell whether the last two zeroes are significant or only indicate the magnitude of the numerical value of l. However, in the expression l = 1.200 km, which uses the SI prefix 3 symbol for 10 (kilo, symbol k), the two zeroes are assumed to be significant because if they were not, the value of l would have been written l = 1.2 km. It is often recommended that, for ease of understanding, prefix symbols should be chosen in such a way that numerical values are between 0.1 and 1000, and that only prefix symbols that represent the number 10 raised to a power that is a multiple of 3 should be used. 7 6 Examples: 3.3 × 10 Hz may be written as 33 × 10 Hz = 33 MHz −3 0.009 52 g may be written as 9.52 × 10 g = 9.52 mg 3 2703 W may be written as 2.703 × 10 W = 2.703 kW −8−9 5.8 × 10 m may be written as 58 × 10 m = 58 nm However, the values of quantities do not always allow this recommendation to be followed, nor is it mandatory to try to do so. In a table of values of the same kind of quantities or in a discussion of such values, it is usually recommended that only one prefix symbol should be used even if some of the numerical values are not between 0.1 and 1000. For example, it is often considered preferable to write “the size of the sample is 10 mm × 3 mm × 0.02 mm” rather than “the size of the sample is 1 cm × 3 mm × 20 µm.” In certain kinds of engineering drawings it is customary to express all dimensions in millimeters. This is an example of selecting a prefix based on the practice in a particular field of science or technology. 7.10 Values of quantities expressed simply as numbers: the unit one, symbol 1 Certain quantities, such as refractive index, relative permeability, and mass fraction, are defined as the ratio of two mutually comparable quantities and thus are of dimension one (see Sec. 7.14). The coherent SI unit for such a quantity is the ratio of two identical SI units and may be expressed by the number 1. However, the number 1 generally does not appear in the expression for the value of a quantity of dimension one. For example, the value of the refractive index of a given medium is expressed as n = 1.51 × 1 = 1.51. On the other hand, certain quantities of dimension one have units with special names and symbols which can be used or not depending on the circumstances. Plane angle and solid angle, for which the SI units are the radian (rad) and steradian (sr), respectively, are examples of such quantities (see Sec. 4.2.1). 7.10.1 Decimal multiples and submultiples of the unit one Because SI prefix symbols cannot be attached to the unit one (see Sec. 6.2.6), powers of 10 are used to express decimal multiples and submultiples of the unit one. -6 Example: µ = 1.2 × 10 but not: µ = 1.2 µ r r Note: µ is the quantity symbol for relative permeability. r 7.10.2 %, percentage by, fraction In keeping with Ref. 4: ISO 31-0, this Guide takes the position that it is acceptable to use the internationally recognized symbol % (percent) for the number 0.01 with the SI and thus to express the values of quantities of dimension one (see Sec. 7.14) with its aid. When it is used, a space is left between 19

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