Table Of Content)
c
(
,
7
4 BRITISH STANDARD BS ISO
:
3
80000-2:2009
0
0
1
0
2
/
1 Quantities and units
0
/
4
2
f
o
s Part 2: Mathematical signs and
a
t symbols to be used in the natural
c
e
r sciences and technology
r
o
c
n
o
i
s
r
e
V
,
g
n
o
K
g
n
o
H
f
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
ICS 01.060
e
h
T
:
y
p
o
c
d
e
s
n NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAW
e
c
i
L
)
c
(
,
7
4
BS ISO 80000-2:2009
:
3
0
0
1 National foreword
0
2
/
1
0 This British Standard is the UK implementation of ISO 80000-2:2009. It
4/ supersedes BS ISO 31-11:1992 which is withdrawn.
2
The UK participation in its preparation was entrusted to Technical
f
o Committee SS/7, General metrology, quantities, units and symbols.
s
a A list of organizations represented on this committee can be obtained on
request to its secretary.
t
c
e This publication does not purport to include all the necessary provisions
r
r of a contract. Users are responsible for its correct application.
o
c Compliance with a British Standard cannot confer immunity
n from legal obligations.
o
i
s
r
e
V
,
g
n
o
K
g
n
o
H
f
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t This British Standard Amendments/corrigenda issued since publication
i
s was published under the
r
e authority of the Standards
v Policy and Strategy Date Comments
i Committee on 31 January
n
U 2010
© BSI 2010
e
h
T
:
y
p
o ISBN 978 0 580 54864 2
c
d
e
s
n
e
c 标准分享网 www.bzfxw.com 免费下载
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 INTERNATIONAL ISO
:
3
0 STANDARD 80000-2
0
1
0
2
/
1
0 First edition
4/ 2009-12-01
2
f
o
s
a
t
c
e
r
r
o
c
n
o
Quantities and units —
i
s
r
e
Part 2:
V
, Mathematical signs and symbols to be
g
n used in the natural sciences and
o
K technology
g
n
o
Grandeurs et unités —
H
Partie 2: Signes et symboles mathématiques à employer dans les
f
o sciences de la nature et dans la technique
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
e
h
T
:
y
p
o Reference number
c ISO 80000-2:2009(E)
d
e
s
n
e © ISO 2009
c
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
0 PDF disclaimer
2
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
/
1 shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
0 downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
/
4 accepts no liability in this area.
2
Adobe is a trademark of Adobe Systems Incorporated.
f
o Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
s parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
a the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.
t
c
e
rr
o
c
n
o
i
s
r
e
V
,
g
n
o
K
g
n
o
H
f
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
ty COPYRIGHT PROTECTED DOCUMENT
i
s
er © ISO 2009
v All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
ni electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
U ISO's member body in the country of the requester.
ISO copyright office
e
Case postale 56 • CH-1211 Geneva 20
h
T Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
y: E-mail [email protected]
p Web www.iso.org
o Published in Switzerland
c
d
e
s
n
ii © ISO 2009 – All rights reserved
e
c 标准分享网 www.bzfxw.com 免费下载
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
0
2
/
1
0
4/ Contents Page
2
f
o
Foreword............................................................................................................................................................iv
s
a
Introduction........................................................................................................................................................vi
t
c 1 Scope......................................................................................................................................................1
e
r
r 2 Normative references............................................................................................................................1
o
c 3 Variables, functions, and operators....................................................................................................1
n
o 4 Mathematical logic................................................................................................................................3
i
s 5 Sets.........................................................................................................................................................4
r
e
V 6 Standard number sets and intervals...................................................................................................6
g, 7 Miscellaneous signs and symbols......................................................................................................8
n
8 Elementary geometry..........................................................................................................................10
o
K
9 Operations............................................................................................................................................11
g
n 10 Combinatorics.....................................................................................................................................14
o
H 11 Functions..............................................................................................................................................15
f 12 Exponewntial wand lowgari.thmicb funcztionsf........x........w...................c........o.........m..........................................18
o
y SI 13 Circular and hyperbolic functions.....................................................................................................19
tB
i
s 14 Complex numbers...............................................................................................................................21
r
e
15 Matrices................................................................................................................................................22
v
i
n 16 Coordinate systems............................................................................................................................24
U
17 Scalars, vectors, and tensors............................................................................................................26
e
h 18 Transforms...........................................................................................................................................30
T
, 19 Special functions.................................................................................................................................31
g
n Annex A (normative) Clarification of the symbols used...............................................................................36
o
K Bibliography......................................................................................................................................................40
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
e
h
T
:
y
p
o
c
d
e
s
n
© ISO 2009 – All rights reserved iii
e
c
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
0
2
/
1
0
4/ Foreword
2
f
o ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
s (ISO member bodies). The work of preparing International Standards is normally carried out through ISO
a
technical committees. Each member body interested in a subject for which a technical committee has been
ct established has the right to be represented on that committee. International organizations, governmental and
e non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
r
r International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
o
c
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
n
o
si The main task of technical committees is to prepare International Standards. Draft International Standards
r adopted by the technical committees are circulated to the member bodies for voting. Publication as an
e
V International Standard requires approval by at least 75 % of the member bodies casting a vote.
,
g Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
n
rights. ISO shall not be held responsible for identifying any or all such patent rights.
o
K
ISO 80000-2 was prepared by Technical Committee ISO/TC 12, Quantities and units, in collaboration with
g
n IEC/TC 25, Quantities and units.
o
H
This first edition cancels and replaces ISO 31-11:1992, which has been technically revised. The major
f technical changes from wthe prweviouws stan.dardb are thze folflowingx: w.com
o
y SI
tB⎯ Four clauses have been added, i.e. “Standard number sets and intervals”, “Elementary geometry”,
i
s “Combinatorics” and “Transforms”.
r
e
v ISO 80000 consists of the following parts, under the general title Quantities and units:
i
n
U
⎯ Part 1: General
e
h
T ⎯ Part 2: Mathematical signs and symbols to be used in the natural sciences and technology1)
,
g ⎯ Part 3: Space and time
n
o
K ⎯ Part 4: Mechanics
g
n ⎯ Part 5: Thermodynamics
o
H
⎯ Part 7: Light
f
o
⎯ Part 8: Acoustics
y
t
i
rs ⎯ Part 9: Physical chemistry and molecular physics
e
v
i ⎯ Part 10: Atomic and nuclear physics
n
U
⎯ Part 11: Characteristic numbers
e
h
T ⎯ Part 12: Solid state physics
:
y
p
o
c 1) Title to be shortened to read “Mathematics” in the second edition of ISO 80000-2.
d
e
s
n
iv © ISO 2009 – All rights reserved
e
c 标准分享网 www.bzfxw.com 免费下载
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
IEC 80000 consists of the following parts, under the general title Quantities and units:
0
2
/ ⎯ Part 6: Electromagnetism
1
0
4/ ⎯ Part 13: Information science and technology
2
f ⎯ Part 14: Telebiometrics related to human physiology
o
s
a
t
c
e
r
r
o
c
n
o
i
s
r
e
V
,
g
n
o
K
g
n
o
H
f www.bzfxw.com
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
e
h
T
:
y
p
o
c
d
e
s
n
© ISO 2009 – All rights reserved v
e
c
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
0
2
/
1
0
4/ Introduction
2
f
o Arrangement of the tables
s
a
The first column “Item No.” of the tables contains the number of the item, followed by either the number of the
ct corresponding item in ISO 31-11 in parentheses, or a dash when the item in question did not appear in
e ISO 31-11.
r
r
o
The second column “Sign, symbol, expression” gives the sign or symbol under consideration, usually in the
c
context of a typical expression. If more than one sign, symbol or expression is given for the same item, they
n
o are on an equal footing. In some cases, e.g. for exponentiation, there is only a typical expression and no
si symbol.
r
e
V
The third column “Meaning, verbal equivalent” gives a hint on the meaning or how the expression may be read.
, This is for the identification of the concept and is not intended to be a complete mathematical definition.
g
n
o The fourth column “Remarks and examples” gives further information. Definitions are given if they are short
K
enough to fit into the column. Definitions need not be mathematically complete.
g
n
The arrangement of the table in Clause 16 “Coordinate systems” is somewhat different.
o
H
f www.bzfxw.com
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
e
h
T
:
y
p
o
c
d
e
s
n
vi © ISO 2009 – All rights reserved
e
c 标准分享网 www.bzfxw.com 免费下载
i
L
)
c
(
, BS ISO 80000-2:2009
7
4
: INTERNATIONAL STANDARD ISO 80000-2:2009(E)
3
0
0
1
0
2
/
1
0
4/ Quantities and units —
2
f
o Part 2:
s
a Mathematical signs and symbols to be used in the natural
t
c sciences and technology
e
r
r
o
c
n
o
si 1 Scope
r
e
V
ISO 80000-2 gives general information about mathematical signs and symbols, their meanings, verbal
,
g equivalents and applications.
n
o
The recommendations in ISO 80000-2 are intended mainly for use in the natural sciences and technology, but
K
also apply to other areas where mathematics is used.
g
n
o
H 2 Normative references
f www.bzfxw.com
o
y SI The following referenced documents are indispensable for the application of this document. For dated
tB references, only the edition cited applies. For undated references, the latest edition of the referenced
i
s document (including any amendments) applies.
r
e
v ISO 80000-1:—2) , Quantities and units — Part 1: General
i
n
U
e
3 Variables, functions and operators
h
T
g, Variables such as x, y, etc., and running numbers, such as i in Σi xi are printed in italic (sloping) type.
Parameters, such as a, b, etc., which may be considered as constant in a particular context, are printed in
n
o italic (sloping) type. The same applies to functions in general, e.g. f, g.
K
g An explicitly defined function not depending on the context is, however, printed in Roman (upright) type, e.g.
n sin, exp, ln, Γ. Mathematical constants, the values of which never change, are printed in Roman (upright) type,
o e.g. e = 2,718 218 8…; π = 3,141 592…; i2 = −1. Well-defined operators are also printed in Roman (upright)
H
style, e.g. div, δ in δx and each d in df/dx.
f
o
Numbers expressed in the form of digits are always printed in Roman (upright) style, e.g. 351 204; 1,32; 7/8.
y
t
i
s The argument of a function is written in parentheses after the symbol for the function, without a space
r
e between the symbol for the function and the first parenthesis, e.g. f(x), cos(ωt + ϕ). If the symbol for the
v function consists of two or more letters and the argument contains no operation symbol, such as +, −, ×, ⋅ or / ,
i
n
the parentheses around the argument may be omitted. In these cases, there should be a thin space between
U
the symbol for the function and the argument, e.g. int 2,4; sin nπ; arcosh 2A; Ei x.
e
h If there is any risk of confusion, parentheses should always be inserted. For example, write cos(x) + y; do not
T
write cos x + y, which could be mistaken for cos(x + y).
:
y
p
o
c 2) To be published. (Revision of ISO 31-0:1992)
d
e
s
n © ISO 2009 – All rights reserved 1
e
c
i
L
)
c
(
, BS ISO 80000-2:2009
7
4 ISO 80000-2:2009(E)
:
3
0
0
1
A comma, semicolon or other appropriate symbol can be used as a separator between numbers or
0
2 expressions. The comma is generally preferred, except when numbers with a decimal comma are used.
/
1
0 If an expression or equation must be split into two or more lines, one of the following methods shall be used.
/
4
2 a) Place the line breaks immediately after one of the symbols =, +, −, ± or ∓, or, if necessary, immediately
of after one of the symbols ×, ⋅, or /. In this case, the symbol indicates that the expression continues on the
next line or next page.
s
a
ct b) Place the line breaks immediately before one of the symbols =, +, −, ± or ∓, or, if necessary, immediately
e before one of the symbols ×, ⋅, or /. In this case, the symbol indicates that the expression is a continuation
r
r of the previous line or page.
o
c
The symbol shall not be given twice around the line break; two minus signs could for example give rise to sign
n
o errors. Only one of these methods should be used in one document. If possible, the line break should not be
si inside of an expression in parentheses.
r
e
V It is customary 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,
,
g however, be explained if necessary.
n
o
K
g
n
o
H
f www.bzfxw.com
o
y SI
tB
i
s
r
e
v
i
n
U
e
h
T
,
g
n
o
K
g
n
o
H
f
o
y
t
i
s
r
e
v
i
n
U
e
h
T
:
y
p
o
c
d
e
s
n 2 © ISO 2009 – All rights reserved
e
c 标准分享网 www.bzfxw.com 免费下载
i
L
