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INTERNATIONAL ISO
STANDARD 80000-2
Second edition
2019-08
Quantities and units —
Part 2:
Mathematics
Grandeurs et unités —
Partie 2: Mathématiques
Reference number
©
ISO 2019
© ISO 2019
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ii © ISO 2019 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Variables, functions and operators . 1
5 Mathematical logic . 2
6 Sets . 3
7 Standard number sets and intervals. 4
8 Miscellaneous symbols . 6
9 Elementary geometry . 7
10 Operations . 8
11 Combinatorics .10
12 Functions .11
13 Exponential and logarithmic functions .15
14 Circular and hyperbolic functions .16
15 Complex numbers.18
16 Matrices .18
17 Coordinate systems .19
18 Scalars, vectors and tensors .21
19 Transforms .25
20 Special functions .26
Bibliography .32
Alphabetical index .33
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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso
.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 12, Quantities and units, in collaboration
with Technical Committee IEC/TC 25, Quantities and units.
This second edition cancels and replaces the first edition (ISO 80000-2:2009), which has been
technically revised.
The main changes compared to the previous edition are as follows:
— Clause 4 revised to add clarification about writing of font types; revised rule for splitting equations
over two or more lines;
— Clause 18 revised to include clarification on scalars, vectors and tensors;
— missing symbols and expressions added in the second column "Symbol, expression" of the tables,
and additional clarifications given in the fourth column “Remarks and examples” when necessary;
— Annex A deleted.
NOTE Although missing symbols and expressions have been added in this second edition of ISO 80000-1, the
document remains non exhaustive.
A list of all parts in the ISO 80000 and IEC 80000 series can be found on the ISO and IEC websites.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/members .html.
iv © ISO 2019 – All rights reserved
Introduction
Arrangement of the tables
Each table of symbols and expressions (except Table 13) gives hints (in the third column) about the
meaning or how the expression may be read for each item (numbered in the first column) of the
symbol under consideration, usually in the context of a typical expression (second column). If more
than one 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 purpose of the entries is
identification of each concept and is not intended to be a complete mathematical definition. The fourth
column “Remarks and examples” gives further information and is not normative.
Table 13 has a different format. It gives the symbols of coordinates, as well as the position vectors and
their differentials, for coordinate systems in three-dimensional spaces.
INTERNATIONAL STANDARD ISO 80000-2:2019(E)
Quantities and units —
Part 2:
Mathematics
1 Scope
This document specifies mathematical symbols, explains their meanings, and gives verbal equivalents
and applications.
This document is intended mainly for use in the natural sciences and technology, but also applies to
other areas where mathematics is used.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements 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, Quantities and units — Part 1: General
3 Terms and definitions
Tables 1 to 16 give the symbols and expressions used in the different fields of mathematics.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
4 Variables, functions and operators
It is customary to use different sorts of letters for different sorts of entities, e.g. x, y, … for numbers
or elements of some given set, f, g for functions, etc. This makes formulas more readable and helps in
setting up an appropriate context.
Variables such as x, y, etc., and running numbers, such as i in x are printed in italic type. Parameters,
∑
i
i
such as a, b, etc., which may be considered as constant in a particular context, are printed in italic 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 upright type, e.g.
sin, exp, ln, Γ. Mathematical constants, the values of which never change, are printed in upright type,
e.g. e = 2,718 281 828 …; π = 3,141 592 …; i = −1. Well-defined operators are also printed in upright
type, e.g. div, δ in δx and each d in df/dx. Some transforms use special capital letters (see Clause 19,
Transforms).
Numbers expressed in the form of digits are always printed in upright type, e.g. 351 204; 1,32; 7/8.
Binary operators, for example +, −, /, shall be preceded and followed by thin spaces. This rule does not
apply in case of unary operators, as in −17,3.
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(ω t + 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 shall 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).
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, the following method shall be used:
— Place the line breaks immediately before one of the symbols =, +, −, ±, or , or, if necessary,
immediately before one of the symbols ×, ⋅, or /.
The symbol shall not be given twice around the line break; two minus signs could for example give rise
to sign errors. If possible, the line break should not be inside of an expression in parentheses.
5 Mathematical logic
Table 1 — Symbols and expressions in mathematical logic
Symbol,
Item No. Meaning, verbal equivalent Remarks and examples
expression
2-5.1 p ∧ q conjunction of p and q,
p and q
2-5.2 p ∨ q disjunction of p and q, This “or” is inclusive, i.e. p ∨ q is true, if
either p or q, or both are true.
p or q
2-5.3 ¬ p negation of p,
not p
2-5.4 p ⇒ q p implies q, q ⇐ p has the same meaning as p ⇒ q.
if p, then q ⇒ is the implication symbol.
→ is also used as implication symbol.
2-5.5 p ⇔ q p is equivalent to q (p ⇒ q) ∧ (q ⇒ p) has the same meaning as
p ⇔ q.
⇔ is the equivalence symbol.
↔ is also used as equivalence symbol.
2-5.6 ∀x ∈ A p(x) for e
...