EN ISO 80000-2:2013

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.Größen und Einheiten - Teil 2: Mathematische Zeichen für Naturwissenschaft und Technik (ISO 80000-2:2009)Grandeurs et unités - Partie 2: Signes et symboles mathématiques à employer dans les sciences de la nature et dans la technique (ISO 80000-2:2009)Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology (ISO 80000-2:2009)07.020MatematikaMathematics01.075Simboli za znakeCharacter symbols01.060Quantities and unitsICS:Ta slovenski standard je istoveten z:EN ISO 80000-2:2013SIST EN ISO 80000-2:2013en01-junij-2013SIST EN ISO 80000-2:2013SLOVENSKI
STANDARDSIST ISO 31-11:1995SIST ISO 31-11:19951DGRPHãþD

EUROPEAN STANDARD NORME EUROPÉENNE EUROPÄISCHE NORM
EN ISO 80000-2
April 2013 ICS 01.060 English Version
Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology (ISO 80000-2:2009)
Grandeurs et unités - Partie 2: Signes et symboles mathématiques à employer dans les sciences de la nature et dans la technique (ISO 80000-2:2009)
Größen und Einheiten - Teil 2: Mathematische Zeichen für Naturwissenschaft und Technik (ISO 80000-2:2009) This European Standard was approved by CEN on 14 March 2013.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management Centre has the same status as the official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and United Kingdom.
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Reference numberISO 80000-2:2009(E)© ISO 2009
INTERNATIONAL STANDARD ISO80000-2First edition2009-12-01Quantities 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
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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. ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel.
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ii © ISO 2009 – All rights reserved
ISO 80000-2:2009(E) © ISO 2009 – All rights reserved iii 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 80000-2:2009(E) iv © ISO 2009 – All rights reserved 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 ⎯ Part 2: Mathematical signs and symbols to be used in the natural sciences and technology1) ⎯ Part 3: Space and time ⎯ Part 4: Mechanics ⎯ Part 5: Thermodynamics ⎯ Part 7: Light ⎯ Part 8: Acoustics ⎯ Part 9: Physical chemistry and molecular physics ⎯ Part 10: 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. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved v 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 80000-2:2009(E) vi © ISO 2009 – All rights reserved 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.
INTERNATIONAL STANDARD ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 1 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. ISO 80000-1:—2) , Quantities and units — Part 1: General 3 Variables, functions and operators Variables such as x, y, etc., and running numbers, such as i in Σi xi are printed in italic (sloping) type. Parameters, such as a, b, etc., which may be considered as constant in a particular context, are printed in 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, e.g. e = 2,718 218 8…; π = 3,141 592…; i2 = −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) SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) 2 © ISO 2009 – All rights reserved 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 Ï, 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 Ï, 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. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 3 4 Mathematical logic Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-4.1 (11-3.1) p ß q conjunction of p and q, p and q
2-4.2 (11-3.2) p à q disjunction of p and q, p or q This “or” is inclusive, i.e.
p à q is true, if either p or q, or both are true. 2-4.3 (11-3.3) ¬ p negation of p, not p
2-4.4 (11-3.4) p ≈ q p implies q, if p, then q q √ p has the same meaning as p ≈ q. ≈ is the implication symbol. 2-4.5 (11-3.5) p ª q p is equivalent to q (p ≈ q) ∧ (q ≈ p) has the same meaning as p ª q. ª is the equivalence symbol. 2-4.6 (11-3.6) Áx Ç A
p(x) for every x belonging to A, the proposition p(x) is true lf it is clear from the context which set A is being considered, the notation Áx p(x) can be used. Á is the universal quantifier. For x Ç A, see 2-5.1. 2-4.7 (11-3.7) Ãx Ç A
p(x) there exists an x belonging to A for which p(x) is true lf it is clear from the context which set A is being considered, the notation Ãx p(x) can be used. Ã is the existential quantifier. For x Ç A, see 2-5.1. Ã1x p(x) is used to indicate that there is exactly one element for which p(x) is true. Ã! is also used for Ã1.
ISO 80000-2:2009(E) 4 © ISO 2009 – All rights reserved 5 Sets Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-5.1 (11-4.1) x Ç A x belongs to A, x is an element of the set A A Ê x has the same meaning as x Ç A. 2-5.2 (11-4.2) y È A y does not belong to A, y is not an element of the set A A Ë y has the same meaning as y È A. The negating stroke may also be vertical. 2-5.3 (11-4.5) {x1, x2, …, xn} set with elements x1, x2, …, xn Also {xi | i Ç I}, where I denotes a set of subscripts. 2-5.4 (11-4.6) {x Ç A | p(x)} set of those elements of A for which the proposition p(x) is true EXAMPLE
{x Ç R | x u 5} lf it is clear from the context which set A is being considered, the notation {x | p(x)} can be used (for example {x | x u 5}, if it is clear that x is a variable for real numbers). 2-5.5 (11-4.7) card A A number of elements in A, cardinality of A The cardinality can be a transfinite number.See also 2-9.16. 2-5.6 (11-4.8) Å the empty set
2-5.7 (11-4.18) B ; A
B is included in A, B is a subset of A Every element of B belongs to A. 7 is also used, but see remark to 2-5.8. A < B has the same meaning as B ; A. 2-5.8 (11-4.19) B 7 A B is properly included in A, B is a proper subset of A Every element of B belongs to A, but at least one element of A does not belong to B. lf 7 is used for 2-5.7, then ? shall be used for 2-5.8. A 8 B has the same meaning as B 7 A. 2-5.9 (11-4.24) A â B union of A and B The set of elements which belong to A or to B or to both A and B. A â B = {x | x Ç A à x Ç B} 2-5.10 (11-4.26) A á B
intersection of A and B The set of elements which belong to both A and B. A á B = {x | x Ç A ß x Ç B} 2-5.11 (11-4.25) 1niiA=∪ A1 â A2 â … â An union of the sets A1, A2, …, An The set of elements belonging to at least one of the sets A1, A2, ., An 1ni=∪, iI∈∪ and iI∈∪ are also used, where I denotes a set of subscripts. 2-5.12 (11-4.27) 1niiA=∩ A1 á A2 á … á An intersection of the sets A1, …, An The set of elements belonging to all sets A1, A2, ., An 1ni=∩, iI∈∩ and iI∈∩ are also used, where I denotes a set of subscripts. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 5 Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-5.13 (11-4.28) A Ñ B difference of A and B, A minus B The set of elements which belong to A but not to B. A Ñ B = {x | x Ç A ß x È B} A − B should not be used. ÂA B is also used. ÂA B is mainly used 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 (11-4.30) (a, b) ordered pair a, b, couple a, b (a, b) = (c, d) if and only if a = c and b = d. If the comma can be mistaken as the decimal sign, then the semicolon (;) or a stroke (⏐) may be used as separator. 2-5.15 (11-4.31) (a1, a2, …, an) ordered n-tuple See remark to 2-5.14. 2-5.16 (11-4.32) A × B Cartesian product of the sets A and B The set of ordered pairs (a, b) such that a Ç A and b Ç B. A × B = {(x, y) | x Ç A ß y Ç B} 2-5.17 (—) 1niiA=Π 12.nAAA××× Cartesian product of the sets A1, A2, …, An The set of ordered n-tuples (x1, x2, …, xn) such that x1 Ç A1, x2 Ç A2, …, xn Ç An. A × A × . × A is denoted by An, where n is the number of factors in the product. 2-5.18 (11-4.33) idA identity relation on A, diagonal of A × A idA is the set of all pairs (x, x) where x Ç A. If the set A is clear from the context, the subscript A can be omitted.
ISO 80000-2:2009(E) 6 © ISO 2009 – All rights reserved 6 Standard number sets and intervals Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-6.1 (11.4.9) N the set of natural numbers, the set of positive integers and zero N = {0, 1, 2, 3, …} N* = {1, 2, 3, …} Other restrictions can be indicated in an obvious way, as shown below. N>5 = {n Ç N | n > 5} The symbols IN and B are also used. 2-6.2 (11.4.10) Z the set of integers Z = {…, −2, −1, 0, 1, 2, …} Z* = {n Ç Z | n ≠ 0} Other restrictions can be indicated in an obvious way, as shown below. ZW−3 = {n Ç Z | n W −3} The symbol P is also used. 2-6.3 (11.4.11) Q the set of rational numbers Q* = {r Ç Q | r ≠ 0} Other restrictions can be indicated in an obvious way, as shown below. Q<0 = {r Ç Q | r < 0} The symbols
and G are also used.
2-6.4 (11.4.12) R the set of real numbers R* = {x Ç R | x ≠ 0} Other restrictions can be indicated in an obvious way, as shown below. RW0 = {x Ç R | x W 0} The symbols IR and J are also used. 2-6.5 (11.4.13) C the set of complex numbers C* = {z Ç C | z ≠ 0} The symbols . and . are also used. 2-6.6 (—) P the set of prime numbers P = {2, 3, 5, 7, 11, 13, 17, …} The symbols F and F are also used. 2-6.7 (11.4.14) [a, b] closed interval from a included to b included [a, b] = {x Ç R | a u x u b} 2-6.8 (11.4.15) (a, b] left half-open interval from a excluded to b included (a, b] = {x Ç R | a < x u b} The notation ]a, b] is also used. 2-6.9 (11.4.16) [a, b) right half-open interval from a included to b excluded [a, b) = {x Ç R | a u x < b} The notation [a, b[ is also used. 2-6.10 (11.4.17) (a, b) open interval from a excluded to b excluded (a, b) = {x Ç R | a < x < b} The notation ]a, b[ is also used. 2-6.11 (—) (−∞, b] closed unbounded interval up to b included (−∞, b] = {x Ç R | x u b} The notation ]−∞, b] is also used. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 7 Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-6.12 (—) (−∞, b) open unbounded interval up to b excluded (−∞, b) = {x Ç R | x < b} The notation ]−∞, b[ is also used. 2-6.13 (—) [a, +∞) closed unbounded interval onward from a included [a, +∞) = {x Ç R | a u x} The notations [a, ∞ [,
[a, +∞ [ and [a, ∞) are also used. 2-6.14 (—) (a, +∞) open unbounded interval onward from a excluded (a, +∞) = {x Ç R | a < x} The notations ]a, +∞[, ]a, ∞ [ and (a, ∞) are also used.
ISO 80000-2:2009(E) 8 © ISO 2009 – All rights reserved 7 Miscellaneous signs and symbols Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-7.1 (11-5.1) a = b a is equal to b The symbol
may be used to emphasize that a particular equality is an identity. See also 2-7.18. 2-7.2 (11-5.2) a ≠ b a is not equal to b The negating stroke may also be vertical. 2-7.3 (11-5.3) a := b a is by definition equal to b EXAMPLE p := mv, where p is momentum, m is mass and v is velocity. The symbols =def and
are also used. 2-7.4 (11-5.4) a
b a corresponds to b EXAMPLES When E = kT, then 1 eV
11 604,5 K 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 (11-5.5) a ≈ b a is approximately equal to b It depends on the user whether an approximation is sufficiently good. Equality is not excluded. 2-7.6 (11-7.7) a û b a is asymptotically equal to b EXAMPLE 11sin()xaxa−− as x → a (For
x → a, see 2-7.16.) 2-7.7 (11-5.6) a ó b a is proportional to b The symbol ó is also used for equivalence relations. The notation a Ö b is also used. 2-7.8 (—) M ý N M is congruent to N, M is isomorphic to N M and N are point sets (geometrical figures). This symbol is also used for isomorphisms of mathematical structures. 2-7.9 (11-5.7) a < b a is less than b
2-7.10 (11-5.8) b > a b is greater than a
2-7.11 (11-5.9) a u b a is less than or equal to b
2-7.12 (11-5.10) b W a b is greater than or equal to a
2-7.13 (11-5.11) a
b a is much less than b It depends on the user whether a is sufficiently small as compared to b. 2-7.14 (11-5.12) b
a b is much greater than a It depends on the user whether b is sufficiently great as compared to a. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 9 Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-7.15 (11-5.13) ∞ infinity This symbol does not denote a number but is often part of various expressions dealing with limits. The notations +∞, -∞ are also used. 2-7.16 (11-7.5) x → a x tends to a This symbol occurs as part of various expressions dealing with limits. a may be also ∞, +∞, or -∞. 2-7.17 (—) m⏐n m divides n For integers m and n: ∃ k Ç Z m⋅k = n 2-7.18 (—) n
k mod m n is congruent to k modulo m For integers n, k and m: m⏐(n − k) See also 2-7.1. 2-7.19 (1-5.14) (a + b) [a + b] {a + b} 〈a + b〉 parentheses square brackets braces angle brackets It is recommended to use only parentheses for grouping, since brackets and braces often have a specific meaning in particular fields. Parentheses can be nested without ambiguity.
ISO 80000-2:2009(E) 10 © ISO 2009 – All rights reserved 8 Elementary geometry Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-8.1 (11-5.15) ABÝCD the straight line AB is parallel to the straight line CD It is written g Ý h if g and h are the straight lines determined by the points A and B, and the points C and D, respectively. AB//CD is also used. 2-8.2 (11-5.16) ABZCD the straight line AB is perpendicular to the straight line CD It is written g Z h if g and h are the straight lines determined by the points A and B, and the points C and D, respectively. In a plane, the straight lines must intersect. 2-8.3 (—) ÚABC angle at vertex B in the triangle ABC
The angle is not oriented, it holds that ÚABC = ÚCBA and 0 u ÚABC u π rad. 2-8.4 (—) AB line segment from A to B The line segment is the set of points between A and B on the straight line AB. 2-8.5 (—) AB→ vector from A to B 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. 2-8.6 (—) d(A, B) distance between points A and B The distance is the length of the line segment ABand also the magnitude of the vector AB→. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 11 9 Operations Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-9.1 (11-6.1) a + b a plus b This operation is named addition. The symbol + is the addition symbol. 2-9.2 (11-6.2) a − b a minus b This operation is named subtraction. The symbol − is the subtraction symbol. 2-9.3 (11-6.3) a ± b a plus or minus b This is a combination of two values into one expression. 2-9.4 (11-6.4) a Ï b a minus or plus b −(a ± b) = −a Ï b 2-9.5 (11-6.5) a ⋅ b a × b a b ab a multiplied by b, a times b This operation is named multiplication. The symbol for multiplication is a half-high dot (·) or a cross (×). Either may be omitted if no 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 (11-6.6) ab a/b a divided by b ab = a ⋅ b–1 See also ISO 80000-1:—, 7.1.3. 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. 2-9.7 (11-6.7) 1niia=∑ a1 + a2 + … + an, sum of a1, a2, …, an The notations 1niia=∑, iia∑, iia∑ and ia∑are also used. 2-9.8 (11-6.8) 1niia=∏ a1 ⋅ a2 ⋅ … ⋅ an, product of a1, a2, …, an The notations 1niia=∏, iia∏, iia∏ and ia∏are also used. 2-9.9 (11-6.9) ap a to the power p The verbal equivalent of a2 is a squared; the verbal equivalent of a3 is a cubed. 2-9.10 (11-6.10) a1/2 a a to the power 1/2, square root of a lf a W 0, then aW 0. The symbol √a should be avoided. See remark to 2-9.11. 2-9.11 (11-6.11) a1/n na a to the power 1/n, nth root of a lf a W 0, then naW 0. The symbol n√a should be avoided. lf the symbol n√ or √ acts on a composite expression, parentheses shall be used to avoid ambiguity. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) 12 © ISO 2009 – All rights reserved Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-9.12 (11-6.14) x 〈x〉 ax mean value of x, arithmetic mean of x Mean values obtained by other methods are the
- harmonic mean denoted by subscript h,- 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 (11-6.13) sgn a signum a For real a: sgn a = 1if00if01if0aaa>⎧⎪=⎨⎪−<⎩ 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 (11-6.12) |a| absolute value of a, modulus of a, magnitude of a The notation abs a is also used. Absolute value of real number a. Modulus of complex number a; see 2-14.4. Magnitude of vector a; see 2-17.4. See also 2-5.5. 2-9.17 (11-6.17) ⎣a⎦ floor a, the greatest integer less than or equal to the real number a The notation ent a is also used. EXAMPLES ⎣2,4⎦ = 2 ⎣−2,4⎦ = −3 2-9.18 (—) ⎡a⎤ ceil a, the least integer greater than or equal to the real number a “ceil” is an abbreviation of the word “ceiling”. EXAMPLES ⎡2,4⎤ = 3 ⎡−2,4⎤ = −2 2-9.19 (—) int a integer part of the real number a int a = sgn a ⋅ ⎣⏐a⏐⎦ EXAMPLES int(2,4) = 2 int(−2,4) = −2 2-9.20 (—) frac a fractional part of the real number a frac a = a − int a EXAMPLES frac(2,4) = 0,4 frac(−2,4) = −0,4 SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 13 Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 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. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) 14 © ISO 2009 – All rights reserved 10 Combinatorics In this clause, n and k are natural numbers, with k u n. Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-10.1 (11-6.15) n! factorial n! = 1nkk=∏ = 1 ⋅ 2 ⋅ 3 ⋅ … ⋅ n (n > 0) 0! = 1 2-10.2 (—) ka ka⎡⎤⎣⎦ falling factorial ka = a⋅(a − 1)⋅…⋅(a − k + 1) (k > 0) 0a = 1 a may be a complex number.
For a natural number n: kn = !()!nnk− 2-10.3 (—) ka ()ka rising factorial ka = a⋅(a + 1)⋅…⋅(a + k − 1) (k > 0) 0a = 1 a may be a complex number.
For a natural number n: kn = (1)!(1)!nkn+−− ()ka is called Pochhammer symbol in the theory of special functions. In combinatorics and statistics, however, the same symbol is often used for the falling factorial. 2-10.4 (11-6.16) nk⎛⎞⎜⎟⎝⎠ binomial coefficient !!()!nnkknk⎛⎞=⎜⎟−⎝⎠
(0 u k u n) 2-10.5 (—) Bn Bernoulli numbers Bn = 1011 B1nkknkn−=+⎛⎞−⎜⎟+⎝⎠∑ (n > 0) B0 = 1 1B12=−, 23B0n+= 2-10.6 (11-6.16) Ckn number of combinations without repetition Ckn = nk⎛⎞⎜⎟⎝⎠ = !!()!nknk− 2-10.7 (—) RCkn number of combinations with repetition RCkn = 1nkk+−⎛⎞⎜⎟⎝⎠ 2-10.8 (—) Vkn number of variations without repetition Vkn = kn = !()!nnk− The term “permutation” is used when n = k. 2-10.9 (—) RVkn number of variations with repetitionRVkknn= SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 15 11 Functions Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-11.1 (11-7.1) f, g, h, … functions A function assigns to any argument in its domain a unique value in its range. 2-11.2 (11-7.2) f(x) f(x1, …, xn) value of function f for argument x or for argument (x1, …, xn), respectively A function having a set of n-tuples as its domain is an n-place function. 2-11.3 (—) f : A g B f maps A into B The function f has domain A and range included in B. 2-11.4 (—) f : x{T(x), x Ç A f is the function that maps any x Ç A to T(x) T(x) is a defining term denoting the value of the function f for the argument x. Since f(x) = T(x), the defining term is often used as a symbol for the function f. EXAMPLE 2:3,0;2fxxyx∈⎡⎤⎣⎦6 f is the function (depending on the parameter y) defined on the stated interval by the term 23xy.2-11.5 (—) fxy→ f(x) = y, f maps x onto y EXAMPLE π cos⎯⎯⎯→ −1 2-11.6 (11-7.3) baf (.,,.)ubuafu== ()()fbfa− ()().,,.,,.fbfa− This notation is used mainly when evaluating definite integrals. 2-11.7 (11-7.4) gfD composite function of f and g, g circle f ()()(())gfxgfx=D In the composite gfD, the function g is
applied after function f has been applied. 2-11.8 (11-7.6) limxa→f(x) limx→af(x) limit of f(x) as x tends to a f(x) → b as x → a may be written for limx→af(x) = b. Limits “from the right” (x > a) and “from the left” (x < a) are denoted by lim x→a+ f(x) and lim x→a− f(x), respectively. 2-11.9 (11-7.8) f(x) = O(g(x)) f(x) is big-O of g(x), Ûf(x)/g(x)Û is bounded from above in the limit implied by the context, f(x) is of the order comparable with or inferior to g(x) The symbol “=” here is used for historical reasons and does not have the meaning of equality, because transitivity does not apply. EXAMPLE ()sinOxx=, when 0x→ 2-11.10 (11-7.9) f(x) = o(g(x)) f(x) is little-o of g(x), f(x)/g(x) → 0 in the limit implied by the context, f(x) is of the order inferior to g(x) The symbol “=” here is used for historical reasons and does not have the meaning of equality, because transitivity does not apply. EXAMPLE ()cos1oxx=+, when 0x→ SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) 16 © ISO 2009 – All rights reserved Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-11.11 (11-7.10) ∆f delta f, finite increment of f Difference of two function values implied by the context. EXAMPLES 21xxx∆=− ()()21ffxfx∆=− 2-11.12 (11-7.11) ddfx df∕dx f′ derivative of f with respect to x Only to be used for functions of one variable. d()dfxx, d f(x)∕dx, ()fx′ and Df are also used. lf the independent variable is time t, fis also used for f′. 2-11.13 (11-7.12) ddxafx=⎛⎞⎜⎟⎝⎠ (df∕dx)x = a f′ (a) value of the derivative of f for x = a
2-11.14 (11-7.13) ddnnfx dnf∕dxn f (n) nth derivative of f with respect to xOnly to be used for functions of one variable.
d()dnnfxx, dnf(x)∕dxn, f(n)(x) and Dnf are also used. f′′and f′′′ are also used for f (2) and f (3), respectively. lf the independent variable is time t, f is also used for f′′. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 17 Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-11.15 (11-7.14) fx∂∂ ∂f∕∂x ∂x f partial derivative of f
with respect to x Only to be used for functions of several variables. (,,.)fxyx∂∂, ∂f (x, y, …)∕∂ x,
∂x f (x, y, ….) and Dxf(x, y, …) are also used. The other independent variables may be shown as subscripts, e.g. .yfx∂⎛⎞⎜⎟∂⎝⎠. This partial-derivative notation is extended to derivatives of higher order, e.g. 22fx∂∂ = fxx∂∂⎛⎞⎜⎟∂∂⎝⎠ 2ffxyxy⎛⎞∂∂∂=⎜⎟∂∂∂∂⎝⎠ Other notations, e.g. xyffxy⎛⎞∂∂=⎜⎟∂∂⎝⎠, are also used. 2-11.16 (11-7.15) d f total differential of f d f(x, y, …) = dd.ffxyxy∂∂++∂∂ 2-11.17 (11-7.16) δ f infinitesimal variation of f
2-11.18 (11-7.17) ()dfxx∫ indefinite integral of f
2-11.19 (11-7.18) ()dbafxx∫
definite integral of f
from a to b This is the simple case of a function defined on an interval. Integration of functions defined on more general domains may also be defined. Special notations, e.g. CSV,,,∫∫∫å, are used for integration over a curve C, a surface S, a three-dimensional domain V, and a closed curve or surface, respectively. Multiple integrals are also denoted ,∫∫æ, etc. 2-11.20 (—) ()dbafxx−∫ Cauchy principal value of the integral of f with f singular at c 0lim()d()dcbacfxxfxxδδδ−→++⎛⎞⎜⎟+⎜⎟⎝⎠∫∫ where acb<< 2-11.21 (—) ()dfxx∞−∞−∫ Cauchy principal value of the integral of f ()limdaaafxx→∞−−∫
ISO 80000-2:2009(E) 18 © ISO 2009 – All rights reserved 12 Exponential and logarithmic functions Complex arguments can be used, in particular for the base e. Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-12.1 (11-8.2) e base of natural logarithm e := lim n→∞ 11nn⎛⎞+⎜⎟⎝⎠ = 2,718 281 8… 2-12.2 (11-8.1) ax a to the power of x,
exponential function to the base a of argument x See also 2-9.9. 2-12.3 (11-8.3) ex exp x e to the power of x, exponential function to the base e of argument x See 2-14.5. 2-12.4 (11-8.4) loga x logarithm to the base a of argument x log x is used when the base does not need to be specified. 2-12.5 (11-8.5) In x natural logarithm of x In x = loge x log x shall not be used in place of In x, Ig x, lb x, or loge x, log10 x, log2 x. 2-12.6 (11-8.6) Ig x decimal logarithm of x, common logarithm of x Ig x = log10 x See remark to 2-12.5. 2-12.7 (11-8.7) lb x binary logarithm of x lb x = log2 x See remark to 2-12.5.
ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 19 13 Circular and hyperbolic functions Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-13.1 (11-9.1) π ratio of the circumference of a circle to its diameter π = 3,141 592 6… 2-13.2 (11-9.2) sin x sine of x iieesin2ix-xx−=, sin x = x − x3/3! + x5/5! − … (sin x)n, (cos x)n, etc., are often written
sinn x, cosn x, etc. 2-13.3 (11-9.3) cos x cosine of x cos x = sin(x + π/2) 2-13.4 (11-9.4) tan x tangent of x tan x = sin x/cos x tg x should not be used. 2-13.5 (11-9.5) cot x cotangent of x cot x = 1/tan x ctg x should not be used. 2-13.6 (11-9.6) sec x secant of x sec x = 1/cos x 2-13.7 (11-9.7) csc x cosecant of x csc x = 1/sin x cosec x is also used. 2-13.8 (11-9.8) arcsin x arc sine of x y = arcsin x ⇔ x = sin y, −π/2 u y u π/2 The function arcsin is the inverse of the function sin with the restriction mentioned above. 2-13.9 (11-9.9) arccos x arc cosine of x y = arccos x ⇔ x = cos y, 0 u y u π The function arccos is the inverse of the function cos with the restriction mentioned above. 2-13.10 (11-9.10) arctan x arc tangent of x y = arctan x ⇔ x = tan y, −π/2 < y < π/2 The function arctan is the inverse of the function tan with the restriction mentioned above. arctg x should not be used. 2-13.11 (11-9.11) arccot x arc cotangent of x y = arccot x ⇔ x = cot y, 0 < y < π The function arccot is the inverse of the function cot with the restriction mentioned above. arcctg x should not be used. 2-13.12 (11-9.12) arcsec x arc secant of x y = arcsec x ⇔ x = sec y, 0 u y u π, y ≠ π/2
The function arcsec is the inverse of the function sec with the restriction mentioned above. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) 20 © ISO 2009 – All rights reserved Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-13.13 (11-9.13) arccsc x arc cosecant of x y = arccsc x ⇔ x = csc y, −π/2 u y u π/2, y ≠ 0 The function arccsc is the inverse of the function csc with the restriction mentioned above. arccosec x should be avoided. 2-13.14 (11-9.14) sinh x hyperbolic sine of x eesinh2x-xx−= sinh x = x + x3/3! + … sh x should be avoided. 2-13.15 (11-9.15) cosh x hyperbolic cosine of x cosh2 x = sinh2 x + 1 ch x should be avoided. 2-13.16 (11-9.16) tanh x hyperbolic tangent of x tanh x = sinh x/cosh x th x should be avoided. 2-13.17 (11-9.17) coth x hyperbolic cotangent of x coth x = 1/tanh x 2-13.18 (11-9.18) sech x hyperbolic secant of x sech x = 1/cosh x 2-13.19 (11-9.19) csch x hyperbolic cosecant of x csch x = 1/sinh x cosech x should be avoided. 2-13.20 (11-9.20) arsinh x inverse hyperbolic sine of x, area hyperbolic sine of x y = arsinh x ⇔ x = sinh y The function arsinh is the inverse of the function sinh. arsh x should be avoided. 2-13.21 (11-9.21) arcosh x inverse hyperbolic cosine of x, area hyperbolic cosine of x y = arcosh x ⇔ x = cosh y, y W 0 The function arcosh is the inverse of the function cosh with the restriction mentioned above. arch x should be avoided. 2-13.22 (11-9.22) artanh x inverse hyperbolic tangent of x, area hyperbolic tangent of x y = artanh x ⇔ x = tanh y The function artanh is the inverse of the function tanh. arth x should be avoided. 2-13.23 (11-9.23) arcoth x inverse hyperbolic cotangent of x,area hyperbolic cotangent of x y = arcoth x ⇔ x = coth y, y ≠ 0 The function arcoth is the inverse of the function coth with the restriction mentioned above. 2-13.24 (11-9.24) arsech x inverse hyperbolic secant of x, area hyperbolic secant of x y = arsech x ⇔ x = sech y, y W 0 The function arsech is the inverse of the function sech with the restriction mentioned above. 2-13.25 (11-9.25) arcsch x inverse hyperbolic cosecant of x, area hyperbolic cosecant of x y = arcsch x ⇔ x = csch y, y W 0 The function arcsch is the inverse of the function csch with the restriction mentioned above. arcosech x should be avoided. SIST EN ISO 80000-2:2013

ISO 80000-2:2009(E) © ISO 2009 – All rights reserved 21 14 Complex numbers Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-14.1 (11-10.1) i j imaginary unit i2 = j2 = −1 i is used in mathematics and in physics, j is used in electrotechnology. 2-14.2 (11-10.2) Re z real part of z z = x + i y where x and y are real numbers. x = Re z and y = Im z. 2-14.3 (11-10.3) Im z imaginary part of z See 2-14.2. 2-14.4 (11-10.4) |z| modulus of z |z| = 22xy+ where x = Re z and y = Im z. See also 2-9.16. 2-14.5 (11-10.5) arg z argument of z z = r eiϕ where r = |z| and ϕ = arg z, −π < ϕ u π i.e. Re z = r cos ϕ and Im z = r sin ϕ. 2-14.6 (11-10.6) z z* complex conjugate of z z is mainly used in mathematics, z* mainly in physics and engineering. 2-14.7 (11-10.7) sgn z signum z sgn z = z / |z| = exp(i arg z) (z ≠ 0)sgn z = 0 for z = 0 See also item 2-9.13.
ISO 80000-2:2009(E) 22 © ISO 2009 – All rights reserved 15 Matrices Matrices are usually written with boldface italic capital letters and their elements with thin italic lower case letters, but other typefaces may also be used. Item No. Sign, symbol, expression Meaning, verbal equivalent Remarks and examples 2-15.1 (11-11.1) A 1111nmmnaaaa⎛⎞⎜⎟⎜⎟⎝⎠"###" matrix A of type m by n A is the matrix with the elements aij = (A)ij. m is the number of rows and n is the number of columns. A = (aij) is also used. Square brackets are also used instead of parentheses. 2-15.2 (—) A + B sum of matrices A and B (A + B)ij = (A)ij + (B)ij The matrices A and B must have the same number of columns and rows. 2-15.3 (—) x A product of scalar x and matrix A (x A)ij = x (A)ij 2-15.4 (11-11.2) AB product of matrices A and B (AB)ik = j∑(A)ij(B) jk The number of c
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