1、BS EN ISO80000-2:2013ICS 01.060NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAWBRITISH STANDARDQuantities and unitsPart 2: Mathematical signs andsymbols to be used in the naturalsciences and technologyNational forewordThis British Standard is the UK implementation of EN ISO 800
2、00-2:2013. It supersedes BS ISO 80000-2:2009, which is withdrawn.The UK participation in its preparation was entrusted to Technical Committee SS/7, General metrology, quantities, units and symbols.A list of organizations represented on this committee can be obtained on request to its secretary.This
3、publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application.Compliance with a British Standard cannot confer immunity from legal obligations.BS EN ISO 80000-2:2013This British Standard was published under the authority of the
4、Standards Policy and Strategy Committee on 31 January 2010 The British Standards Institution 2013. Published by BSI Standards Limited 2013Amendments/corrigenda issued since publicationDate Comments 31 May 2013 This corrigendum renumbers BS ISO 80000-2:2009 as BS EN ISO 80000-2:2013ISBN 978 0 580 790
5、18 8EUROPEAN STANDARD NORME EUROPENNE EUROPISCHE NORM EN ISO 80000-2 April 2013 ICS 01.060 English Version Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology (ISO 80000-2:2009) Grandeurs et units - Partie 2: Signes et symboles mathmatiques
6、 employer dans les sciences de la nature et dans la technique (ISO 80000-2:2009) Gren und Einheiten - Teil 2: Mathematische Zeichen fr Naturwissenschaft und Technik (ISO 80000-2:2009) This European Standard was approved by CEN on 14 March 2013. CEN members are bound to comply with the CEN/CENELEC In
7、ternal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to a
8、ny CEN member. This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management Centre has the same status as the official
9、versions. CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norw
10、ay, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and United Kingdom. EUROPEAN COMMITTEE FOR STANDARDIZATION COMIT EUROPEN DE NORMALISATION EUROPISCHES KOMITEE FR NORMUNG Management Centre: Avenue Marnix 17, B-1000 Brussels 2013 CEN All rights of exploitation in a
11、ny form and by any means reserved worldwide for CEN national Members. Ref. No. EN ISO 80000-2:2013: EForeword The text of ISO 80000-2:2009 has been prepared by Technical Committee ISO/TC 12 “Quantities and units” of the International Organization for Standardization (ISO) and has been taken over as
12、EN ISO 80000-2:2013. This European Standard shall be given the status of a national standard, either by publication of an identical text or by endorsement, at the latest by October 2013, and conflicting national standards shall be withdrawn at the latest by October 2013. Attention is drawn to the po
13、ssibility that some of the elements of this document may be the subject of patent rights. CEN and/or CENELEC shall not be held responsible for identifying any or all such patent rights. According to the CEN-CENELEC Internal Regulations, the national standards organizations of the following countries
14、 are bound to implement this European Standard: Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland,
15、 Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the United Kingdom. Endorsement notice The text of ISO 80000-2:2009 has been approved by CEN as EN ISO 80000-2:2013 without any modification. BS EN ISO 80000-2:2013 ISO 80000-2:2013 (E)EN ISO 20 All rights reserved iiiCon
16、tents Page Foreword iv Introductionvi 1 Scope1 2 Normative references1 3 Variables, functions, and operators 1 4 Mathematical logic 3 5 Sets .4 6 Standard number sets and intervals .6 7 Miscellaneous signs and symbols 8 8 Elementary geometry10 9 Operations11 10 Combinatorics .14 11 Functions15 12 Ex
17、ponential and logarithmic functions .18 13 Circular and hyperbolic functions .19 14 Complex numbers .21 15 Matrices 22 16 Coordinate systems 24 17 Scalars, vectors, and tensors 26 18 Transforms.30 19 Special functions.31 Annex A (normative) Clarification of the symbols used.36 Bibliography40 13BS EN
18、 ISO 80000-2:2013 ISO 80000-2:2009 (E)iv ISO 2013 All rights reservedForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technic
19、al committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely wi
20、th the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft Inte
21、rnational Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this document ma
22、y be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. ISO 80000-2 was prepared by Technical Committee ISO/TC 12, Quantities and units, in collaboration with IEC/TC 25, Quantities and units. This first edition cancels and replaces ISO 31-1
23、1:1992, which has been technically revised. The major technical changes from the previous standard are the following: Four clauses have been added, i.e. “Standard number sets and intervals”, “Elementary geometry”, “Combinatorics” and “Transforms”. ISO 80000 consists of the following parts, under the
24、 general title Quantities and units: Part 1: General Part 2: Mathematical signs and symbols to be used in the natural sciences and technology1) Part 3: Space and time Part 4: Mechanics Part 5: Thermodynamics Part 7: Light Part 8: Acoustics Part 9: Physical chemistry and molecular physics Part 10: At
25、omic and nuclear physics Part 11: Characteristic numbers Part 12: Solid state physics 1) Title to be shortened to read “Mathematics” in the second edition of ISO 80000-2. BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)vIEC 80000 consists of the following parts, under the general title Quantities and uni
26、ts: Part 6: Electromagnetism Part 13: Information science and technology Part 14: Telebiometrics related to human physiology ISO 20 All rights reserved 13BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)vi ISO 2013 All rights reservedIntroduction Arrangement of the tables The first column “Item No.” of th
27、e tables contains the number of the item, followed by either the number of the corresponding item in ISO 31-11 in parentheses, or a dash when the item in question did not appear in ISO 31-11. The second column “Sign, symbol, expression” gives the sign or symbol under consideration, usually in the co
28、ntext of a typical expression. If more than one sign, symbol or expression is given for the same item, they are on an equal footing. In some cases, e.g. for exponentiation, there is only a typical expression and no symbol. The third column “Meaning, verbal equivalent” gives a hint on the meaning or
29、how the expression may be read. This is for the identification of the concept and is not intended to be a complete mathematical definition. The fourth column “Remarks and examples” gives further information. Definitions are given if they are short enough to fit into the column. Definitions need not
30、be mathematically complete. The arrangement of the table in Clause 16 “Coordinate systems” is somewhat different. BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)INTERNATIONAL STANDARD 1Quantities and units Part 2: Mathematical signs and symbols to be used in the natural sciences and technology 1 Scope I
31、SO 80000-2 gives general information about mathematical signs and symbols, their meanings, verbal equivalents and applications. The recommendations in ISO 80000-2 are intended mainly for use in the natural sciences and technology, but also apply to other areas where mathematics is used. 2 Normative
32、references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 80000-1:2), Quantities and units P
33、art 1: General 3 Variables, functions and operators Variables such as x, y, etc., and running numbers, such as i in ixiare printed in italic (sloping) type. Parameters, such as a, b, etc., which may be considered as constant in a particular context, are printed in italic (sloping) type. The same app
34、lies to functions in general, e.g. f, g. An explicitly defined function not depending on the context is, however, printed in Roman (upright) type, e.g. sin, exp, ln, . Mathematical constants, the values of which never change, are printed in Roman (upright) type, e.g. e = 2,718 218 8; = 3,141 592; i2
35、= 1. Well-defined operators are also printed in Roman (upright) style, e.g. div, in x and each d in df/dx. Numbers expressed in the form of digits are always printed in Roman (upright) style, e.g. 351 204; 1,32; 7/8. The argument of a function is written in parentheses after the symbol for the funct
36、ion, without a space between the symbol for the function and the first parenthesis, e.g. f(x), cos(t + ). If the symbol for the function consists of two or more letters and the argument contains no operation symbol, such as +, , , or / , the parentheses around the argument may be omitted. In these c
37、ases, there should be a thin space between the symbol for the function and the argument, e.g. int 2,4; sin n; arcosh 2A; Ei x. If there is any risk of confusion, parentheses should always be inserted. For example, write cos(x) + y; do not write cos x + y, which could be mistaken for cos(x + y). 2) T
38、o be published. (Revision of ISO 31-0:1992) ISO 20 All rights reserved13BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)2 ISO 2013 All rights reservedA comma, semicolon or other appropriate symbol can be used as a separator between numbers or expressions. The comma is generally preferred, except when num
39、bers with a decimal comma are used. If an expression or equation must be split into two or more lines, one of the following methods shall be used. a) Place the line breaks immediately after one of the symbols =, +, , or , or, if necessary, immediately after one of the symbols , , or /. In this case,
40、 the symbol indicates that the expression continues on the next line or next page. b) Place the line breaks immediately before one of the symbols =, +, , or , or, if necessary, immediately before one of the symbols , , or /. In this case, the symbol indicates that the expression is a continuation of
41、 the previous line or page. The symbol shall not be given twice around the line break; two minus signs could for example give rise to sign errors. Only one of these methods should be used in one document. If possible, the line break should not be inside of an expression in parentheses. It is customa
42、ry to use different sorts of letters for different sorts of entities. This makes formulas more readable and helps in setting up an appropriate context. There are no strict rules for the use of letter fonts which should, however, be explained if necessary. BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)3
43、4 Mathematical logic Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-4.1 (11-3.1) p q conjunction of p and q, p and q 2-4.2 (11-3.2) p q disjunction of p and q, p or q This “or” is inclusive, i.e. p q is true, if either p or q, or both are true. 2-4.3 (11-3.3) p n
44、egation of p, not p 2-4.4 (11-3.4) p q p implies q, if p, then q q p has the same meaning as p q. is the implication symbol. 2-4.5 (11-3.5) p q p is equivalent to q (p q) (q p) has the same meaning as p q. is the equivalence symbol. 2-4.6 (11-3.6) x A p(x) for every x belonging to A, the proposition
45、 p(x) is true lf it is clear from the context which set A is being considered, the notation x p(x) can be used. is the universal quantifier. For x A, see 2-5.1. 2-4.7 (11-3.7) x A p(x) there exists an x belonging to A for which p(x) is true lf it is clear from the context which set A is being consid
46、ered, the notation x p(x) can be used. is the existential quantifier. For x A, see 2-5.1. 1x p(x) is used to indicate that there is exactly one element for which p(x) is true. ! is also used for 1. ISO 20 All rights reserved13BS EN ISO 80000-2:2013 ISO 80000-2:2009 (E)4 ISO 2013 All rights reserved5
47、 Sets Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-5.1 (11-4.1) x A x belongs to A, x is an element of the set A A x has the same meaning as x A. 2-5.2 (11-4.2) y A y does not belong to A, y is not an element of the set A A y has the same meaning as y A. The ne
48、gating stroke may also be vertical. 2-5.3 (11-4.5) x1, x2, , xn set with elements x1, x2, , xnAlso xi| i I, where I denotes a set of subscripts. 2-5.4 (11-4.6) x A | p(x) set of those elements of A for which the proposition p(x) is true EXAMPLE x R | x u 5 lf it is clear from the context which set A
49、 is being considered, the notation x | p(x) can be used (for example x | x u 5, if it is clear that x is a variable for real numbers). 2-5.5 (11-4.7) card A A number of elements in A, cardinality of A The cardinality can be a transfinite number.See also 2-9.16. 2-5.6 (11-4.8) the empty set 2-5.7 (11-4.18) B A B is included in A, B is a subset of A Every element of B belongs to A. is also used, but
