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IS0
STANDARD 31-11
Second edition
1992-12-15
Quantities and units -
Part 11:
Mathematical signs and symbols for use in the
physical sciences and technology
Grandeurs et unit& -
Partie I 1: Signes et symboles mathgmatiques 9 employer dans /es
sciences physiques et dans la technique
Reference number
IS0 31-I 1:1992(E)
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IS0 31-l 1:1992(E)
Foreword
IS0 (the International Organization for Standardization) is a worldwide
federation of national standards bodies (IS0 member bodies). The work
of preparing International Standards is normally carried out through IS0
technical 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. IS0
collaborates closely with the International Electrotechnical Commission
(IEC) on all matters of electrotechnical standardization.
Draft International 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.
International Standard IS0 31-11 was prepared by Technical Committee
lSO/TC 12, Quantities, units, symbols, conversion factors.
This second edition cancels and replaces the first edition
e major technical changes from the first edition are
(IS0 31-11 :I 978). Th
the following:
- a new clause on coordinate systems has been added;
- some new items have been added in the old clauses.
The scope of Technical Committee lSO/TC 12 is standardization of units
and symbols for quantities and units (and mathematical symbols) used
within the different fields of science and technology, giving, where
necessary, definitions of the quantities and units. Standard conversion
factors for converting between the various units also come under the
scope of the TC. In fulfilment of this responsibility, lSO/TC 12 has pre-
pared IS0 31.
0 IS0 1992
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced
or utilized in any form or by any means, electronic or mechanical, including photocopying and
microfilm, without permission in writing from the publisher.
International Organization for Standardization
Case Postale 56 l CH-1211 Geneve 20 l Switzerland
Printed in Switzerland
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IS0 31-11:1992(E)
0 IS0
IS0 31 consists of the following parts, under the general title Quantities
and units:
- Part 0: General principles
- Part I: Space and time
- Part 2: Periodic and related phenomena
- Part 3: Mechanics
- Part 4: Heat
- Part 5: Electricity and magnetism
- Part 6: Light and related electromagnetic radiations
- Part 7: Acoustics
- Part 8: Physical chemistry and molecular physics
- Part 9: Atomic and nuclear physics
- Par? 10: Nuclear reactions and ionizing radiations
- Part 11: Mathematical signs and symbols for use in the physical
sciences and technology
- Part 12: Characteristic numbers
- Pati 13: Solid state physics
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IS0 31-11:1992(E) 0 IS0
Introduction
0.1 General
If more than one sign, symbol or expression is given for the same item,
they are on an equal footing. Signs, symbols and expressions in the “Re-
marks” column are given for information.
Where the numbering of an item has been changed in the revision of a
part of IS0 31, the number in the preceding edition is shown in parenth-
eses below the new number for the item; a dash is used to indicate that
the item in question did not appear in the preceding edition.
0.2 Variables, functions and operators
Variables, such as x, y, etc., and running numbers, such as i in Ci xi, are
printed in italic (sloping) type. Also parameters, such as a, b, etc., which
may be considered as constant in a particular context, are printed in italic
(sloping) type. The same applies to functions in general, e.g.5 g.
An explicitly defined function is, however, printed in Roman (upright) type,
e.g. sin, exp, In, r. Mathematical constants, the values of which never
change, are printed in Roman (upright) type, e.g. e = 2,718 281 8.;
IT = 3,141 592 6.; i* = - 1. Well defined operators are also printed in up-
right style, e.g. div, 6 in 6~ and each d in dfldx.
Numbers expressed in the form of digits are always printed upright, e.g.
351 204; 1,32; 718.
The argument of a function is written in parentheses after the symbol for
the function, without a space between the symbol for the function and the
first parenthesis, e.g. f(x), cos(wt + q). If the symbol for the function
consists of two or more letters and the argument contains no operation
sign, such as +; -; x; 9; or /, the parentheses around the argument may
be omitted. In these cases, there should be a thin space between the
symbol for the function and the argument, e.g. ent 2,4; sin no;
arcosh 2A; Ei X.
If there is any risk of confusion, parentheses should always be inserted.
For example, write COS(X) + y or (cos x) + y; do not write cos x + y, which
could be mistaken for COS(X + y) .
If an expression or equation must be split into two or more lines, the
line-breaks should preferably be immediately after one of the signs =; +;
-; -t-; or T; or, if necessary, immediately after one of the signs x; l ; or /.
In this case, the sign works like a hyphen at the end of the first line, in-
forming the reader that the rest will follow on the next line or even on the
next page. The sign should not be repeated at the beginning of the fol-
lowing line; two minus signs could for example give rise to sign errors.
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0 IS0
IS0 31-11:1992(E)
0.3 Scalars, vectors and tensors
Scalars, vectors and tensors are used to denote certain physical quantities.
They are as such independent of the particular choice of coordinate sys-
tem, whereas each component of a vector or a tensor depends on that
choice.
It is important to distinguish between the “components of a vector” a, i.e.
an, a, and az, and the “component vectors ”, i.e. axex, a,e, and azez.
position vector are equal to the Cartesian
The cartesia n corn ents of the
Pan
the position vector.
coordinates of the nt given by
PO1
Instead of treating each component as a physical quantity (i.e. numerical
value x unit), the vector could be written as a numerical-value vector
multiplied by the unit. All units are scalars.
EXAMPLE
numerical-value vector
component F,
I I
F= (3 N, -2 N, 5 N) = (3, -2, 5) N
I
1 \
numerical value unit
unit
The same considerations apply to tensors of second and higher orders.
V
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ISO 31-l 1:1992(E)
INTERNATIONAL STANDARD 0 ISO
Quantities and units -
Part 11:
Mathematical signs and symbols for use in the physical
sciences and technology
of this part of IS0 31. At the time of publication, the
1 Scope
edition indicated was valid. All standards are subject
to revision, and parties to agreements based on this
This part of IS0 31 gives general information about
part of IS0 31 are encouraged to investigate the
mathematical signs and symbols, their meanings,
possibility of applying the most recent edition of the
verbal equivalents and applications.
standard indicated below. Members of IEC and IS0
The recommendations in this part of IS0 31 are in- maintain registers of currently valid International
tended mainly for use in the physical sciences and
Standards.
technology.
IS0 31-0:1992, Quantities and units - Part 0: Gen-
era/principles.
2 Normative reference
The following standard contains provisions which,
through reference in this text, constitute provisions
1
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0 IS0
IS0 31-l 1:1992(E)
3 MATHEMATICAL LOGIC
Symbol,
Meaning, verbal equivalent and remarks
Item No. Application Name of symbol
sign
11-3.1 A conjunction sign
P and 4
PA4
(I I-2.1)
11-3.2 v disjunction sign p or 4 (or both)
PV
(I l-2.2)
11-3.3 1 negation sign negation of p; not p; non p
1P
(11-2.3)
11-3.4 * implication sign if p then 4; p implies 4
p-4
(11-2.4)
Can also be written 4 = p.
Sometimes -+ is used.
p = 4 and 4 ap; p is equivalent to 4
11-3.5 - equivalence sign
P*4
(11-2.5)
Sometimes w is used.
universal quantifier for every x belonging to A, the proposition
11-3.6 v VXEA p(x)
p(x) is true
(M-2-6)
(vx EA) PM
If it is clear from the context which set A
is being considered, the notation V x p(x)
can be used.
ForxEA, see 11-4.1.
there exists an x belonging to A for which
3xeA p(x) existential quantifier
11-3.7 3
p(x) is true
(U-2.7)
(3-A) PM
If it is clear from the context which set A
is being considered, the notation 3 x p(x)
can be used.
ForxEA, see W-4.1.
1
Y or 3 is used to indicate the existence
of one and only one element for which
p(x) is true.
2
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Q IS0
IS0 31=11:1992(E)
4 SETS
Symbol,
Item No. Application Meaning, verbal equivalent Remarks and examples
sign
XEA
11-4.1 E x belongs to A; x is an
(I I-1.1) element of the set A
11-4.2 $
y does not belong to A; The symbol & is also used.
YW
(1 I-1.2)
y is not an element of the
set A
11-4.3 3 A3X the set A contains x (as A 3 x has the same meaning as x f A.
(I I-1.3) element)
11-4.4 $ the set A does not contain A $ y has the same meaning as y 4 A.
A$Y
(I I-1.4) y (as element) The symbol $ is also used.
11-4.5 { ) set with elements Also [xi5 e I), where I denotes a set of
[x,1 $1 l **I XpJ
(I l-1.5) indices.
XII $1 . ‘I xn
11-4.6 set of those elements of EXAMPLE
( I I 1-A IP(
(I I-1.6) A for which the {XE[WIX<5)
proposition p(x) is true If it is clear from the context which set A
is being considered, the notation
(X 1 p(x)] can be used.
EXAMPLE
{xIxG5}
11-4.7 card card (A) number of elements in A;
cardinal of A
(4
11-4.8 0 the empty set
(11-W)
11-4.9 rid N the set of natural N = [O, 1, 2,3, . .}
(I I-1.8) numbers; Exclusion of zero from the sets 1 l-4.9
the set of positive
to 1 l-4.13 is denoted by an asterisk,
integers and zero
e.g. Ill*.
. . . . k- I]
&= I I
(0 I
11-4.10 z z
the set of integers -2, -1, 0, 1, 2,.)
z I
(I I-1.9)
Se: remark to 1 l-4.9.
11-4.11 Q Q
the set of rational See remark to 1 l-4.9.
(I I-l. 10)
numbers
11-4.12 R R the set of real numbers
See remark to 1 l-4.9.
(I l-l. 1 I)
11-4.13 a= c the set of complex See remark to 1 l-4.9.
(I l-l. Ii?) numbers
3
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IS0 31-l 1:1992(E)
0 IS0
4 SETS (continued)
Symbol,
Item No. Application Meaning, verbal equivalent Remarks and examples
sign
11-4.14 [,I
closed interval in R from a [a, b] = (x E R 1 a < x < b}
Cal 4
(included) to b (included)
(-1
11-4.15 I,]
left half-open interval in IR ]a, b] = (x E R 1 a < x < b}
14 4
from a (excluded) to b
t-1 I
( 1 (a, 4
(included)
11-4.16 [ ,[ right half-open interval in R [a, b[ = (x E R 1 a < x < b}
Cal bC
from a (included) to b
(-1 I
c > Cal b)
(excluded)
11-4.17 ] ,[
open interval in R from a ]a, b[ = (x E R I a < x < b}
14 bC
(-1 (excluded) to b (excluded)
I
( 1 (4 b)
11-4.18 c BcA B is included in A;
Every element of B belongs to A.
(11-1.13)
B is a subset of A c is also used, but see remark to 11-4.19.
11-4.19 c BcA B is properly included in A; Every element of B belongs to A, but B is
(I I-1.14) B is a proper subset of A not equal to A.
If c is used for 1 I-4.1 8, then s shall be
used for 11-4.19.
C is not included in A;
11-4.20 $ q is also used.
(11-1.75) C is not a subset of A The symbols $ and $ are also used.
11-4.21 2 AzB A includes B (as subset) A contains every element of B.
(I 7-I. 16) I is also used, but see remark to 1 l-4.22.
A 2 B has the same meaning as B c A.
11-4.22 I AxB A includes B properly A contains every element of B, but A is
(I I-I. 17) not equal to B.
If =) is used for 11-4.21, then 2 shall be
used for 1 l-4.22.
A 3 B has the same meaning as B c A.
A does not include C (as
11-4.23 2 a is also used.
AW
(1 l-l. 18) subset) The symbols 2 and * are also used.
A 2 C has the same meaning as C $ A.
11-4.24 u union of A and B The set of elements which belong to A
AuB
(1 l-l. 19) or to B or to both A and B.
AuB=(xlx~Avx~B}
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0 IS0 IS0 3141:1992(E)
4 SETS (continued)
Symbol,
Application Meaning, verbal equivalent Remarks and examples
Item No.
sign
n
cAi=A, UA,U.**UAn,
11-4.25 U union of a collection of sets
UA i
i=l
i=l
(I l-1.20)
A,r em-1 An
the set of elements belonging to at least
one of the sets A,, . . . . An.
n
U i=l and Uf Uiel
id
are also used, where I denotes a set of
indices.
intersection of A and B, The set of elements which belong to both
11-4.26 n AnB
read as A inter B A and B.
(I l-1.21)
A~B={xIxEAAxEB)
n
intersection of a collection fiAi=A, nA,n.nA,,
11-4.27 n
nA
i
i=l i=l
of sets A,, . . . . An
(1 l-1.22)
the set of elements belonging to all sets
A,, A*, . . . and A,.
n
n and nl fli,z
i=l
id
are also used, where I denotes a set of
indices.
difference between A and The set of elements which belong to A,
11-4.28 \
A\B
but not to B.
(11-1.23)
B;
A\B= (xIxEAAx~B)
A minus B
A -B should not be used.
complement of subset B The set of those elements of A which do
11-4.29 [
CB
A
not belong to the subset B.
(11-1.24) of A
If it is clear from the context which set A
is being considered, the symbol A is often
omitted.
Also CAB = A \ B
11-4.30 ( ,) ordered pair a, b; (a, b) = (c, d) if and only if a = c and
(a, b)
(11-1.25) couple a, b b = d.
(a, b) is also used.
11-4.31 ( , . . . . ) (a,, a2, . . . . a,) ordered n-tuplet a,) is also used.
(al, 3, .-,
(11-1.26)
5
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0 IS0
IS0 31-l 1:1992(E)
4 SETS (concluded)
Symbol,
Application Meaning, verbal equivalent Remarks and examples
Item No.
sign
11-4.32 x AxB Cartesian product of A The set of ordered pairs (a, b) such that
(I I-1.27) and B aeAand bEB.
AxB=((a,b) )aEAAbeB)
x A is denoted by An, where yt
A xA x . . .
is the number of factors in the product.
11-4.33 A set of pairs (x, X) of A x A,
AA = {(XIX) 1-A)
AA
where x E A; idA is also used.
(4
diagonal of the set A x A
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63 IS0 IS0 31=11:1992(E)
5 MISCELLANEOUS SIGNS AND SYMBOLS
Symbol,
Item No. Application Meaning, verbal equivalent Remarks and examples
sign
=
a is equal to b = may be used to emphasize that a
11-5.1 = a b
particular equality is an identity.
( I l-3. I)
11-5.2 # a is not equal to b The symbol + is also used.
a#b
(V-3.2)
def
11-5.3 = a def b a is by definition equal EXAMPLE
to b def mv, where p is momentum, m is
(W-3.3)
P
mass and v is velocity.
2 and
:= are also used.
a corresponds to b EXAMPLES
11-5.4 s aeb
When E = kT, 1 eV G 11 604,5 K.
(11-3.4)
When 1 cm on a map corresponds
to a length of 10 km, one may write
Icm~lOkm.
11-5.5 x a x b a is approximately equal The symbol = is reserved for “is
(W-3.5) to b asymptotically equal to,,. See 1 l-7.7.
11-5.6 - a- b a is proportional to b
(M-3.6) oc aocb
11-5.7 < a (W-3.7)
11-5.8 > b>a b is greater than a
(11-3.8)
a is less than or equal The symbols rs and 5 are also used.
11-5.9 < a (I l-3.9) to b
11-5.10 > baa b is greater than or equal The symbols 2 and 2 are also used.
(11-310) to a
11-5.11 < a << b a is much less than b
(II-3 11)
11-5.12 > b is much greater
b > a
than a
(11-3.12)
11-5.13 00 infinity
(II-3 13)
7
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IS0 31-I 1:1992(E) Q IS0
5 MISCELLANEOUS SIGNS AND SYMBOLS (concluded)
Symbol,
Item No. Application
Meaning, verbal equivalent Remarks and examples
sign
11-5.14 ( ) ac + bc, parentheses
(a + b)c In ordinary algebra the sequence of ( ),
t-1 [a + b]c ac + bc, square brackets [ 1, ( ) and ( ) in order of nesting is not
Cl
(a + b)c ac + bc, braces standardized. Special uses are made of
0
(a + b)c ac + bc, angle brackets ( ), [ 1, ( ) and ( ) in particular fields.
0
1 I-5.15 // AB // CD the line AB is parallel to the
line CD
(4
11-5.16 1 AB L CD the line AB is perpendicular
to the line CD
i-1
i
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IS0 31=11:1992(E)
0 IS0
6 OPERATIONS
Meaning, verbal
Remarks and examples
Symbol, application
Item No.
equivalent
a plus b
11-6.1 a+b
(U-4.1)
a minus b
11-6.2 a - b
(11-4.2)
a plus or minus b
11-6.3 a&b
(4
a minus or plus-b -(a&b)=-arb
1 l-6.4 aTb
(-1
See also 1 l-4.32, 1 I-13.6 and 1 l-l 3.7.
a multiplied by b
axb ab
11-6.5 amb
The sign for multiplication of numbers is a
(11-4.3)
cross (x) or a dot half high (m). If a dot is used
as the decimal sign, only the cross shall be
used for multiplication of numbers. For
decimal sign see IS0 31-0:1992,
subclause 3.3.2.
See also IS0 31-0:1992, subclause 3.1.3.
a divided by b
11-6.6 $ a/b ab-’
(U-4.4)
n
al + a2 + . . . + a,
11-6.7
ai
c
(U-4.5) i=l
n
al 8 a2 m . . . . a,
11-6.8 Also ‘nn= , ail nail ‘n i ail ‘nai
ai
I-I
i
(11-4.6) i=l
a to the power p
11-6.9 a’
(11-4.7)
a to the power l/2;
If a >/ 0, then a 30.
11-6.10 a1/2 a+
d-
Ja fi
square root of a
See remark to 1 l-6.1 1.
(W-4.8)
a to the power l/n;
11-6.11
l/n a+ n a
If a > 0, then n a > 0.
J
nth root of a II-
(17-4.9) a
n
If the symbol J or “J acts on a composite
a
II- expression, parentheses shall be used to
avoid ambiguity.
absolute value of a; abs a is also used.
11-6.12 Ial
magnitude of a;
(11-4.10)
modulus of a
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0 IS0
IS0 31-11:1992(E)
6 OPERATIONS (concluded)
Meaning, verbal
Remarks and examples
Item No. Symbol, application
equivalent
11-6.13 sgn a Signum a For real a:
(11-4.11)
1 ifa>O
sgn a = Oifa=O
-1 ifa<
I
For complex a, see 1 l-l 0.7.
mean value of a The method of forming the mean shall be
11-6.14 Zi (a)
stated if not clear from the context.
(11-4.12)
n
factorial yt For yt 3 1: n! = k=l x2x3x.xn
11-6.15 n!
I-I
k=l
(11-4.13)
For n = 0: n! = 1
11-6.16 yt binomial coefficient n, p n
nl .
=
(W-4.14) p cfi
! n-p)!
P
( 1 ( 1
PC
11-6.17 ent a the greatest integer less ent 2,4 = 2
(W4.?5) E(a) than or equal to a;
ent( -2,4) = -3
characteristic of a
[a] or int a is sometimes used for ent a, but
is now often used with the meaning “integer
part of a,,, e.g.
= int 2,4 = 2
[2 41
[ 12,4] = int( -2,4) = -2
10
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IS0 31=11:1992(E)
0 IS0
7 FUNCTIONS
Meaning, verbal
Remarks and examples
Symbol, application
Item No.
equivalent
function f A function may also be denoted by x Ha.
11-7.1 f
Letters other than fare also used.
(I I-5.1)
value of the function f at
11-7.2 x
f( >
f(x, y, . .) x or at (x, y, . .)
(11-5.2)
respectively
This notation is used mainly when evaluating
11-7.3 x 4:
f(b) -f(a)
f( >I
definite integrals.
(1 l-5.3)
b
H x >I a
the composite function of (g oj) (x) = g@(x))
11-7.4 gof
f and g, read as g circle f
(11-54)
x tends to a
11-7.5 x+a
(11-5.5)
X--,a f(x) = b may be written f(x) + b as
lim f(x) limit of f(x) as x tends to Irm ’
11-7.6
a x --+ a.
(11-5.6) x4a
.
Limits “from the right,, (x > a) and “from the
Irm X
x-+a f( >
left,, (X < a) may be denoted by
.
Itm X.+a+ f(x) and lim x+a- f(x) respectively.
EXAMPLE
is asymptotically equal to
11-7.7 N
(11-57)
1
1
- - as x + a,
sin(x-a) - x - a
If(x)/g(x) I is bounded
11-7.8 O(g(x))
above in the limit implied
(1 l-5.6) f(x) = O(g(x))
by the context;
f is of the order of g
f(x)/g(x) + 0 in the limit
1 l-7.9 o(g(x))
implied by the context;
(~~-5.9) f(x) = o(g(x))
f is of lower order than g
(finite) increme
...