ISO 80000-2:2009
(Main)Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology
Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology
ISO 80000-2:2009 gives general information about mathematical signs and symbols, their meanings, verbal equivalents and applications. The recommendations in ISO 80000-2:2009 are intended mainly for use in the natural sciences and technology, but also apply to other areas where mathematics is used.
Grandeurs et unités — Partie 2: Signes et symboles mathématiques à employer dans les sciences de la nature et dans la technique
L'ISO 80000-2:2009 donne des informations générales sur les signes et symboles mathématiques, leurs sens, leurs énoncés et leurs applications. Les recommandations données dans l'ISO 80000-2:2009 sont principalement destinées à être utilisées dans les sciences de la nature et dans la technique. Cependant, elles s'appliquent également à d'autres domaines utilisant les mathématiques.
Veličine in enote - 2. del: Matematični znaki in simboli za uporabo v naravoslovnih vedah in tehniki
Standard ISO 80000-2 podaja splošne informacije o matematičnih znakih in simbolih, njihovem pomenu, besednih ekvivalentih in uporabi. Priporočila v standardu ISO 80000-2 so namenjena zlasti uporabi v naravoslovnih vedah in tehniki, vendar se uporabljajo tudi na drugih področjih, kjer se uporablja matematika.
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INTERNATIONAL ISO
STANDARD 80000-2
First edition
2009-12-01
Quantities and units —
Part 2:
Mathematical signs and symbols to be
used in the natural sciences and
technology
Grandeurs et unités —
Partie 2: Signes et symboles mathématiques à employer dans les
sciences de la nature et dans la technique
Reference number
ISO 80000-2:2009(E)
©
ISO 2009
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ISO 80000-2:2009(E)
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ii © ISO 2009 – All rights reserved
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ISO 80000-2:2009(E)
Contents Page
Foreword .iv
Introduction.vi
1 Scope.1
2 Normative references.1
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 geometry.10
9 Operations.11
10 Combinatorics .14
11 Functions.15
12 Exponential 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
Bibliography.40
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ISO 80000-2:2009(E)
Foreword
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
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. ISO collaborates closely with 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 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.
Attention is drawn to the possibility that some of the elements of this document may 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 31-11:1992, 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 general title Quantities and units:
⎯ Part 1: General
1)
⎯ Part 2: Mathematical signs and symbols to be used in the natural sciences and technology
⎯ 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: Atomic 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.
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ISO 80000-2:2009(E)
IEC 80000 consists of the following parts, under the general title Quantities and units:
⎯ Part 6: Electromagnetism
⎯ Part 13: Information science and technology
⎯ Part 14: Telebiometrics related to human physiology
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ISO 80000-2:2009(E)
Introduction
Arrangement of the tables
The first column “Item No.” of the 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
context 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 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 be mathematically complete.
The arrangement of the table in Clause 16 “Coordinate systems” is somewhat different.
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INTERNATIONAL STANDARD ISO 80000-2:2009(E)
Quantities and units —
Part 2:
Mathematical signs and symbols to be used in the natural
sciences and technology
1 Scope
ISO 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 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.
2)
ISO 80000-1:— , Quantities and units — Part 1: General
3 Variables, functions and operators
Variables such as x, y, etc., and running numbers, such as i in Σ x are printed in italic (sloping) type.
i i
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. f, g.
An explicitly defined function not depending on the context is, however, printed in Roman (upright) type, e.g.
sin, exp, ln, Γ. Mathematical constants, the values of which never change, are printed in Roman (upright) type,
2
e.g. e = 2,718 218 8…; π = 3,141 592…; i = −1. Well-defined operators are also printed in Roman (upright)
style, e.g. div, δ in δx 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 function, without a space
between the symbol for the function and the first parenthesis, e.g. f(x), cos(ωt + ϕ). 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 cases, there should be a thin space between
the symbol for the function and the argument, e.g. int 2,4; sin nπ; 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) To be published. (Revision of ISO 31-0:1992)
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ISO 80000-2:2009(E)
A comma, semicolon or other appropriate symbol can be used as a separator between numbers or
expressions. The comma is generally preferred, except when numbers 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, 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 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 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,
however, be explained if necessary.
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ISO 80000-2:2009(E)
4 Mathematical logic
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-4.1 conjunction of p and q,
p ∧ q
(11-3.1) p and q
disjunction of p and q,
2-4.2 This “or” is inclusive, i.e.
p ∨ q
(11-3.2) p or q p ∨ q is true, if either p or q, or both are
true.
negation of p,
2-4.3
¬ p
(11-3.3) not p
2-4.4 p implies q,
p ⇒ q q ⇐ p has the same meaning as p ⇒ q.
(11-3.4) if p, then q
⇒ is the implication symbol.
2-4.5 p is equivalent to q
p ⇔ q (p ⇒ q) ∧ (q ⇒ p) has the same meaning as
(11-3.5)
p ⇔ q.
⇔ is the equivalence symbol.
2-4.6 for every x belonging to A, the lf it is clear from the context which set A is
∀x ∈ A p(x)
proposition p(x) is true
(11-3.6) being considered, the notation ∀x p(x) can
be used.
∀ is the universal quantifier.
For x ∈ A, see 2-5.1.
2-4.7 there exists an x belonging to A lf it is clear from the context which set A is
∃x ∈ A p(x)
for which p(x) is true
(11-3.7) being considered, the notation ∃x p(x) can
be used.
∃ is the existential quantifier.
For x ∈ A, see 2-5.1.
1
∃ x p(x) is used to indicate that there is
exactly one element for which p(x) is true.
1
∃! is also used for ∃ .
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ISO 80000-2:2009(E)
5 Sets
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-5.1 x belongs to A,
x ∈ A A ∋ x has the same meaning as x ∈ A.
(11-4.1) x is an element of the set A
y does not belong to A,
2-5.2
y ∉ A A ∌ y has the same meaning as y ∉ A.
(11-4.2) y is not an element of the set A
The negating stroke may also be vertical.
2-5.3 {x , x , …, x } set with elements x , x , …, x
Also {x | i ∈ I}, where I denotes a set of
1 2 n 1 2 n
i
(11-4.5) subscripts.
EXAMPLE {x ∈ R | x u 5}
2-5.4 set of those elements of A for
{x ∈ A | p(x)}
lf it is clear from the context which set A is being
which the proposition p(x) is
(11-4.6)
considered, the notation {x | p(x)} can be used
true
(for example {x | x u 5}, if it is clear that x is a
variable for real numbers).
2-5.5 card A number of elements in A, The cardinality can be a transfinite number.
(11-4.7) cardinality of A See also 2-9.16.
A
2-5.6 the empty set
∅
(11-4.8)
2-5.7 B is included in A, Every element of B belongs to A.
B ⊆ A
(11-4.18) B is a subset of A
⊂ is also used, but see remark to 2-5.8.
A ⊇ B has the same meaning as B ⊆ A.
2-5.8 B is properly included in A, Every element of B belongs to A, but at
B ⊂ A
least one element of A does not belong
(11-4.19) B is a proper subset of A
to B.
lf ⊂ is used for 2-5.7, then ⊊ shall be used
for 2-5.8.
A ⊃ B has the same meaning as B ⊂ A.
2-5.9 union of A and B The set of elements which belong to A or
A ∪ B
to B or to both A and B.
(11-4.24)
A ∪ B = {x | x ∈ A ∨ x ∈ B}
2-5.10 intersection of A and B The set of elements which belong to both A
A ∩ B
and B.
(11-4.26)
A ∩ B = {x | x ∈ A ∧ x ∈ B}
2-5.11 n union of the sets A , A , …, A The set of elements belonging to at least
1 2 n
A one of the sets A , A , ., A
∪ i 1 2 n
(11-4.25)
i=1
n
, and are also used,
∪ ∪ ∪
A ∪ A ∪ …
i=1 iI∈
1 2
iI∈
∪ A
n
where I denotes a set of subscripts.
2-5.12 n intersection of the sets A , …, A The set of elements belonging to all sets
1 n
A A , A , ., A
∩ i 1 2 n
(11-4.27)
i=1
n
, and are also used,
∩ ∩ ∩
A ∩ A ∩ … i=1 iI∈
1 2
iI∈
∩ A
n
where I denotes a set of subscripts.
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ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-5.13 difference of A and B, The set of elements which belong to A but
A ∖ B
not to B.
(11-4.28) A minus B
A ∖ B = {x | x ∈ A ∧ x ∉ B}
A − B should not be used.
∁ B is also used. ∁ B is mainly used
A A
when B is a subset of A, and the symbol A
may be omitted if it is clear from the
context which set A is being considered.
2-5.14 (a, b) ordered pair a, b, (a, b) = (c, d) if and only if a = c and b = d.
(11-4.30) couple a, b
If the comma can be mistaken as the
decimal sign, then the semicolon (;) or a
stroke (⏐) may be used as separator.
2-5.15 (a , a , …, a) ordered n-tuple See remark to 2-5.14.
1 2 n
(11-4.31)
2-5.16 A × B Cartesian product of the sets A The set of ordered pairs (a, b) such that
and B
(11-4.32) a ∈ A and b ∈ B.
A × B = {(x, y) | x ∈ A ∧ y ∈ B}
n The set of ordered n-tuples (x , x , …, x )
2-5.17 Cartesian product of the sets
1 2 n
such that x ∈ A , x ∈ A , …, x ∈ A .
A A , A , …, A
i 1 1 2 2 n n
1 2 n
(—) Π
i=1
n
A × A × . × A is denoted by A , where n is
A××AA.× the number of factors in the product.
12 n
id identity relation on A,
2-5.18
id is the set of all pairs (x, x) where x ∈ A.
A
A
(11-4.33) If the set A is clear from the context, the
diagonal of A × A
subscript A can be omitted.
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ISO 80000-2:2009(E)
6 Standard number sets and intervals
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-6.1 N the set of natural numbers, N = {0, 1, 2, 3, …}
(11.4.9) the set of positive integers and
N* = {1, 2, 3, …}
zero
Other restrictions can be indicated in an
obvious way, as shown below.
N = {n ∈ N | n > 5}
>5
The symbols IN and ℕ are also used.
Z
2-6.2 the set of integers Z = {…, −2, −1, 0, 1, 2, …}
(11.4.10)
Z* = {n ∈ Z | n ≠ 0}
Other restrictions can be indicated in an
obvious way, as shown below.
Z = {n ∈ Z | n W −3}
W−3
The symbol ℤ is also used.
2-6.3 Q the set of rational numbers
Q* = {r ∈ Q | r ≠ 0}
(11.4.11)
Other restrictions can be indicated in an
obvious way, as shown below.
Q = {r ∈ Q | r < 0}
<0
The symbols and ℚ are also used.
2-6.4 R the set of real numbers
R* = {x ∈ R | x ≠ 0}
(11.4.12)
Other restrictions can be indicated in an
obvious way, as shown below.
R = {x ∈ R | x W 0}
W0
The symbols IR and ℝ are also used.
2-6.5 C the set of complex numbers
C* = {z ∈ C | z ≠ 0}
(11.4.13)
The symbols ℂ and ℂ are also used.
2-6.6 P the set of prime numbers P = {2, 3, 5, 7, 11, 13, 17, …}
(—)
The symbols ℙ and ℙ are also used.
2-6.7 [a, b] closed interval from a included
[a, b] = {x ∈ R | a u x u b}
to b included
(11.4.14)
2-6.8 (a, b] left half-open interval from a
(a, b] = {x ∈ R | a < x u b}
excluded to b included
(11.4.15)
The notation ]a, b] is also used.
2-6.9 [a, b) right half-open interval from a
[a, b) = {x ∈ R | a u x < b}
included to b excluded
(11.4.16)
The notation [a, b[ is also used.
2-6.10 (a, b) open interval from a excluded to b
(a, b) = {x ∈ R | a < x < b}
(11.4.17) excluded
The notation ]a, b[ is also used.
closed unbounded interval up to b
2-6.11 (−∞, b]
(−∞, b] = {x ∈ R | x u b}
included
(—)
The notation ]−∞, b] is also used.
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ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-6.12 open unbounded interval up to b
(−∞, b)
(−∞, b) = {x ∈ R | x < b}
excluded
(—)
The notation ]−∞, b[ is also used.
2-6.13 [a, +∞) closed unbounded interval
[a, +∞) = {x ∈ R | a u x}
onward from a included
(—)
The notations [a, ∞ [, [a, +∞ [ and [a, ∞) are
also used.
2-6.14 (a, +∞) open unbounded interval onward
(a, +∞) = {x ∈ R | a < x}
from a excluded
(—)
The notations ]a, +∞[, ]a, ∞ [ and (a, ∞) are
also used.
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ISO 80000-2:2009(E)
7 Miscellaneous signs and symbols
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-7.1 a = b a is equal to b
The symbol ≡ may be used to emphasize
(11-5.1) that a particular equality is an identity.
See also 2-7.18.
2-7.2 a is not equal to b The negating stroke may also be vertical.
a ≠ b
(11-5.2)
a is by definition equal to b EXAMPLE
2-7.3 a := b
p := mv, where p is momentum, m is mass and v
(11-5.3)
is velocity.
The symbols = and ≝ are also used.
def
EXAMPLES
2-7.4 a corresponds to b
a ≙ b
When E = kT, then 1 eV ≙ 11 604,5 K
(11-5.4)
When 1 cm on a map corresponds to a length of
10 km, one may write 1 cm ≙ 10 km.
The correspondence is not symmetric.
2-7.5 a is approximately equal to b It depends on the user whether an
a ≈ b
approximation is sufficiently good. Equality
(11-5.5)
is not excluded.
EXAMPLE
2-7.6 a is asymptotically equal to b
a ≃ b
11
(11-7.7)
� as x → a
sin(xa−−) x a
(For x → a, see 2-7.16.)
2-7.7 a is proportional to b
a ∼ b The symbol ∼ is also used for equivalence
(11-5.6) relations.
The notation a ∝ b is also used.
2-7.8 M is congruent to N, M and N are point sets (geometrical
M ≅ N
figures).
(—) M is isomorphic to N
This symbol is also used for isomorphisms
of mathematical structures.
2-7.9 a < b a is less than b
(11-5.7)
2-7.10 b > a b is greater than a
(11-5.8)
2-7.11 a is less than or equal to b
a u b
(11-5.9)
2-7.12 b is greater than or equal to a
b W a
(11-5.10)
2-7.13 a is much less than b It depends on the user whether a is
a � b
sufficiently small as compared to b.
(11-5.11)
2-7.14 b is much greater than a It depends on the user whether b is
b � a
sufficiently great as compared to a.
(11-5.12)
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ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-7.15 infinity This symbol does not denote a number but
∞
is often part of various expressions dealing
(11-5.13)
with limits.
The notations +∞, -∞ are also used.
2-7.16 x → a x tends to a This symbol occurs as part of various
expressions dealing with limits.
(11-7.5)
a may be also ∞, +∞, or -∞.
2-7.17 m⏐n m divides n For integers m and n:
(—) ∃ k ∈ Z m⋅k = n
n is congruent to k modulo m For integers n, k and m:
2-7.18
n ≡ k mod m
m⏐(n − k)
(—)
See also 2-7.1.
(a + b) parentheses
2-7.19 It is recommended to use only parentheses
for grouping, since brackets and braces
[a + b] square brackets
(1-5.14)
often have a specific meaning in particular
{a + b} braces
fields. Parentheses can be nested without
〈a + b〉 angle brackets
ambiguity.
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ISO 80000-2:2009(E)
8 Elementary geometry
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-8.1 the straight line AB is parallel to
AB∥CD It is written g∥ h if g and h are the straight
the straight line CD
(11-5.15) lines determined by the points A and B,
and the points C and D, respectively.
AB//CD is also used.
2-8.2 the straight line AB is
AB⊥CD It is written g⊥ h if g and h are the straight
perpendicular to the straight
(11-5.16) lines determined by the points A and B,
line CD
and the points C and D, respectively. In a
plane, the straight lines must intersect.
2-8.3 angle at vertex B in the triangle The angle is not oriented, it holds that
∢ABC
ABC
(—) ∢ABC = ∢CBA and
0 u ∢ABC u π rad.
2-8.4 line segment from A to B The line segment is the set of points
AB
between A and B on the straight line AB.
(—)
2-8.5 → vector from A to B → →
AB If AB = CD then B, seen from A, is in the
(—)
same direction and distance as D is, seen
from C. It does not follow that A = C and
B = D.
d(A, B)
2-8.6 distance between points A and B The distance is the length of the line
(—) segment AB and also the magnitude of the
→
vector AB .
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ISO 80000-2:2009(E)
9 Operations
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-9.1 a + b a plus b This operation is named addition. The
symbol + is the addition symbol.
(11-6.1)
2-9.2 a minus b This operation is named subtraction. The
a − b
symbol − is the subtraction symbol.
(11-6.2)
2-9.3 a plus or minus b This is a combination of two values into
a ± b
one expression.
(11-6.3)
2-9.4 a minus or plus b
a ∓ b −(a ± b) = −a ∓ b
(11-6.4)
a multiplied by b,
2-9.5 a ⋅ b This operation is named multiplication. The
symbol for multiplication is a half-high dot
(11-6.5) a times b
a × b
(·) or a cross (×).
a b
Either may be omitted if no
ab
misunderstanding is possible.
See also 2-5.16, 2-5.17, 2-17.11, 2-17.12,
2-17.23 and 2-17.24 for the use of the dot
and cross in various products.
2-9.6 a divided by b
a a
–1
= a ⋅ b
(11-6.6) b b
See also ISO 80000-1:—, 7.1.3.
a/b
For ratios, the symbol : is also used.
EXAMPLE The ratio of height h to breadth b
of an A4 sheet is h : b = 2 .
The symbol ÷ should not be used.
n n
2-9.7 a + a + … + a ,
1 2 n
The notations a , a , a and
∑ i ∑ i ∑ i
a
i=1 i
∑ i
(11-6.7)
sum of a , a , …, a i
1 2 n
i=1
a are also used.
∑ i
n n
2-9.8
a ⋅ a ⋅ … ⋅ a ,
1 2 n
The notations a , a , a and
∏ i ∏ i ∏ i
a
i=1 i
∏ i
(11-6.8)
product of a , a , …, a i
1 2 n
i=1
a are also used.
∏ i
p 2
2-9.9 a to the power p
a The verbal equivalent of a is
3
a squared; the verbal equivalent of a is a
(11-6.9)
cubed.
1/2
a to the power 1/2,
2-9.10 a
lf a W 0, then a W 0.
(11-6.10) square root of a
a
The symbol √a should be avoided.
See remark to 2-9.11.
1/n
n
2-9.11 a to the power 1/n,
a
lf a W 0, then a W 0.
(11-6.11) nth root of a
n
n
a
The symbol √a should be avoided.
n
lf the symbol √ or √ acts on a composite
expression, parentheses shall be used to
avoid ambiguity.
© ISO 2009 – All rights reserved 11
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ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
Mean values obtained by other methods
2-9.12
x mean value of x,
are the
(11-6.14)
〈x〉 arithmetic mean of x
- harmonic mean denoted by subscript h,
x
a
- geometric mean denoted by subscript g,
- quadratic mean, often called “root mean
square”, denoted by subscript q or rms.
The subscript may only be omitted for the
arithmetic mean.
In mathematics x is also used for the
complex conjugate of x; see 2-14.6.
2-9.13 sgn a signum a For real a:
(11-6.13) 1if a >0
⎧
⎪
sgn a = 0if a =0
⎨
⎪
−<1if a 0
⎩
See also item 2-14.7.
2-9.14 inf M infimum of M
Greatest lower bound of a non-empty set
of numbers bounded from below.
(—)
2-9.15 sup M supremum of M
Smallest upper bound of a non-empty set
of numbers bounded from above.
(—)
2-9.16 absolute value of a, The notation abs a is also used.
|a|
Absolute value of real number a.
(11-6.12) modulus of a,
Modulus of complex number a; see 2-14.4.
magnitude of a
Magnitude of vector a; see 2-17.4.
See also 2-5.5.
floor a, The notation ent a is also used.
2-9.17 ⎣a⎦
EXAMPLES
(11-6.17) the greatest integer less than or
equal to the real number a
⎣2,4⎦ = 2
⎣−2,4⎦ = −3
2-9.18 ceil a, “ceil” is an abbreviation of the word
⎡a⎤
“ceiling”.
(—) the least integer greater than or
EXAMPLES
equal to the real number a
⎡2,4⎤ = 3
⎡−2,4⎤ = −2
int a integer part of the real number a
2-9.19 int a = sgn a ⋅ ⎣⏐a⏐⎦
EXAMPLES
(—)
int(2,4) = 2
int(−2,4) = −2
2-9.20 frac a fractional part of the real frac a = a − int a
number a
EXAMPLES
(—)
frac(2,4) = 0,4
frac(−2,4) = −0,4
12 © ISO 2009 – All rights reserved
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ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-9.21 min(a, b) minimum of a and b The operation generalizes to more
numbers and to sets of numbers. However,
(—)
an infinite set of numbers need not have a
smallest element.
2-9.22 max(a, b) maximum of a and b The operation generalizes to more
numbers and to sets of numbers. However,
(—)
an infinite set of numbers need not have a
greatest element.
© ISO 2009 – All rights reserved 13
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ISO 80000-2:2009(E)
10 Combinatorics
In this clause, n and k are natural numbers, with k u n.
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
n
2-10.1 n! factorial
n! = k = 1 ⋅ 2 ⋅ 3 ⋅ … ⋅ n (n > 0)
∏
(11-6.15)
...
SLOVENSKI STANDARD
SIST ISO 80000-2:2013
01-maj-2013
1DGRPHãþD
SIST ISO 31-11:1995
9HOLþLQHLQHQRWHGHO 0DWHPDWLþQL]QDNLLQVLPEROL]DXSRUDERYQDUDYRVORYQLK
YHGDKLQWHKQLNL
Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural
sciences and technology
Grandeurs et unités - Partie 2: Signes et symboles mathématiques à employer dans les
sciences de la nature et dans la technique
Ta slovenski standard je istoveten z: ISO 80000-2:2009
ICS:
01.060 9HOLþLQHLQHQRWH Quantities and units
01.075 Simboli za znake Character symbols
07.020 Matematika Mathematics
SIST ISO 80000-2:2013 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 80000-2:2013
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SIST ISO 80000-2:2013
INTERNATIONAL ISO
STANDARD 80000-2
First edition
2009-12-01
Quantities and units —
Part 2:
Mathematical signs and symbols to be
used in the natural sciences and
technology
Grandeurs et unités —
Partie 2: Signes et symboles mathématiques à employer dans les
sciences de la nature et dans la technique
Reference number
ISO 80000-2:2009(E)
©
ISO 2009
---------------------- Page: 3 ----------------------
SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
PDF disclaimer
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parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.
COPYRIGHT PROTECTED DOCUMENT
© ISO 2009
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 either ISO at the address below or
ISO's member body in the country of the requester.
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Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
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Web www.iso.org
Published in Switzerland
ii © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Contents Page
Foreword .iv
Introduction.vi
1 Scope.1
2 Normative references.1
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 geometry.10
9 Operations.11
10 Combinatorics .14
11 Functions.15
12 Exponential 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
Bibliography.40
© ISO 2009 – All rights reserved iii
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Foreword
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
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. ISO collaborates closely with 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 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.
Attention is drawn to the possibility that some of the elements of this document may 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 31-11:1992, 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 general title Quantities and units:
⎯ Part 1: General
1)
⎯ Part 2: Mathematical signs and symbols to be used in the natural sciences and technology
⎯ 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: Atomic 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.
iv © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
IEC 80000 consists of the following parts, under the general title Quantities and units:
⎯ Part 6: Electromagnetism
⎯ Part 13: Information science and technology
⎯ Part 14: Telebiometrics related to human physiology
© ISO 2009 – All rights reserved v
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Introduction
Arrangement of the tables
The first column “Item No.” of the 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
context 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 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 be mathematically complete.
The arrangement of the table in Clause 16 “Coordinate systems” is somewhat different.
vi © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
INTERNATIONAL STANDARD ISO 80000-2:2009(E)
Quantities and units —
Part 2:
Mathematical signs and symbols to be used in the natural
sciences and technology
1 Scope
ISO 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 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.
2)
ISO 80000-1:— , Quantities and units — Part 1: General
3 Variables, functions and operators
Variables such as x, y, etc., and running numbers, such as i in Σ x are printed in italic (sloping) type.
i i
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. f, g.
An explicitly defined function not depending on the context is, however, printed in Roman (upright) type, e.g.
sin, exp, ln, Γ. Mathematical constants, the values of which never change, are printed in Roman (upright) type,
2
e.g. e = 2,718 218 8…; π = 3,141 592…; i = −1. Well-defined operators are also printed in Roman (upright)
style, e.g. div, δ in δx 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 function, without a space
between the symbol for the function and the first parenthesis, e.g. f(x), cos(ωt + ϕ). 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 cases, there should be a thin space between
the symbol for the function and the argument, e.g. int 2,4; sin nπ; 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) To be published. (Revision of ISO 31-0:1992)
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
A comma, semicolon or other appropriate symbol can be used as a separator between numbers or
expressions. The comma is generally preferred, except when numbers 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, 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 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 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,
however, be explained if necessary.
2 © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
4 Mathematical logic
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-4.1 conjunction of p and q,
p ∧ q
(11-3.1) p and q
disjunction of p and q,
2-4.2 This “or” is inclusive, i.e.
p ∨ q
(11-3.2) p or q p ∨ q is true, if either p or q, or both are
true.
negation of p,
2-4.3
¬ p
(11-3.3) not p
2-4.4 p implies q,
p ⇒ q q ⇐ p has the same meaning as p ⇒ q.
(11-3.4) if p, then q
⇒ is the implication symbol.
2-4.5 p is equivalent to q
p ⇔ q (p ⇒ q) ∧ (q ⇒ p) has the same meaning as
(11-3.5)
p ⇔ q.
⇔ is the equivalence symbol.
2-4.6 for every x belonging to A, the lf it is clear from the context which set A is
∀x ∈ A p(x)
proposition p(x) is true
(11-3.6) being considered, the notation ∀x p(x) can
be used.
∀ is the universal quantifier.
For x ∈ A, see 2-5.1.
2-4.7 there exists an x belonging to A lf it is clear from the context which set A is
∃x ∈ A p(x)
for which p(x) is true
(11-3.7) being considered, the notation ∃x p(x) can
be used.
∃ is the existential quantifier.
For x ∈ A, see 2-5.1.
1
∃ x p(x) is used to indicate that there is
exactly one element for which p(x) is true.
1
∃! is also used for ∃ .
© ISO 2009 – All rights reserved 3
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
5 Sets
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-5.1 x belongs to A,
x ∈ A A ∋ x has the same meaning as x ∈ A.
(11-4.1) x is an element of the set A
y does not belong to A,
2-5.2
y ∉ A A ∌ y has the same meaning as y ∉ A.
(11-4.2) y is not an element of the set A
The negating stroke may also be vertical.
2-5.3 {x , x , …, x } set with elements x , x , …, x
Also {x | i ∈ I}, where I denotes a set of
1 2 n 1 2 n
i
(11-4.5) subscripts.
EXAMPLE {x ∈ R | x u 5}
2-5.4 set of those elements of A for
{x ∈ A | p(x)}
lf it is clear from the context which set A is being
which the proposition p(x) is
(11-4.6)
considered, the notation {x | p(x)} can be used
true
(for example {x | x u 5}, if it is clear that x is a
variable for real numbers).
2-5.5 card A number of elements in A, The cardinality can be a transfinite number.
(11-4.7) cardinality of A See also 2-9.16.
A
2-5.6 the empty set
∅
(11-4.8)
2-5.7 B is included in A, Every element of B belongs to A.
B ⊆ A
(11-4.18) B is a subset of A
⊂ is also used, but see remark to 2-5.8.
A ⊇ B has the same meaning as B ⊆ A.
2-5.8 B is properly included in A, Every element of B belongs to A, but at
B ⊂ A
least one element of A does not belong
(11-4.19) B is a proper subset of A
to B.
lf ⊂ is used for 2-5.7, then ⊊ shall be used
for 2-5.8.
A ⊃ B has the same meaning as B ⊂ A.
2-5.9 union of A and B The set of elements which belong to A or
A ∪ B
to B or to both A and B.
(11-4.24)
A ∪ B = {x | x ∈ A ∨ x ∈ B}
2-5.10 intersection of A and B The set of elements which belong to both A
A ∩ B
and B.
(11-4.26)
A ∩ B = {x | x ∈ A ∧ x ∈ B}
2-5.11 n union of the sets A , A , …, A The set of elements belonging to at least
1 2 n
A one of the sets A , A , ., A
∪ i 1 2 n
(11-4.25)
i=1
n
, and are also used,
∪ ∪ ∪
A ∪ A ∪ …
i=1 iI∈
1 2
iI∈
∪ A
n
where I denotes a set of subscripts.
2-5.12 n intersection of the sets A , …, A The set of elements belonging to all sets
1 n
A A , A , ., A
∩ i 1 2 n
(11-4.27)
i=1
n
, and are also used,
∩ ∩ ∩
A ∩ A ∩ … i=1 iI∈
1 2
iI∈
∩ A
n
where I denotes a set of subscripts.
4 © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-5.13 difference of A and B, The set of elements which belong to A but
A ∖ B
B.
not to
(11-4.28) A minus B
A ∖ B = {x | x ∈ A ∧ x ∉ B}
A − B should not be used.
∁ B is also used. ∁ B is mainly used
A A
when B is a subset of A, and the symbol A
may be omitted if it is clear from the
context which set A is being considered.
2-5.14 (a, b) ordered pair a, b, (a, b) = (c, d) if and only if a = c and b = d.
(11-4.30) couple a, b
If the comma can be mistaken as the
decimal sign, then the semicolon (;) or a
stroke (⏐) may be used as separator.
2-5.15 (a , a , …, a) ordered n-tuple See remark to 2-5.14.
1 2 n
(11-4.31)
2-5.16 A × B Cartesian product of the sets A The set of ordered pairs (a, b) such that
and B
(11-4.32) a ∈ A and b ∈ B.
A × B = {(x, y) | x ∈ A ∧ y ∈ B}
n The set of ordered n-tuples (x , x , …, x )
2-5.17 Cartesian product of the sets
1 2 n
such that x ∈ A , x ∈ A , …, x ∈ A .
A A , A , …, A
i 1 1 2 2 n n
1 2 n
(—) Π
i=1
n
A × A × . × A is denoted by A , where n is
A××AA.× the number of factors in the product.
12 n
id identity relation on A,
2-5.18
id is the set of all pairs (x, x) where x ∈ A.
A
A
(11-4.33) If the set A is clear from the context, the
diagonal of A × A
subscript A can be omitted.
© ISO 2009 – All rights reserved 5
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
6 Standard number sets and intervals
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-6.1 N the set of natural numbers, N = {0, 1, 2, 3, …}
(11.4.9) the set of positive integers and
N* = {1, 2, 3, …}
zero
Other restrictions can be indicated in an
obvious way, as shown below.
N = {n ∈ N | n > 5}
>5
The symbols IN and ℕ are also used.
Z
2-6.2 the set of integers Z = {…, −2, −1, 0, 1, 2, …}
(11.4.10)
Z* = {n ∈ Z | n ≠ 0}
Other restrictions can be indicated in an
obvious way, as shown below.
Z = {n ∈ Z | n W −3}
W−3
The symbol ℤ is also used.
2-6.3 Q the set of rational numbers
Q* = {r ∈ Q | r ≠ 0}
(11.4.11)
Other restrictions can be indicated in an
obvious way, as shown below.
Q = {r ∈ Q | r < 0}
<0
The symbols and ℚ are also used.
2-6.4 R the set of real numbers
R* = {x ∈ R | x ≠ 0}
(11.4.12)
Other restrictions can be indicated in an
obvious way, as shown below.
R = {x ∈ R | x W 0}
W0
The symbols IR and ℝ are also used.
2-6.5 C the set of complex numbers
C* = {z ∈ C | z ≠ 0}
(11.4.13)
The symbols ℂ and ℂ are also used.
2-6.6 P the set of prime numbers P = {2, 3, 5, 7, 11, 13, 17, …}
(—)
The symbols ℙ and ℙ are also used.
2-6.7 [a, b] closed interval from a included
[a, b] = {x ∈ R | a u x u b}
to b included
(11.4.14)
2-6.8 (a, b] left half-open interval from a
(a, b] = {x ∈ R | a < x u b}
excluded to b included
(11.4.15)
The notation ]a, b] is also used.
2-6.9 [a, b) right half-open interval from a
[a, b) = {x ∈ R | a u x < b}
included to b excluded
(11.4.16)
The notation [a, b[ is also used.
2-6.10 (a, b) open interval from a excluded to b
(a, b) = {x ∈ R | a < x < b}
(11.4.17) excluded
The notation ]a, b[ is also used.
closed unbounded interval up to b
2-6.11 (−∞, b]
(−∞, b] = {x ∈ R | x u b}
included
(—)
The notation ]−∞, b] is also used.
6 © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-6.12 open unbounded interval up to b
(−∞, b)
(−∞, b) = {x ∈ R | x < b}
excluded
(—)
The notation ]−∞, b[ is also used.
2-6.13 [a, +∞) closed unbounded interval
[a, +∞) = {x ∈ R | a u x}
onward from a included
(—)
The notations [a, ∞ [, [a, +∞ [ and [a, ∞) are
also used.
2-6.14 (a, +∞) open unbounded interval onward
(a, +∞) = {x ∈ R | a < x}
from a excluded
(—)
The notations ]a, +∞[, ]a, ∞ [ and (a, ∞) are
also used.
© ISO 2009 – All rights reserved 7
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
7 Miscellaneous signs and symbols
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-7.1 a = b a is equal to b
The symbol ≡ may be used to emphasize
(11-5.1) that a particular equality is an identity.
See also 2-7.18.
2-7.2 a is not equal to b The negating stroke may also be vertical.
a ≠ b
(11-5.2)
a is by definition equal to b EXAMPLE
2-7.3 a := b
p := mv, where p is momentum, m is mass and v
(11-5.3)
is velocity.
The symbols = and ≝ are also used.
def
EXAMPLES
2-7.4 a corresponds to b
a ≙ b
When E = kT, then 1 eV ≙ 11 604,5 K
(11-5.4)
When 1 cm on a map corresponds to a length of
10 km, one may write 1 cm ≙ 10 km.
The correspondence is not symmetric.
2-7.5 a is approximately equal to b It depends on the user whether an
a ≈ b
approximation is sufficiently good. Equality
(11-5.5)
is not excluded.
EXAMPLE
2-7.6 a is asymptotically equal to b
a ≃ b
11
(11-7.7)
� as x → a
sin(xa−−) x a
(For x → a, see 2-7.16.)
2-7.7 a is proportional to b
a ∼ b The symbol ∼ is also used for equivalence
(11-5.6) relations.
The notation a ∝ b is also used.
2-7.8 M is congruent to N, M and N are point sets (geometrical
M ≅ N
figures).
(—) M is isomorphic to N
This symbol is also used for isomorphisms
of mathematical structures.
2-7.9 a < b a is less than b
(11-5.7)
2-7.10 b > a b is greater than a
(11-5.8)
2-7.11 a is less than or equal to b
a u b
(11-5.9)
2-7.12 b is greater than or equal to a
b W a
(11-5.10)
2-7.13 a is much less than b It depends on the user whether a is
a � b
sufficiently small as compared to b.
(11-5.11)
2-7.14 b is much greater than a It depends on the user whether b is
b � a
sufficiently great as compared to a.
(11-5.12)
8 © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-7.15 infinity This symbol does not denote a number but
∞
is often part of various expressions dealing
(11-5.13)
with limits.
The notations +∞, -∞ are also used.
2-7.16 x → a x tends to a This symbol occurs as part of various
expressions dealing with limits.
(11-7.5)
a may be also ∞, +∞, or -∞.
2-7.17 m⏐n m divides n For integers m and n:
(—) ∃ k ∈ Z m⋅k = n
n is congruent to k modulo m For integers n, k and m:
2-7.18
n ≡ k mod m
m⏐(n − k)
(—)
See also 2-7.1.
(a + b) parentheses
2-7.19 It is recommended to use only parentheses
for grouping, since brackets and braces
[a + b] square brackets
(1-5.14)
often have a specific meaning in particular
{a + b} braces
fields. Parentheses can be nested without
〈a + b〉 angle brackets
ambiguity.
© ISO 2009 – All rights reserved 9
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
8 Elementary geometry
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-8.1 the straight line AB is parallel to
AB∥CD It is written g∥ h if g and h are the straight
the straight line CD
(11-5.15) lines determined by the points A and B,
and the points C and D, respectively.
AB//CD is also used.
2-8.2 the straight line AB is
AB⊥CD It is written g⊥ h if g and h are the straight
perpendicular to the straight
(11-5.16) lines determined by the points A and B,
line CD
and the points C and D, respectively. In a
plane, the straight lines must intersect.
2-8.3 angle at vertex B in the triangle The angle is not oriented, it holds that
∢ABC
ABC
(—) ∢ABC = ∢CBA and
0 u ∢ABC u π rad.
2-8.4 line segment from A to B The line segment is the set of points
AB
between A and B on the straight line AB.
(—)
2-8.5 → vector from A to B → →
AB If AB = CD then B, seen from A, is in the
(—)
same direction and distance as D is, seen
from C. It does not follow that A = C and
B = D.
d(A, B)
2-8.6 distance between points A and B The distance is the length of the line
(—) segment AB and also the magnitude of the
→
vector AB .
10 © ISO 2009 – All rights reserved
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
9 Operations
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-9.1 a + b a plus b This operation is named addition. The
symbol + is the addition symbol.
(11-6.1)
2-9.2 a minus b This operation is named subtraction. The
a − b
symbol − is the subtraction symbol.
(11-6.2)
2-9.3 a plus or minus b This is a combination of two values into
a ± b
one expression.
(11-6.3)
2-9.4 a minus or plus b
a ∓ b −(a ± b) = −a ∓ b
(11-6.4)
a multiplied by b,
2-9.5 a ⋅ b This operation is named multiplication. The
symbol for multiplication is a half-high dot
(11-6.5) a times b
a × b
(·) or a cross (×).
a b
Either may be omitted if no
ab
misunderstanding is possible.
See also 2-5.16, 2-5.17, 2-17.11, 2-17.12,
2-17.23 and 2-17.24 for the use of the dot
and cross in various products.
2-9.6 a divided by b
a a
–1
= a ⋅ b
(11-6.6) b b
See also ISO 80000-1:—, 7.1.3.
a/b
For ratios, the symbol : is also used.
EXAMPLE The ratio of height h to breadth b
of an A4 sheet is h : b = 2 .
The symbol ÷ should not be used.
n n
2-9.7 a + a + … + a ,
1 2 n
The notations a , a , a and
∑ i ∑ i ∑ i
a
i=1 i
∑ i
(11-6.7)
sum of a , a , …, a i
1 2 n
i=1
a are also used.
∑ i
n n
2-9.8
a ⋅ a ⋅ … ⋅ a ,
1 2 n
The notations a , a , a and
∏ i ∏ i ∏ i
a
i=1 i
∏ i
(11-6.8)
product of a , a , …, a i
1 2 n
i=1
a are also used.
∏ i
p 2
2-9.9 a to the power p
a The verbal equivalent of a is
3
a squared; the verbal equivalent of a is a
(11-6.9)
cubed.
1/2
a to the power 1/2,
2-9.10 a
lf a W 0, then a W 0.
(11-6.10) square root of a
a
The symbol √a should be avoided.
See remark to 2-9.11.
1/n
n
2-9.11 a to the power 1/n,
a
lf a W 0, then a W 0.
(11-6.11) nth root of a
n
n
a
The symbol √a should be avoided.
n
lf the symbol √ or √ acts on a composite
expression, parentheses shall be used to
avoid ambiguity.
© ISO 2009 – All rights reserved 11
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SIST ISO 80000-2:2013
ISO 80000-2:2009(E)
Sign, symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
Mean values obtained by other methods
2-9.12
x mean value of x,
are the
(11-6.14)
〈x〉 arithmetic mean of x
- harmonic mean denoted by subscript h,
x
a
- geometric mean denoted by subscript g,
- quadratic mean, often called “root mean
square”, denoted by subscript q or rms.
The subscript may only be omitted for the
arithmetic mean.
In mathematics x is also used for the
complex conjugate of x; see 2-14.6.
2-9.13 sgn a signum a For real a:
(11-6.13) 1if a >0
⎧
⎪
sgn a = 0if a =0
⎨
⎪
−<1if a 0
⎩
See also item 2-14.7.
2-9.14 inf M infimum of M
Greatest lower bound of a non-empty set
of numbers bounded from below.
(—)
2-9.15 sup M supremum of M
Smallest upper bound of a non-empty set
of numbers bounded from above.
(—)
2-9.16 absolute value of a, The notation abs a is also used.
|a|
Absolute value of real number a.
(11-6.12) modulus of a,
Modulus of complex number a; see 2-14.4.
magnitude of a
Magnitude of vector a; see 2-17.4.
See also 2-5.5.
floor a, The notation ent a is also used.
2-9.17 ⎣a⎦
EXAMPLES
(11-6.17) the greatest integer less than or
equal to the real number a
⎣2,4⎦ = 2
⎣−2,4⎦ = −3
2-9.18
...
NORME ISO
INTERNATIONALE 80000-2
Première édition
2009-12-01
Grandeurs et unités —
Partie 2:
Signes et symboles mathématiques à
employer dans les sciences de la nature
et dans la technique
Quantities and units —
Part 2: Mathematical signs and symbols to be used in the natural
sciences and technology
Numéro de référence
ISO 80000-2:2009(F)
©
ISO 2009
---------------------- Page: 1 ----------------------
ISO 80000-2:2009(F)
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© ISO 2009
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quelque forme que ce soit et par aucun procédé, électronique ou mécanique, y compris la photocopie et les microfilms, sans l'accord écrit
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Publié en Suisse
ii © ISO 2009 – Tous droits réservés
---------------------- Page: 2 ----------------------
ISO 80000-2:2009(F)
Sommaire Page
Avant-propos .iv
Introduction.vi
1 Domaine d'application .1
2 Références normatives.1
3 Variables, fonctions et opérateurs .1
4 Logique mathématique .3
5 Ensembles.4
6 Ensembles normalisés de nombres et intervalles.6
7 Signes et symboles divers .8
8 Géométrie élémentaire.10
9 Opérations.11
10 Combinatoire .14
11 Fonctions.15
12 Fonctions exponentielles et logarithmiques .18
13 Fonctions circulaires et hyperboliques .19
14 Nombres complexes .21
15 Matrices .22
16 Systèmes de coordonnées.24
17 Scalaires, vecteurs et tenseurs.26
18 Transformées.30
19 Fonctions spéciales .31
Annexe A (normative) Clarification des symboles utilisés .36
Bibliographie.41
© ISO 2009 – Tous droits réservés iii
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ISO 80000-2:2009(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée
aux comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du
comité technique créé à cet effet. Les organisations internationales, gouvernementales et non
gouvernementales, en liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec
la Commission électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 2.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur
publication comme Normes internationales requiert l'approbation de 75 % au moins des comités membres
votants.
L'attention est appelée sur le fait que certains des éléments du présent document peuvent faire l'objet de
droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de ne
pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO 80000-2 a été élaborée par le comité technique ISO/TC 12, Grandeurs et unités, en collaboration avec
le comité d'études CEI/CE 25, Grandeurs et unités.
Cette première édition annule et remplace l'ISO 31-11:1992, qui a fait l'objet d'une révision technique. Les
principales modifications techniques par rapport à la norme précédente sont les suivantes:
⎯ Quatre articles ont été ajoutés: «Ensembles normalisés de nombres et intervalles», «Géométrie
élémentaire», «Combinatoire», et «Transformées».
L'ISO 80000 comprend les parties suivantes, présentées sous le titre général Grandeurs et unités:
⎯ Partie 1: Généralités
⎯ Partie 2: Signes et symboles mathématiques à employer dans les sciences de la nature et dans la
1)
technique
⎯ Partie 3: Espace et temps
⎯ Partie 4: Mécanique
⎯ Partie 5: Thermodynamique
⎯ Partie 7: Lumière
⎯ Partie 8: Acoustique
⎯ Partie 9: Chimie physique et physique moléculaire
⎯ Partie 10: Physique atomique et nucléaire
1) Le titre sera abrégé en «Mathématiques» dans la seconde édition de l'ISO 80000-2.
iv © ISO 2009 – Tous droits réservés
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ISO 80000-2:2009(F)
⎯ Partie 11: Nombres caractéristiques
⎯ Partie 12: Physique de l'état solide
La CEI 80000 comprend les parties suivantes, présentées sous le titre général Grandeurs et unités:
⎯ Partie 6: Électromagnétisme
⎯ Partie 13: Science et technologies de l'information
⎯ Partie 14: Télébiométrique relative à la physiologie humaine
© ISO 2009 – Tous droits réservés v
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ISO 80000-2:2009(F)
Introduction
Disposition des tableaux
La première colonne, «N°», des tableaux comporte le numéro du concept; le numéro correspondant dans
l'ISO 31-11 est indiqué entre parenthèses; un tiret est utilisé pour indiquer que le concept en question ne
figurait pas dans l'édition précédente.
La seconde colonne, «Signe, symbole, expression», indique le signe ou le symbole considéré, généralement
dans le contexte d'une expression type. Lorsque plusieurs signes, symboles ou expressions sont indiqués
pour le même concept, ils sont également admissibles. Dans certains cas, par exemple pour l'élévation à une
puissance, il n'existe qu'une expression type, mais pas de symbole.
La troisième colonne, «Sens, énoncé», donne une information d'aide sur le sens ou sur la manière dont
l'expression peut être lue. Cela aide à l'identification du concept mais n'est pas une définition mathématique
complète.
La quatrième colonne, «Remarques et exemples», donne des informations complémentaires. Des définitions
sont données si elles sont assez courtes pour tenir dans la colonne. Il n'est pas nécessaire qu'elles soient
mathématiquement complètes.
La disposition du tableau de l'Article 16, «Systèmes de coordonnées», est légèrement différente.
vi © ISO 2009 – Tous droits réservés
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NORME INTERNATIONALE ISO 80000-2:2009(F)
Grandeurs et unités —
Partie 2:
Signes et symboles mathématiques à employer dans les
sciences de la nature et dans la technique
1 Domaine d'application
L'ISO 80000-2 donne des informations générales sur les signes et symboles mathématiques, leurs sens, leurs
énoncés et leurs applications.
Les recommandations données dans l'ISO 80000-2 sont principalement destinées à être utilisées dans les
sciences de la nature et dans la technique. Cependant, elles s'appliquent également à d'autres domaines
utilisant les mathématiques.
2 Références normatives
Les documents de référence suivants sont indispensables pour l'application du présent document. Pour les
références datées, seule l'édition citée s'applique. Pour les références non datées, la dernière édition du
document de référence s'applique (y compris les éventuels amendements).
2)
ISO 80000-1:— , Grandeurs et unités — Partie 1: Généralités
3 Variables, fonctions et opérateurs
Les variables, telles que x, y, etc., et les indices tels que i dans Σ x sont imprimés en caractères italiques
i i
(penchés). Il en est de même pour les paramètres tels que a, b, etc., qui peuvent être considérés comme
constants dans un contexte particulier. La même règle s'applique aussi aux fonctions en général, par
exemple f, g.
Cependant, on écrit en caractères romains (droits) une fonction explicitement définie qui ne dépend pas du
contexte, par exemple sin, exp, In, Γ. Les constantes mathématiques dont la valeur ne change jamais sont
2
imprimées en caractères romains (droits), par exemple: e = 2,718 281 8.; π = 3,141 592 …; i = −1. Les
opérateurs bien définis sont aussi imprimés en droit, par exemple: div, δ dans δx et chaque d dans df/dx.
Les nombres exprimés par des chiffres sont toujours écrits en droit, par exemple: 351 204; 1,32; 7/8.
L'argument d'une fonction est écrit entre parenthèses après le symbole de la fonction, sans espace entre le
symbole de la fonction et la première parenthèse, par exemple: f(x), cos(ωt + ϕ). Si le symbole de la fonction
comporte deux lettres ou plus et si l'argument ne contient pas de signe d'opération tel que +; −; ×; ⋅; ou /, les
parenthèses autour de l'argument peuvent être omises. Dans ce cas, il convient de laisser un léger espace
entre le symbole de la fonction et l'argument, par exemple: int 2,4; sin nπ; arcosh 2A; Ei x.
2) À publier. (Révision de l'ISO 31-0:1992)
© ISO 2009 – Tous droits réservés 1
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ISO 80000-2:2009(F)
S'il existe un risque de confusion, il convient de toujours insérer des parenthèses. Par exemple, écrire
cos(x) + y; ne pas écrire cos x + y qui pourrait être compris comme cos(x + y).
Une virgule, un point-virgule ou un autre symbole approprié peut être utilisé comme séparateur entre les
nombres ou expressions. La virgule est généralement préconisée, sauf dans le cas de nombres comportant
une virgule comme signe décimal.
S'il faut écrire une expression ou une équation sur deux lignes ou plus, l'une des méthodes suivantes doit être
utilisée:
a) Effectuer la coupure immédiatement après l'un des symboles =, +, −, ±, ou ∓, ou, si nécessaire,
immédiatement après l'un des symboles ×, ⋅, ou /. Dans ce cas, le symbole indique que l'expression
continue à la ligne ou la page suivante.
b) Effectuer la coupure immédiatement avant l'un des symboles =, +, −, ±, ou ∓, ou, si nécessaire,
immédiatement avant l'un des symboles ×, ⋅, ou /. Dans ce cas, le symbole indique que l'expression est la
continuation de la précédente ligne ou page.
Le symbole ne doit pas être répété au début de la ligne suivante, deux signes moins pourraient par exemple
entraîner des erreurs de signe. Il convient de n'utiliser qu'une seule de ces méthodes dans un même
document. Il convient, si possible, que la coupure de ligne ne se trouve pas dans une expression entre
parenthèses.
Il est de règle d'utiliser différents types de caractères pour différentes entités. Cela facilite la lecture des
formules et la mise en place d'un contexte approprié. Il n'existe aucune règle stricte relative à l'utilisation de
polices de caractères, dont il convient cependant d'expliquer l'utilisation si nécessaire.
2 © ISO 2009 – Tous droits réservés
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ISO 80000-2:2009(F)
4 Logique mathématique
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-4.1 conjonction de p et q,
p ∧ q
(11-3.1) p et q
2-4.2 disjonction de p et q, Ce «ou» est inclusif, c'est-à-dire p ∨ q est
p ∨ q
vrai si p ou q ou les deux sont vrais.
(11-3.2) p ou q
2-4.3 négation de p,
¬ p
(11-3.3) non p
2-4.4 p implique q, p entraîne q, q ⇐ p a le même sens que p ⇒ q.
p ⇒ q
(11-3.4) si p alors q ⇒ est le symbole d'implication.
2-4.5 p équivaut à q, (p ⇒ q) ∧ (q ⇒ p) a le même sens que
p ⇔ q
p est équivalent à q p ⇔ q.
(11-3.5)
⇔ est le symbole d'équivalence.
2-4.6 pour tout x appartenant à A, la Si le contexte permet de savoir clairement
∀x ∈ A p(x)
proposition p(x) est vraie quel est l'ensemble A considéré, on peut
(11-3.6)
utiliser la notation ∀x p(x).
∀ est le quantificateur universel.
Pour x ∈ A, voir 2-5.1.
2-4.7 il existe un x appartenant à A Si le contexte permet de savoir clairement
∃x ∈ A p(x)
pour lequel p(x) est vrai quel est l'ensemble A considéré, on peut
(11-3.7)
utiliser la notation ∃x p(x).
∃ est le quantificateur existentiel.
Pour x ∈ A, voir 2-5.1.
1
∃ x p(x) est utilisé pour indiquer l'existence
d'un élément et d'un seul pour lequel p(x)
est vrai.
1
∃! est aussi utilisé pour ∃ .
© ISO 2009 – Tous droits réservés 3
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ISO 80000-2:2009(F)
5 Ensembles
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-5.1 x appartient à A, A ∋ x a le même sens que x ∈ A.
x ∈ A
(11-4.1) x est un élément de l'ensemble A
2-5.2 y n'appartient pas à A, A ∌ y a le même sens que y ∉ A.
y ∉ A
(11-4.2) y n'est pas un élément de La barre de négation peut aussi être
l'ensemble A verticale.
{x , x , …, x }
2-5.3 ensemble dont les éléments sont S'écrit aussi {x | i ∈ I}, où I est un
1 2 n
i
x , x , …, x ensemble d'indices.
1 2 n
(11-4.5)
2-5.4 ensemble des éléments de A EXEMPLE {x ∈ R | x u 5}
{x ∈ A | p(x)}
pour lesquels la proposition Si le contexte permet de savoir clairement quel
(11-4.6)
p(x) est vraie est l'ensemble A considéré, on peut utiliser la
notation {x | p(x)} (par exemple {x | x u 5}
s'il est clair que x est une variable représentant
un nombre réel).
2-5.5 card A nombre des éléments de A, Le cardinal peut être un nombre transfini.
(11-4.7) cardinal de A Voir aussi 2-9.16.
A
2-5.6 ∅ l'ensemble vide
(11-4.8)
2-5.7 B ⊆ A B est inclus dans A, Tout élément de B appartient à A.
(11-4.18) B est un sous-ensemble de A ⊂ est aussi utilisé, mais voir la remarque
au 2-5.8.
A ⊇ B a le même sens que B ⊆ A.
2-5.8 B ⊂ A B est strictement inclus dans A, Tout élément de B appartient à A, mais au
moins un élément de A n'appartient pas à
(11-4.19) B est un sous-ensemble strict
B.
de A
Si ⊂ est utilisé pour 2-5.7, alors ⊊ doit
être utilisé pour 2-5.8.
A ⊃ B a le même sens que B ⊂ A.
2-5.9 A ∪ B réunion de A et de B Ensemble des éléments appartenant à A
ou à B ou aux deux.
(11-4.24)
A ∪ B = {x | x ∈ A ∨ x ∈ B}
2-5.10 A ∩ B intersection de A et de B Ensemble des éléments appartenant à la
fois à A et à B.
(11-4.26)
A ∩ B = {x | x ∈ A ∧ x ∈ B}
2-5.11 n réunion des ensembles Ensemble des éléments appartenant au
A A , A , …, A moins à un des ensembles A , A , ., A
∪ i 1 2 n 1 2 n
(11-4.25)
i=1
n
, , et sont aussi
∪ ∪ ∪
A ∪ A ∪ … i=1 iI∈
1 2
iI∈
∪ A
n
utilisés, où I est un ensemble d'indices.
4 © ISO 2009 – Tous droits réservés
---------------------- Page: 10 ----------------------
ISO 80000-2:2009(F)
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-5.12 n intersection des ensembles Ensemble des éléments appartenant à la
A A , …, A fois à A , A , ., A
∩ i 1 n 1 2 n
(11-4.27)
i=1
n
, et sont aussi utilisés,
∩ ∩ ∩
A ∩ A ∩ … i=1 iI∈
1 2
iI∈
∩ A
n
où I est un ensemble d'indices.
2-5.13 différence de A et de B, Ensemble des éléments de A
A ∖ B
n'appartenant pas à B.
(11-4.28) A moins B
A ∖ B = {x | x ∈ A ∧ x ∉ B}
Il convient de ne pas utiliser A − B.
∁ B est aussi utilisé. ∁ B est surtout
A A
utilisé lorsque B est un sous-ensemble
de A, et le symbole A peut être omis si le
contexte permet de savoir clairement quel
est l'ensemble A considéré.
2-5.14 (a, b) couple a, b, (a, b) = (c, d) si et seulement si a = c et
b = d.
(11-4.30) paire ordonnée a, b
Si la virgule peut être confondue avec le
signe décimal, alors le point virgule (;) ou
la barre (⏐) peuvent être utilisés comme
séparateur.
2-5.15 (a , a , …, a ) n-uple, n-uplet, multiplet Voir remarque au 2-5.14.
1 2 n
(11-4.31)
2-5.16 A × B produit cartésien des ensembles Ensemble des couples (a, b) pour lesquels
A et B a ∈ A et b ∈ B.
(11-4.32)
A × B = {(x, y) | x ∈ A ∧ y ∈ B}
2-5.17 n produit cartésien des ensembles Ensembles des n-uples (x , x , …, x ) pour
1 2 n
A A , A , …, A lesquels x ∈ A , x ∈ A , …, x ∈ A .
i 1 2 n 1 1 2 2 n n
Π
(—)
i=1
n
A × A × . × A est noté A , où n est le
nombre de facteurs dans le produit.
A××AA.×
12 n
2-5.18 id application identité sur A, id est l'ensemble de toutes les paires
A A
(x, x) où x ∈ A. Si le contexte permet de
(11-4.33)
diagonale de A × A
savoir clairement quel est l'ensemble A
considéré, l'indice A peut être omis.
© ISO 2009 – Tous droits réservés 5
---------------------- Page: 11 ----------------------
ISO 80000-2:2009(F)
6 Ensembles normalisés de nombres et intervalles
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-6.1 N ensemble des entiers naturels, N = {0, 1, 2, 3, …}
(11.4.9) ensemble des entiers positifs et
N* = {1, 2, 3, …}
de zéro
D'autres restrictions peuvent être
indiquées de manière évidente comme
indiqué ci-dessous.
N = {n ∈ N | n > 5}
>5
Les symboles IN et ℕ sont aussi utilisés.
2-6.2 Z ensemble des entiers,
Z = {…, −2, −1, 0, 1, 2, …}
(11.4.10)
ensemble des entiers relatifs
Z* = {n ∈ Z | n ≠ 0}
D'autres exclusions peuvent être
indiquées de manière évidente comme
indiqué ci-dessous.
Z = {n ∈ Z | n W −3}
W−3
Le symbole ℤ est aussi utilisé.
2-6.3 Q ensemble des nombres
Q* = {r ∈ Q | r ≠ 0}
rationnels, ensemble des
(11.4.11)
D'autres exclusions peuvent être
rationnels
indiquées de manière évidente comme
indiqué ci-dessous.
Q = {r ∈ Q | r < 0}
<0
Les symboles et ℚ sont aussi utilisés.
2-6.4 R ensemble des nombres réels,
R* = {x ∈ R | x ≠ 0}
ensemble des réels
(11.4.12)
D'autres exclusions peuvent être
indiquées de manière évidente comme
indiqué ci-dessous.
R = {x ∈ R | x W 0}
W0
Les symboles IR et ℝ sont aussi utilisés.
2-6.5 C ensemble des nombres
C* = {z ∈ C | z ≠ 0}
complexes
(11.4.13)
Les symboles ℂ et ℂ sont aussi utilisés.
2-6.6 P ensemble des nombres premiers P = {2, 3, 5, 7, 11, 13, 17, …}
(—)
Les symboles ℙ et ℙ sont aussi utilisés.
2-6.7 [a, b] intervalle fermé de a inclus à b
[a, b] = {x ∈ R | a u x u b}
inclus
(11.4.14)
2-6.8 (a, b] intervalle semi-ouvert à gauche
(a, b] = {x ∈ R | a < x u b}
de a exclu à b inclus
(11.4.15)
La notation ]a, b] est aussi utilisée.
2-6.9 [a, b) intervalle semi-ouvert à droite de
[a, b) = {x ∈ R | a u x < b}
a inclus à b exclu
(11.4.16)
La notation [a, b[ est aussi utilisée.
2-6.10 (a, b) intervalle ouvert de a exclu à b
(a, b) = {x ∈ R | a < x < b}
(11.4.17) exclu
La notation ]a, b[ est aussi utilisée.
6 © ISO 2009 – Tous droits réservés
---------------------- Page: 12 ----------------------
ISO 80000-2:2009(F)
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-6.11 intervalle illimité fermé finissant
(−∞, b]
(−∞, b] = {x ∈ R | x u b}
en b inclus
(—)
La notation ]−∞, b] est aussi utilisée.
2-6.12 (−∞, b) intervalle illimité ouvert finissant
(−∞, b) = {x ∈ R | x < b}
en b exclu
(—)
La notation ]−∞, b[ est aussi utilisée.
2-6.13 intervalle illimité fermé
[a, +∞)
[a, +∞) = {x ∈ R | a u x}
commençant en a inclus
(—)
Les notations [a, ∞ [, [a, +∞ [ et [a, ∞)
sont aussi utilisées.
2-6.14 (a, +∞) intervalle illimité ouvert
(a, +∞) = {x ∈ R | a < x}
commençant en a exclu
(—)
Les notations ]a, +∞[, ]a, ∞ [ et (a, ∞)
sont aussi utilisées.
© ISO 2009 – Tous droits réservés 7
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ISO 80000-2:2009(F)
7 Signes et symboles divers
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-7.1 a = b a est égal à b
Le symbole ≡ peut être utilisé pour
(11-5.1) souligner qu'une égalité particulière est
une identité.
Voir aussi 2-7.18.
2-7.2 a est différent de b La barre de négation peut aussi être
a ≠ b
verticale.
(11-5.2)
EXEMPLE
2-7.3 a := b a est égal par définition à b
p := mv, où p est la quantité de mouvement, m
(11-5.3)
la masse et v la vitesse.
Les symboles = et ≝ sont aussi utilisés.
def
EXEMPLES
2-7.4 a correspond à b
a ≙ b
(11-5.4)
Si E = kT, alors 1 eV ≙ 11 604,5 K
Si 1 cm sur une carte correspond à une
longueur de 10 km, on peut écrire
1 cm ≙ 10 km.
La correspondance n'est pas symétrique.
2-7.5 a est approximativement égal à b Il revient à l'utilisateur de décider si une
a ≈ b
approximation est suffisamment bonne.
(11-5.5)
L'égalité n'est pas exclue.
a est asymptotiquement égal à b EXEMPLE
2-7.6
a ≃ b
11
(11-7.7)
� lorsque x → a
sin(xa−−) x a
(Pour x → a, voir 2-7.16.)
2-7.7 a est proportionnel à b
a ∼ b
Le symbole ∼ est aussi utilisé pour les
(11-5.6) relations d'équivalence.
La notation a ∝ b est aussi utilisée.
2-7.8 M est congruent à N, M et N sont des ensembles de points
M ≅ N
(figures géométriques).
(—) M est isomorphe à N
Ce symbole est aussi utilisé pour les
isomorphismes de structures
mathématiques.
2-7.9 a est inférieur à b
a < b
(11-5.7)
2-7.10 b > a b est supérieur à a
(11-5.8)
a est inférieur ou égal à b
2-7.11 a u b
(11-5.9)
2-7.12 b est supérieur ou égal à a
b W a
(11-5.10)
2-7.13 a est très inférieur à b Il revient à l'utilisateur de décider si a est
a � b
suffisamment petit par rapport à b.
(11-5.11)
8 © ISO 2009 – Tous droits réservés
---------------------- Page: 14 ----------------------
ISO 80000-2:2009(F)
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-7.14 b est très supérieur à a Il revient à l'utilisateur de décider si b est
b � a
suffisamment grand par rapport à a.
(11-5.12)
2-7.15 infini Ce symbole ne désigne pas un nombre
∞
mais fait souvent partie de différentes
(11-5.13)
expressions relatives aux limites.
Les notations +∞, -∞ sont aussi utilisées.
x tend vers a
2-7.16 x → a Ce symbole apparaît dans différentes
expressions relatives aux limites.
(11-7.5)
a peut aussi être ∞, +∞, ou -∞.
m divise n Pour les entiers m et n:
2-7.17 m⏐n
(—) ∃ k ∈ Z m⋅k = n
2-7.18 n est congru à k modulo m Pour les entiers n, k et m:
n ≡ k mod m
m⏐(n − k)
(—)
Voir aussi 2-7.1.
2-7.19 (a + b) parenthèses Il est recommandé d'utiliser, pour les
regroupements, seulement les
[a + b] crochets
(1-5.14)
parenthèses, car les crochets et
{a + b} accolades
accolades ont souvent des significations
〈a + b〉 chevrons, crochets angulaires
spécifiques selon le domaine. Les
parenthèses peuvent être emboîtées sans
ambiguïté.
© ISO 2009 – Tous droits réservés 9
---------------------- Page: 15 ----------------------
ISO 80000-2:2009(F)
8 Géométrie élémentaire
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-8.1 la droite AB est parallèle à la
AB∥CD On écrit g∥ h si g et h sont les droites
droite CD
(11-5.15)
déterminées respectivement par les points
A et B et les points C et D.
AB//CD est aussi utilisé.
2-8.2 la droite AB est perpendiculaire à
AB⊥CD On écrit g⊥ h si g et h sont les droites
la droite CD
(11-5.16) déterminées respectivement par les points
A et B et les points C et D. Dans un plan,
les droites doivent s'intersecter.
2-8.3 angle de sommet B dans le L'angle n'est pas orienté, il vérifie que
∢ ABC
triangle ABC
(—) ∢ABC = ∢CBA et
0 u ∢ABC u π rad.
2-8.4 segment de droite de A à B Le segment de droite est l'ensemble des
AB
points entre A et B sur la droite AB.
(—)
2-8.5 → vecteur de A à B → →
AB Si AB = CD alors B, vu de A, est dans la
(—)
même direction et à égale distance que
D vu de C. Il n'en découle pas que A = C
et B = D.
2-8.6 d(A, B) distance entre les points A et B La distance est la longueur du segment
(—) de droite AB et aussi la norme du vecteur
→
AB .
10 © ISO 2009 – Tous droits réservés
---------------------- Page: 16 ----------------------
ISO 80000-2:2009(F)
9 Opérations
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
2-9.1 a + b a plus b Cette opération est appelée addition. Le
signe + est le symbole de l'addition.
(11-6.1)
2-9.2 a moins b Cette opération est appelée soustraction.
a − b
Le signe − est le symbole de la
(11-6.2)
soustraction.
2-9.3 a plus ou moins b Il s'agit de la combinaison de deux valeurs
a ± b
en une expression.
(11-6.3)
2-9.4 a moins ou plus b
a ∓ b −(a ± b) = −a ∓ b
(11-6.4)
2-9.5 a ⋅ b a multiplié par b, Cette opération est appelée multiplication.
Le signe de la multiplication est un point à
(11-6.5) a fois b
a × b
mi-hauteur (⋅) ou une croix (×).
a b
Il peut être omis si aucune ambiguïté n'est
ab
possible.
Voir aussi 2-5.16, 2-5.17, 2-17.11,
2-17.12, 2-17.23, et 2-17.24 pour
l'utilisation du point et de la croix dans
différents produits.
2-9.6 a divisé par b
a a
–1
= a ⋅ b
(11-6.6) b b
Voir aussi l'ISO 80000-1:—, 7.1.3.
a/b
Pour les rapports, le symbole : est aussi
utilisé.
EXEMPLE Le rapport de la hauteur h à la
largeur b d'une feuille A4 est h : b = 2 .
Il convient de ne pas utiliser le signe ÷.
n n
2-9.7 a + a + … + a ,
1 2 n
Les notations a , a , a , et
∑ i ∑ i ∑ i
a
i=1 i
∑ i
(11-6.7)
somme de a , a , …, a i
1 2 n
i=1
a sont aussi utilisées.
∑ i
n n
2-9.8 a ⋅ a ⋅ … ⋅ a ,
1 2 n
Les notations a , a , a , et
∏ i ∏ i ∏ i
a
i=1 i
∏ i
(11-6.8)
produit de a , a , …, a i
1 2 n
i=1
a sont aussi utilisées.
∏ i
p 2
2-9.9 a puissance p
a L'énoncé de a est a au carré, et celui de
3
a est a au cube.
(11-6.9)
1/2
2-9.10 a puissance 1/2,
a
Si a W 0, alors aW 0.
(11-6.10) racine carrée de a
a
Il convient d'éviter le symbole √a.
Voir remarque en 2-9.11.
© ISO 2009 – Tous droits réservés 11
---------------------- Page: 17 ----------------------
ISO 80000-2:2009(F)
Signe, symbole,
N° Sens, énoncé Remarques et exemples
expression
1/n
2-9.11 a puissance 1/n, n
a
Si a W 0, alors aW 0.
racine n-ième de a
(11-6.11) n
n
a
Il convient d'éviter le symbole √a.
n
Si le symbole √ ou √ agit sur une
expression composée, des parenthèses
doivent être utilisées pour éviter toute
ambiguïté.
2-9.12 La moyenne harmonique indiquée par un
x valeur moyenne de x,
indice h, la moyenne géométrique indiquée
(11-6.14)
〈x〉 moyenne arithmétique de x
par un indice g et la moyenne quadratique,
x
a souvent appelée «root mean square» en
anglais, indiquée par un indice q ou rms,
sont des valeurs moyennes obtenues par
d'autres méthodes.
L'indice ne peut être omis que pour la
moyenne arithmétique.
En mathématique, x est aussi utilisé pour
le conjugué du nombre complexe x;
voir 2-14.6.
2-9.13 sgn a signum a Pour tout a réel:
(11-6.13) ⎧ 1si a >0
⎪
sgn a = 0si a =0
⎨
⎪
−<1si a 0
⎩
Voir aussi 2-14.7.
2-9.14 inf
...
SLOVENSKI SIST ISO 80000-2
STANDARD
maj 2013
Veličine in enote – 2. del: Matematični znaki in simboli za uporabo v
naravoslovnih vedah in tehniki
Quantities and units – Part 2: Mathematical signs and symbols to be used in the
natural sciences and technology
Grandeurs et unités – Partie 2: Signes et symboles mathématiques à employer
dans les sciences de la nature et dans la technique
Referenčna oznaka
ICS 01.060 SIST ISO 80000-2:2013 (sl)
Nadaljevanje na straneh od 2 do 45
© 2013-05. Standard je založil in izdal Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
---------------------- Page: 1 ----------------------
SIST ISO 80000-2 : 2013
NACIONALNI UVOD
Standard SIST ISO 80000-2 (sl), Veličine in enote – 2. del: Matematični znaki in simboli za uporabo v
naravoslovnih vedah in tehniki, maj 2013, ima status slovenskega standarda in je enakovreden
mednarodnemu standardu ISO 80000-2 (en), Quantities and units – Part 2: Mathematical signs and
symbols to be used in the natural sciences and technology, 2009-12.
NACIONALNI PREDGOVOR
Mednarodni standard ISO 80000-2:2009 je pripravil tehnični odbor ISO/TC 12 Veličine, enote, simboli
v sodelovanju s tehničnim odborom IEC/TC 25 Veličine in enote in njihovi črkovni simboli.
Slovenski standard SIST ISO 80000-2:2013 je prevod mednarodnega standarda ISO 80000-2:2009. V
primeru spora glede besedila slovenskega prevoda v tem standardu je odločilen izvirni mednarodni
standard v angleškem jeziku. Slovensko izdajo standarda je pripravil tehnični odbor SIST/TC TRS
Tehnično risanje, veličine, enote, simboli in grafični simboli.
ZVEZA Z NACIONALNIMI STANDARDI
S privzemom tega mednarodnega standarda veljajo za omejeni namen referenčnih standardov vsi
standardi, navedeni v izvirniku, razen standardov, ki so že sprejeti v nacionalno standardizacijo:
SIST ISO/IEC 10646:2008 (en,fr) Informacijska tehnologija – Univerzalni večoktetni nabor znakov
(UCS)
SIST EN 60027-6:2008 (en,fr,de) Črkovni simboli za uporabo v elektrotehniki – 6. del: Krmilna
tehnologija (IEC 60027-6:2006)
PREDHODNA IZDAJA
SIST ISO 31-11:1995 (sl) Veličine in enote – 11. del: Matematični znaki in simboli za
uporabo v fizikalnih in tehničnih vedah
OPOMBE
– Povsod, kjer se v besedilu standarda uporablja izraz “mednarodni standard”, v
SIST ISO 80000-2:2013 to pomeni “slovenski standard”.
– Nacionalni uvod in nacionalni predgovor nista sestavni del standarda.
2
---------------------- Page: 2 ----------------------
SIST ISO 80000-2 : 2013
VSEBINA Stran
Predgovor .4
Uvod .5
1 Področje uporabe .6
2 Zveza z drugimi standardi .6
3 Spremenljivke, funkcije in operatorji.6
4 Matematična logika.8
5 Množice .9
6 Standardne številske množice in intervali .11
7 Drugi znaki in simboli .13
8 Elementarna geometrija .15
9 Operacije .16
10 Kombinatorika.19
11 Funkcije .20
12 Eksponentne in logaritemske funkcije.23
13 Krožne in hiperbolične funkcije.24
14 Kompleksna števila.26
15 Matrike .27
16 Koordinatni sistemi .29
17 Skalarji, vektorji in tenzorji .31
18 Transformi .35
19 Posebne funkcije .36
Dodatek A (normativni): Razlaga uporabljenih simbolov .41
Literatura.45
3
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SIST ISO 80000-2 : 2013
Predgovor
ISO (Mednarodna organizacija za standardizacijo) je svetovna zveza nacionalnih organov za
standarde (članov ISO). Mednarodne standarde navadno pripravljajo tehnični odbori ISO. Vsak član,
ki želi delovati na določenem področju, za katero je bil ustanovljen tehnični odbor, ima pravico biti
zastopan v tem odboru. Pri delu sodelujejo tudi vladne in nevladne mednarodne organizacije,
povezane z ISO. V vseh zadevah, ki so povezane s standardizacijo na področju elektrotehnike, ISO
tesno sodeluje z Mednarodno elektrotehniško komisijo (IEC).
Mednarodni standardi so pripravljeni v skladu s pravili, podanimi v Direktivah ISO/IEC, 2. del.
Glavna naloga tehničnih odborov je priprava mednarodnih standardov. Osnutki mednarodnih
standardov, ki jih sprejmejo tehnični odbori, se pošljejo vsem članom v glasovanje. Za objavo
mednarodnega standarda je treba pridobiti soglasje najmanj 75 % članov, ki se udeležijo glasovanja.
Opozoriti je treba na možnost, da je lahko nekaj elementov tega dokumenta predmet patentnih pravic.
ISO ne prevzema odgovornosti za identifikacijo katerihkoli ali vseh takih patentnih pravic.
ISO 80000-2 je pripravil tehnični odbor ISO/TC 12 Veličine in enote v sodelovanju z IEC/TC 25
Veličine in enote.
Prva izdaja standarda ISO 80000-2 razveljavlja in nadomešča ISO 31-11:1992, ki je tehnično
spremenjen. V primerjavi s prejšnjim standardom so glavne tehnične spremembe naslednje:
– dodane so štiri točke, in sicer "Standardne številske množice in intervali", "Elementarna
geometrija", "Kombinatorika" ter "Transformi".
ISO 80000 s skupnim naslovom Veličine in enote sestavljajo naslednji deli:
– 1. del: Splošno
1
– 2. del: Matematični znaki in simboli za uporabo v naravoslovnih vedah in tehniki
– 3. del: Prostor in čas
– 4. del: Mehanika
– 5. del: Termodinamika
– 7. del: Svetloba
– 8. del: Akustika
– 9. del: Fizikalna kemija in molekulska fizika
– 10. del: Atomska in jedrska fizika
– 11. del: Značilna števila
– 12. del: Fizika trdne snovi
IEC 80000 s skupnim naslovom Veličine in enote sestavljajo naslednji deli:
– 6. del: Elektromagnetizem
– 13. del: Informacijska znanost in tehnologija
– 14. del: Telebiometrija, povezana s fiziologijo človeka
1
Naslov naj se v drugi izdaji ISO 80000-2 skrajšano glasi "Matematika".
4
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SIST ISO 80000-2 : 2013
Uvod
Ureditev preglednic
Prvi stolpec preglednic, "Zap. št.", vsebuje zaporedno številko veličine, ki ji v oklepaju sledi bodisi
številka ustrezne veličine iz ISO 31-11 bodisi pomišljaj, če se ta veličina v ISO 31-11 ni pojavila.
V drugem stolpcu, "Znak, simbol, izraz", je naveden obravnavani znak ali simbol, ponavadi v okviru
značilnega izraza. Če je za isto veličino navedenih več znakov, simbolov ali izrazov, so vsi
enakovredni. V nekaterih primerih, npr. pri potenciranju, je samo en značilen izraz brez simbola.
V tretjem stolpcu, "Pomen, besedni ekvivalent", je nakazan pomen izraza oziroma kako se lahko
prebere. Namenjen je prepoznavanju pojma in ne popolni matematični definiciji.
V četrtem stolpcu, "Opombe in primeri", so podane nadaljnje informacije. Definicije so navedene le, če
so dovolj kratke, da sodijo v stolpec. Ni treba, da so definicije matematično popolne.
Ureditev preglednice "Koordinatni sistemi" v točki 16 je nekoliko drugačna.
5
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SIST ISO 80000-2 : 2013
Veličine in enote
2. del:
Matematični znaki in simboli za uporabo v naravoslovnih vedah in tehniki
1 Področje uporabe
ISO 80000-2 podaja splošne informacije o matematičnih znakih in simbolih, njihovem pomenu,
besednih ekvivalentih in uporabi.
Priporočila iz ISO 80000-2 so v glavnem namenjena za uporabo v naravoslovnih vedah in tehniki,
veljajo pa tudi za druga področja, kjer se matematika uporablja.
2 Zveza z drugimi standardi
Za uporabo tega dokumenta so nujno potrebni spodaj navedeni standardi. Pri datiranem sklicevanju
se upošteva samo navedena izdaja. Pri nedatiranem sklicevanju se upošteva zadnja izdaja
navedenega dokumenta (vključno z morebitnimi dopolnili).
2
ISO 80000-1:– Veličine in enote – 1. del: Splošno
3 Spremenljivke, funkcije in operatorji
Spremenljivke, kot so x, y itd., in tekoče številke, kot je i v ∑ x , so natisnjene s poševnimi črkami. Tudi
i i
parametri, kot so a, b itd., ki se lahko štejejo za konstante v določenem sobesedilu, so natisnjeni poševno.
Enako velja na splošno tudi za funkcije, npr. f, g.
Eksplicitno definirana funkcija, ki ni odvisna od sobesedila, pa je natisnjena s pokončnimi črkami, npr.
sin, exp, ln, Γ. Matematične konstante, katerih vrednosti se nikoli ne spremenijo, so tiskane pokončno,
2
npr. e = 2,718 281 8.; π = 3,141 592.; i = –1. Dobro definirani operatorji so prav tako natisnjeni
pokončno, npr. div, δ v δx in oba d-ja v df /dx.
Števila, izražena s števkami, so vedno natisnjena pokončno, npr. 351 204; 1,32; 7/8.
Argument funkcije se napiše v oklepaju za simbolom funkcije brez presledka med simbolom za
funkcijo in prvim oklepajem, npr. f(x), cos(ωt + φ). Če je simbol za funkcijo sestavljen iz dveh ali več črk
in argument ne vsebuje nobenega operacijskega znaka, kot so +, –, ×, · ali /, se oklepaj argumenta
lahko izpusti. V teh primerih naj bo med simbolom funkcije in argumentom majhen presledek, npr.
int 2,4; sin nπ; arcosh 2A; Ei x.
Ob kakršni koli nevarnosti zamenjave je treba obvezno uporabiti oklepaj. Na primer, piše se cos(x) + y,
ne cos x + y, kar bi se lahko zamenjalo s cos(x + y).
Kot ločilo med števili ali izrazi se lahko uporabi vejica, podpičje ali drug ustrezen simbol. Na splošno
se prednostno uporablja vejica, razen kadar se uporabljajo števila z decimalno vejico.
Če je treba izraz ali enačbo razcepiti v dve ali več vrstic, se uporabi ena od naslednjih metod:
a) prelom vrstice se vstavi takoj za enim od znakov =, +, −, ± ali ∓ ali po potrebi takoj za enim od
znakov ×, · ali /. V tem primeru znak označuje, da se izraz nadaljuje v naslednji vrstici ali na
naslednji strani;
b) prelom vrstice se vstavi takoj pred enim od znakov =, +, −, ± ali ∓ ali po potrebi takoj pred enim
od znakov , · ali /. V tem primeru znak označuje, da je izraz nadaljevanje prejšnje vrstice ali strani.
×
2
V pripravi za izdajo. (Revizija ISO 31-0:1992)
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SIST ISO 80000-2 : 2013
Znak se v naslednji vrstici ne sme ponoviti; dva znaka minus bi npr. lahko povzročila napako v predznaku.
V istem dokumentu naj se uporablja samo ena od teh dveh metod. Prelom vrstice naj po možnosti ne bo
znotraj izraza v oklepaju.
Navadno se za različne vrste osnovnih delcev (edink) uporabljajo različne vrste črk. To pripomore k
čitljivosti formul in k vzpostavljanju ustreznega sobesedila. Glede uporabe črkovne družine ni strogih
pravil, se pa to po potrebi razloži.
7
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SIST ISO 80000-2 : 2013
4 Matematična logika
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-4.1 p ⋀ q konjunkcija p in q,
(11-3.1) p in q
2-4.2 p ⋁ q disjunkcija p in q, Ta "ali" je vključujoč, tj. trditev p ⋁ q
je pravilna, če je pravilen p ali q ali oba.
(11-3.2) p ali q
2-4.3 ¬ p negacija p,
(11-3.3) ni p
2-4.4 p � q p implicira q, q � p ima isti pomen kot p � q.
(11-3.4) če p, potem q � je znak za implikacijo.
2-4.5 p � q p je ekvivalenten q (p � q) � (q � p) ima isti pomen kot
p � q.
(11-3.5)
� je znak za ekvivalenco.
2-4.6 �x � A p(x) za vsak x iz A je trditev p(x) Če je iz sobesedila razvidno, za katero
pravilna množico A gre, se lahko uporablja zapis
(11-3.6)
∀ x p(x).
� je univerzalni kvantifikator.
Za x ∈ A glej 2-5.1.
2-4.7 �x � A p(x) obstaja x iz A, za katerega je Če je iz sobesedila razvidno, za katero
trditev p(x) pravilna množico A gre, se lahko uporablja zapis
(11-3.7)
�x p(x).
� je eksistencialni kvantifikator.
Za x � A glej 2-5.1.
1
Zapis �x p(x) se uporablja za
označitev, da obstaja točno en element,
za katerega je trditev p(x) pravilna.
1
Za �se uporablja tudi zapis �!.
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SIST ISO 80000-2 : 2013
5 Množice
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-5.1 x � A x pripada množici A, A � x ima enak pomen kot x � A.
(11-4.1) x je element množice A
2-5.2 y � A y ne pripada množici A, A � y ima enak pomen kot y � A.
(11-4.2) y ni element množice A Negacija je lahko označena tudi s
pokončnico.
2-5.3 {x , x , ., x } množica z elementi x , x , …, x Tudi {x | i � I }, kjer I označuje
1 2 n 1 2 n i
množico indeksov.
(11-4.5)
2-5.4 {x � A | p(x)} množica tistih elementov A, za PRIMER: {x � R | x ≤ 5}
katere velja trditev p(x)
(11-4.6) Če je iz sobesedila razvidno, za katero
množico A gre, se lahko uporablja zapis
{x | p(x)} (na primer {x | x ≤ 5},
če je jasno, da je x spremenljivka za realna
števila).
2-5.5 card A število elementov v A, Kardinalno število je lahko transfinitno
število.
(11-4.7) |A| kardinalno število množice A
Glej tudi 2-9.16.
2-5.6 ∅ prazna množica
(11-4.8)
2-5.7 B � A B je vsebovan v A, Vsak element iz B pripada A.
(11-4.18) B je podmnožica A Uporablja se tudi �, toda glej opombo
k 2-5.8.
A � B ima enak pomen kot B � A.
2-5.8 B � A B je pravilno vsebovan v A, Vsak element B pripada A, toda vsaj
en element A ne pripada B.
(11-4.19) B je prava podmnožica A
Če se za 2-5.7 uporablja �, potem se
za 2-5.8 uporablja �.
A � B ima enak pomen kot B � A.
2-5.9 A ∪ B unija množic A in B Množica elementov, ki pripadajo A ali
B ali obema.
(11-4.24)
A ∪ B = {x | x � A � x � B}
2-5.10 A ∩ B presek množic A in B Množica elementov, ki pripadajo A in B.
(11-4.26) A ∩ B = {x | x � A � x � B}
n
2-5.11 unija družine množic A , A , …, A Množica elementov, ki pripadajo vsaj
1 2 n
A
U i
1
eni od množic A , A , ., A
(11-4.25) i= 1 2 n
Uporabljajo se tudi
A ∪ A ∪…
1 2
i n
∪ A n 1,U U
n
i∈I
U
i=
i∈I
kjer I označuje množico indeksov.
n
2-5.12 presek družine množic A , …, A Množica elementov, ki pripadajo vsem
1 n
A
I
i
1
množicam A , A , ., A
1 2 n
(11-4.27) i=
Uporabljajo se tudi
A ∩ A ∩ …
1 2
i n
n 1
I I I
∩ A ,
n
i= i∈I
i∈I
kjer I označuje množico indeksov.
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SIST ISO 80000-2 : 2013
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-5.13 A \ B razlika množic A in B, Množica elementov, ki pripadajo A,
toda ne B.
(11-4.28) A brez B
A \ B = {x | x � A � x � B}
A − B naj se ne uporablja.
Uporablja se tudi ∁ B. ∁ B se v
A A
glavnem uporablja, kadar je B
podmnožica A in se znak A lahko izpusti,
če je iz sobesedila razvidno, za katero
množico A gre.
2-5.14 (a, b) urejeni par a, b, (a, b) = (c, d), če in samo če je
a = c in b = d.
(11-4.30) dvojica a, b
Če bi se vejica lahko pomotoma
razumela kot decimalni znak, se lahko
kot ločilo uporabi podpičje (;) ali
pokončnica (|).
2-5.15 (a , a , …, a ) urejena n-terica Glej opombo k 2-5.14.
1 2 n
(11-4.31)
2-5.16 A × B kartezijski produkt množic Množica urejenih parov (a, b), tako da
A in B je a ∈ A in b ∈ B.
(11-4.32)
A × B = {(x, y) | x � A � y � B}
n
2-5.17 kartezijski produkt družine Množica urejenih n-teric
A
i množic A , A , …, A (x , x , …, x ), tako da je x � A ,
Π
(—) 1 1 2 n 1 2 n 1 1
i=
x � A , …, x � A .
2 2 n n
n
A × A ×.× A
1 2 n
A × A × . × A se označi z A , kjer je n
število faktorjev v produktu.
2-5.18 id enakostno razmerje na id je množica vseh parov
A A
množici A, (x, x), kjer je x � A.
(11-4.33)
diagonala množice A × A Če je množica A razvidna iz
sobesedila, se lahko indeks A izpusti.
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SIST ISO 80000-2 : 2013
6 Standardne številske množice in intervali
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-6.1 N množica naravnih števil, N = {0, 1, 2, 3, …}
(11.4.9) množica pozitivnih celih števil in N* = {1, 2, 3, …}
ničle
Vidno so lahko označene tudi druge
omejitve, glej spodaj.
N = {n ∈ N | n > 5}
>5
Uporabljata se tudi znaka in ℕ.
2-6.2 Z množica celih števil Z = {…, −2, −1, 0, 1, 2, …}
Z* = {n ∈ Z | n ≠ 0}
(11.4.10)
Vidno so lahko označene tudi druge
omejitve, glej spodaj.
Z = {n ∈ Z | n ≥ −3}
≥−3
Uporablja se tudi znak ℤ.
Q* = {r ∈ Q | r ≠ 0}
2-6.3 Q množica racionalnih števil
Vidno so lahko označene tudi druge
(11.4.11)
omejitve, glej spodaj.
Q = {r ∈ Q | r < 0}
<0
Uporabljata se tudi znaka in ℚ.
R* = {x ∈ R | x ≠ 0}
2-6.4 R množica realnih števil
Vidno so lahko označene tudi druge
(11.4.12)
omejitve, glej spodaj.
R = {x ∈ R | x ≥ 0}
≥0
Uporabljata se tudi znaka in ℝ.
C* = {z ∈ C | z ≠ 0}
2-6.5 C množica kompleksnih števil
(11.4.13)
Uporabljata se tudi znaka ℂ in ℂ.
2-6.6 P množica praštevil P = {2, 3, 5, 7, 11, 13, 17, …}
(—)
Uporabljata se tudi znaka ℙ in ℙ.
[a, b] = {x ∈ R | a ≤ x ≤ b}
2-6.7 [a, b] zaprti interval od vključno a do
vključno b
(11.4.14)
(a, b] = {x ∈ R | a < x ≤ b}
2-6.8 (a, b] levo polodprti interval od a (brez
a) do vključno b
Uporablja se tudi zapis ]a, b].
(11.4.15)
[a, b) = {x ∈ R | a ≤ x < b}
2-6.9 [a, b) desno polodprti interval od
vključno a do b (brez b)
Uporablja se tudi zapis [a, b[.
(11.4.16)
(a, b) = {x ∈ R | a < x < b}
2-6.10 (a, b) odprti interval od a (brez a) do b
(brez b)
Uporablja se tudi zapis ]a, b[.
(11.4.17)
(−∞, b] = {x ∈ R | x ≤ b}
2-6.11 (−∞, b] zaprti neomejeni interval do
vključno b
Uporablja se tudi zapis ]−∞, b].
(—)
11
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SIST ISO 80000-2 : 2013
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
(−∞, b) = {x ∈ R | x < b}
2-6.12 (−∞, b) odprti neomejeni interval do b
(brez b)
Uporablja se tudi zapis ]−∞, b[.
(—)
[a, +∞) = {x ∈ R | a ≤ x}
2-6.13 [a, +∞) zaprti navzgor neomejeni
interval od vključno a
Uporabljajo se tudi zapisi [a, ∞ [,
(—)
[a, +∞ [ in [a, ∞).
(a, +∞) = {x ∈ R | a < x}
2-6.14 (a, +∞) odprti navzgor neomejeni
interval od a (brez a)
Uporabljajo se tudi zapisi ]a, +∞[,
(—)
]a, ∞ [ in (a, ∞).
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SIST ISO 80000-2 : 2013
7 Drugi znaki in simboli
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-7.1 a = b a je enak b Znak ≡ se lahko uporablja za poudarek,
da je določena enakost identiteta.
(11-5.1)
Glej tudi 2-7.18.
2-7.2 a ≠ b a ni enak b Za negacijo se lahko uporabi tudi
pokončnica.
(11-5.2)
2-7.3 a := b a je po definiciji enak b PRIMER:
p := mυ, kjer je p gibalna količina, m masa
(11-5.3)
in υ hitrost.
Uporabljata se tudi znaka = in ≝.
def
PRIMERA:
2-7.4 a ustreza b
a ≙ b
(11-5.4)
Ko je E = kT, je 1 eV ≙ 11 604,5 K
Če 1 cm na zemljevidu ustreza dolžini
10 km, se lahko zapiše: 1 cm ≙ 10 km.
Ujemanje ni simetrično.
2-7.5 a ≈ b a je približno enak b Od uporabnika je odvisno, ali je
približek dovolj dober. Enakost ni
(11-5.5)
izključena.
PRIMER:
2-7.6 a je asimptotično enak b
a ≃ b
1 1
~
ko gre
(11-7.7)
si n
− x→ a
x− a x− a
(Za x → a glej 2-7.16.)
2-7.7 a ~ b a je sorazmeren b Znak ~ se uporablja tudi za
ekvivalenčne relacije.
(11-5.6)
Uporablja se tudi zapis a ∝ b.
2-7.8 M je kongruenten N, M in N sta točkovni množici
M ≅ N
(geometrični številki).
(—) M je izomorfen N
Ta znak se uporablja tudi za
izomorfizme matematičnih struktur.
2-7.9 a < b a je manjši od b
(11-5.7)
2-7.10 b > a b je večji od a
(11-5.8)
2-7.11 a ≤ b a je manjši ali enak b
(11-5.9)
2-7.12 b ≥ a b je večji ali enak a
(11-5.10)
2-7.13 a « b a je mnogo manjši od b Od uporabnika je odvisno, ali je a
dovolj majhen v primerjavi z b.
(11-5.11)
2-7.14 b » a b je mnogo večji od a Od uporabnika je odvisno, ali je b
dovolj velik v primerjavi z a.
(11-5.12)
13
︵
︶
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SIST ISO 80000-2 : 2013
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
Ta znak ne označuje števila, temveč
2-7.15 ∞ neskončno
je pogosto del različnih izrazov, ki se
(11-5.13)
nanašajo na meje. Uporabljata se tudi
zapisa +∞ in –∞.
2-7.16 x → a x teži proti a Ta znak se pojavlja kot del različnih
izrazov, ki se nanašajo na meje.
(11-7.5)
a je lahko tudi ∞, +∞ ali –∞.
2-7.17 m | n m deli n Za celi števili m in n:
(—)
∃ k ∈ Z m·k = n
2-7.18 n ≡ k mod m n je kongruenten k modulu m Za cela števila n, k in m:
(—) m | (n − k)
Glej tudi 2-7.1.
2-7.19 (a + b) okrogli oklepaj Pri združevanju se priporoča samo
uporaba okroglega oklepaja, saj imajo
(11-5.14) [a + b] oglati oklepaj
oglati, zaviti in lomljeni oklepaji na
{a + b} zaviti oklepaj
določenih področjih poseben pomen.
lomljeni oklepaj
Okrogli oklepaj se lahko nedvoumno
〈a + b〉
ugnezdi.
14
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SIST ISO 80000-2 : 2013
8 Elementarna geometrija
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-8.1 daljica AB je vzporedna z daljico Če sta g in h daljici, ki ju določata
AB║CD
CD točki A in B oziroma C in D, se zapiše
(11-5.15)
g ║ h.
Uporablja se tudi AB//CD.
Če sta g in h daljici, ki ju določata
2-8.2 daljica AB je pravokotna na
AB ⊥ CD
točki A in B oziroma C in D, se zapiše
daljico CD
(11-5.16)
g ⊥ h. V ravnini se morata daljici
sekati.
2-8.3 kot na višinski točki B pri Kot ni usmerjen, velja, da
∢ABC
trikotniku ABC
(—)
je ∢ABC = ∢CBA in
0 ≤ ∢ABC ≤ π rad.
AB
2-8.4 linijski segment od A do B Linijski segment je množica točk med
A in B na daljici AB.
(—)
AB CD
AB
2-8.5 vektor od A do B
Če je = , potem je B, gledan
(—) od A, v isti smeri in razdalji kot je D,
gledan od C. Iz tega ne sledi, da je
A = C in B = D.
2-8.6 d(A, B) razdalja med točkama A in B Razdalja je dolžina linijskega
AB
(—) segmenta ter tudi velikost
AB
vektorja .
15
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SIST ISO 80000-2 : 2013
9 Operacije
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-9.1 a + b a plus b Ta operacija se imenuje seštevanje. Znak +
je znak seštevanja.
(11-6.1)
2-9.2 a − b a minus b Ta operacija se imenuje odštevanje. Znak –
je znak odštevanja.
(11-6.2)
2-9.3 a ± b a plus ali minus b To je kombinacija dveh vrednosti v enem
izrazu.
(11-6.3)
2-9.4 a m b a minus ali plus b −(a ± b) = −a m b
(11-6.4)
2-9.5 a · b a pomnoženo z b, Ta operacija se imenuje množenje. Znak
za množenje je poldvignjena pika (·) ali krat
(11-6.5) a × b a krat b
(×).
a b
Oba se lahko izpustita, če ne more priti do
ab
nesporazuma.
Glej tudi 2-5.16, 2-5.17, 2-17.11,
2-17.12, 2-17.23 in 2-17.24 za uporabo
pike in križca v različnih produktih.
a a
2-9.6 a deljeno z b –1
= a · b
b b
(11-6.6)
Glej tudi ISO 80000-1:—, 7.1.3.
a/b
Za razmerja se uporablja tudi znak :.
PRIMER: Razmerje med višino h in širino b lista
2
formata A4 je h : b = .
Znak ÷ naj se ne uporablja.
n
2-9.7 a + a + … + a , Uporabljajo se tudi zapisi
1 2 n
i n
a
n
∑ i
(11-6.7) 1 vsota a , a , …, a
1 2 n 1
a , a , a a
∑ ∑∑ ∑
i= i i i i
i = i
i
n
2-9.8 a · a · …·a , Uporabljajo se tudi zapisi
1 2 n
i n
a
n
∏ i
(11-6.8) produkt a , a , …, a
1 1 2 n 1
a , a , a a
∏ ∏ ∏ ∏
i = i i i i
i = i
i
p 2
a
2-9.9 a na potenco p Besedni ekvivalent a je a na kvadrat;
3
besedni ekvivalent a je
(11-6.9)
a na kub.
1/2
2-9.10 a a na potenco 1/2,
Če je a ≥ 0, potem je ≥ 0.
a
(11-6.10) kvadratni koren iz a
Znaku √a se je treba izogibati.
Glej opombo k točki 2-9.11.
1/n
2-9.11 a a na potenco 1/n,
Če je a ≥ 0, potem je ≥ 0.
n
(11-6.11) n-ti koren iz a
Znaku √a se je treba izogibati.
n
Če stoji znak √ ali √ pred sestavljenim
izrazom, je treba uporabiti oklepaj, da bi se
izognili dvoumnosti.
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SIST ISO 80000-2 : 2013
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-9.12 srednja vrednost x, Z drugimi metodami dobljene srednje
vrednosti so:
(11-6.14) aritmetična sredina x
〈x〉
– harmonična sredina, označena z
x
indeksom h,
a
– geometrična sredina, označena z
indeksom g,
– kvadratična sredina, pogosto imenovana
"koren srednjega kvadrata", označena z
indeksom q ali rms.
Indeks se lahko izpusti le za aritmetično
sredino.
V matematiki se x uporablja tudi za
kompleksno konjugiran x; glej 2-14.6.
2-9.13 sgn a predznak a; Za realni a:
1 če j e 0
(11-6.13) signum a
⎧
a〉
0 če j e 0
⎪
sgn a = a=
⎨
1 če j e 0
⎪
− a〈
⎩
Glej tudi točko 2-14.7.
2-9.14 inf M natančna spodnja meja M; Največja spodnja meja neprazne navzdol
omejene množice realnih števil.
(—) infimum M
2-9.15 sup M natančna zgornja meja M; Najmanjša zgornja meja neprazne navzgor
omejene množice realnih števil.
(—) supremum M
2-9.16 |a| absolutna vrednost a, Uporablja se tudi zapis abs a.
(11-6.12) modul a, Absolutna vrednost realnega števila a.
velikost a Modul kompleksnega števila a; glej 2-14.4.
Velikost vektorja a; glej 2-17.4.
Glej tudi 2-5.5.
2-9.17 spodnji celi del a, Uporablja se tudi zapis ent a.
⌊a⌋
PRIMERA:
(11-6.17) (floor a)
⌊2,4⌋ = 2
največje celo število, ki je
manjše ali enako realnemu
⌊–2,4⌋ = –3
številu a
2-9.18 zgornji celi del a, "ceil" je okrajšava angleške besede
⌈a⌉
"ceiling".
(—) (ceil a)
PRIMERA:
najmanjše celo število, ki je
⌈2,4⌉ = 3
večje ali enako realnemu
številu a
⌈–2,4⌉ = –2
2-9.19 int a celi del realnega števila a
int a = sgn a ⌊|a|⌋
(—) PRIMERA:
int(2,4) = 2
int(−2,4) = −2
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SIST ISO 80000-2 : 2013
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-9.20 frac a decimalni del realnega frac a = a − int a
števila a
PRIMERA:
(—)
frac(2,4) = 0,4
frac(−2,4) = −0,4
2-9.21 min(a, b) najmanjši del a in b Operacija je splošna za več števil in za
množice števil. Ni pa nujno, da ima
(—)
neskončna množica števil najmanjši
element.
2-9.22 max(a, b) največji del a in b Operacija je splošna za več števil in za
množice števil. Ni pa nujno, da ima
(—)
neskončna množica števil največji element.
18
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SIST ISO 80000-2 : 2013
10 Kombinatorika
V tej točki sta n in k naravni števili, s tem da je k ≤ n.
Znak, simbol,
Zap. št. Pomen, besedni ekvivalent Opombe in primeri
izraz
2-10.1 n! fakulteta
n! = k = 1 · 2 · 3 ·. · n (n > 0)
(11-6.15)
0! = 1
k k
2-10.2 a padajoča fakulteta a = a ·(a – 1) · . · (a – k + 1) (k > 0)
0
(—) [a] a = 1
k
a je lahko kompleksno število.
Za naravno število n je:
!
n
k
n !
=
n− k
k k
2-10.3 rastoča fakulteta
a
a = a ·(a + 1) · . · (a + k – 1) (k > 0)
(—)
(a)
0
k
a = 1
a je lahko kompleksno število.
Za naravno število n je:
V teoriji posebnih funkcij se (a)
k
imenuje Pochhammerjev simbol. V
kombinatoriki in statistiki pa se isti
simbol pogosto uporablja za
padajočo fakulteto.
!
2-10.4 binomski koeficient
n
⎛ n⎞ ⎛n⎞
= ! ! (0 ≤ k ≤ n)
⎜ ⎟ ⎜ ⎟
k k
(11-6.16)
k()n− k
⎝ ⎠ ⎝ ⎠
2-10.5 B Bernoullijeva števila
n
(—)
B = 1
0
B = −1/2 , B = 0
1 2n+3
!
C
k C
2-10.6 število kombinacij brez
n
⎛n⎞
k
n
= = ! !
⎜ ⎟
n
ponavljanja
k
(11-6.16)
⎝ ⎠ k()n− k
1
C
R k
2-10.7 število kombinacij s ponavljanjem C
R ⎛n+ k− ⎞
k
n
⎜ ⎟
n
k
(—)
⎝ ⎠
!
V
V
k
2-10.8 število variacij brez ponavljanja
n
k k
n = n =
!
n
(—)
()n− k
Kadar je
...
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