EN ISO 80000-2:2019

Quantities and units - Part 2: Mathematics (ISO 80000-2:2019, Corrected version 2021-11)

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.

Größen und Einheiten - Teil 2: Mathematik (ISO 80000-2:2019, korrigierte Fassung 2021-11)

Dieses Dokument legt mathematische Zeichen fest, erläutert deren Bedeutung und gibt Sprechweise und Anwendungen an.
Dieses Dokument richtet sich hauptsächlich an Naturwissenschaft und Technik, ist jedoch auch in anderen Bereichen anwendbar, in denen Mathematik verwendet wird.

Grandeurs et unités - Partie 2: Mathématiques (ISO 80000-2:2019, Version corrigée 2021-11)

Le présent document spécifie les symboles mathématiques, explique leurs sens et donne leurs énoncés et leurs applications.
Le présent document est principalement destiné à être utilisé dans les sciences de la nature et dans la technique. Cependant, il s'applique également à d'autres domaines utilisant les mathématiques.

Veličine in enote - 2. del: Matematika (ISO 80000-2:2019, popravljena verzija 2021-11)

Ta dokument določa matematične simbole in razlaga njihov pomen ter podaja besedne ustreznice in načine uporabe. Ta dokument je namenjen zlasti uporabi v naravoslovnih vedah in tehniki, vendar se uporablja tudi na drugih področjih, kjer se uporablja matematika.

General Information

• Status: Published
• Publication Date: 08-Oct-2019
• Withdrawal Date: 29-Apr-2020
• ICS: 01.060 - Quantities and units
• ICS: 01.075 - Character symbols
• Technical Committee: CEN/SS F02 - Units and symbols
• Drafting Committee: CEN/SS F02 - Units and symbols
• Current Stage: 6060 - Definitive text made available (DAV) - Publishing
• Start Date: 09-Oct-2019
• Completion Date: 09-Oct-2019
• Ref Project: SIST EN ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics (ISO 80000-2:2019, Corrected version 2021-11)

Relations

• Replaces: EN ISO 80000-2:2013 - Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology (ISO 80000-2:2009)
• Effective Date: 08-Jun-2022

Standards Content (Sample)

SLOVENSKI STANDARD
01-december-2019
Nadomešča:
SIST EN ISO 80000-2:2013
Veličine in enote - 2. del: Matematika (ISO 80000-2:2019)
Quantities and units - Part 2: Mathematics (ISO 80000-2:2019)
Größen und Einheiten - Teil 2: Mathematik (ISO 80000-2:2019)
Grandeurs et unités - Partie 2: Mathématiques (ISO 80000-2:2019)
Ta slovenski standard je istoveten z: EN ISO 80000-2:2019
ICS:
01.060 Veličine in enote Quantities and units
07.020 Matematika Mathematics
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 80000-2
EUROPEAN STANDARD
NORME EUROPÉENNE
October 2019
EUROPÄISCHE NORM
ICS 01.060; 01.075 Supersedes EN ISO 80000-2:2013
English Version
Quantities and units - Part 2: Mathematics (ISO 80000-
2:2019)
Grandeurs et unités - Partie 2: Mathématiques (ISO Größen und Einheiten - Teil 2: Mathematik (ISO 80000-
80000-2:2019) 2:2019)
This European Standard was approved by CEN on 5 May 2019.

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, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2019 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 80000-2:2019 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
This document (EN ISO 80000-2:2019) has been prepared by Technical Committee ISO/TC 12
"Quantities and units" in collaboration with Technical Committee CEN/SS F02 “Units and symbols” the
secretariat of which is held by CCMC.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by April 2020, and conflicting national standards shall be
withdrawn at the latest by April 2020.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 80000-2:2013.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the
United Kingdom.
Endorsement notice
The text of ISO 80000-2:2019 has been approved by CEN as EN ISO 80000-2:2019 without any
modification.
INTERNATIONAL ISO
STANDARD 80000-2
Second edition
2019-08
Quantities and units —
Part 2:
Mathematics
Grandeurs et unités —
Partie 2: Mathématiques
Reference number
ISO 80000-2:2019(E)
©
ISO 2019
ISO 80000-2:2019(E)
© ISO 2019
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address
below or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Fax: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii © ISO 2019 – All rights reserved

ISO 80000-2:2019(E)
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
ISO 80000-2:2019(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.
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

ISO 80000-2:2019(E)
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 sy
...

SLOVENSKI STANDARD
01-december-2019
Nadomešča:
SIST EN ISO 80000-2:2013
SIST ISO 80000-2:2013
Veličine in enote - 2. del: Matematika (ISO 80000-2:2019, popravljena verzija 2021-
11)
Quantities and units - Part 2: Mathematics (ISO 80000-2:2019, Corrected version 2021-
11)
Größen und Einheiten - Teil 2: Mathematik (ISO 80000-2:2019)
Grandeurs et unités - Partie 2: Mathématiques (ISO 80000-2:2019)
Ta slovenski standard je istoveten z: EN ISO 80000-2:2019
ICS:
01.060 Veličine in enote Quantities and units
07.020 Matematika Mathematics
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 80000-2
EUROPEAN STANDARD
NORME EUROPÉENNE
October 2019
EUROPÄISCHE NORM
ICS 01.060; 01.075 Supersedes EN ISO 80000-2:2013
English Version
Quantities and units - Part 2: Mathematics (ISO 80000-
2:2019, Corrected version 2021-11)
Grandeurs et unités - Partie 2: Mathématiques (ISO Größen und Einheiten - Teil 2: Mathematik (ISO 80000-
80000-2:2019, Version corrigée 2021-11) 2:2019, korrigierte Fassung 2021-11)
This European Standard was approved by CEN on 5 May 2019.

This European Standard was corrected and reissued by the CEN-CENELEC Management Centre on 8 December 2021.

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, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,
Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and
United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATIO N

EUROPÄISCHES KOMITEE FÜR NORMUN G

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2019 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 80000-2:2019 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
This document (EN ISO 80000-2:2019) has been prepared by Technical Committee ISO/TC 12
"Quantities and units" in collaboration with Technical Committee CEN/SS F02 “Units and symbols” the
secretariat of which is held by CCMC.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by April 2020, and conflicting national standards shall be
withdrawn at the latest by April 2020.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 80000-2:2013.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of
North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the
United Kingdom.
Endorsement notice
The text of ISO 80000-2:2019, Corrected version 2021-11 has been approved by CEN as EN ISO 80000-
2:2019 without any modification.

INTERNATIONAL ISO
STANDARD 80000-2
Second edition
2019-08
Corrected version
2021-11
Quantities and units —
Part 2:
Mathematics
Grandeurs et unités —
Partie 2: Mathématiques
Reference number
ISO 80000-2:2019(E)
ISO 80000-2:2019(E)
© ISO 2019
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
ISO 80000-2:2019(E)
Contents Page
Foreword .iv
Introduction . vi
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 .14
14 Circular and hyperbolic functions .15
15 Complex numbers .17
16 Matrices .18
17 Coordinate systems.19
18 Scalars, vectors and tensors .21
19 Transforms .25
20 Special functions .26
Bibliography .31
Alphabetical index .32
iii
ISO 80000-2:2019(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.
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.
This corrected version of ISO 80000-2:2019 incorporates the following corrections:
— in 2-12.20, under "Remarks and examples", last line, f ″′ has been replaced with f ″ ;
— in 2-13.1, under "Remarks and examples", the value 2,718 81 28 . has been replaced with
2,718 281 828 .;
iv
ISO 80000-2:2019(E)
— in 2-20.20, under "Remarks and examples", the first formula has been corrected to read
m
d
m
m
L z =−1 L z ; i.e. addition of +m in the subscript of L;
() () ()
n nm+
m
dz
— in 2-20.21, under "Remarks and
...

SLOVENSKI SIST EN ISO 80000-2

STANDARD
december 2019
Veličine in enote – 2. del: Matematika
(ISO 80000-2:2019, popravljena verzija 2021-11)

Quantities and units – Part 2: Mathematics
(ISO 80000-2:2019, corrected version 2021-11)

Größen und Einheiten – Teil 2: Mathematik
(ISO 80000-2:2019, korrigierte Fassung 2021-11)

Grandeurs et unités – Partie 2: Mathématiques
(ISO 80000-2:2019, version corrigée 2021-11)

Referenčna oznaka
ICS 01.060; 07.020 SIST EN ISO 80000-2:2019 (sl)

Nadaljevanje na strani II ter od 1 do 47

© 2023-10. Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

SIST EN ISO 80000-2 : 2019
NACIONALNI UVOD
Standard SIST EN ISO 80000-2:2019 (sl), Veličine in enote – 2. del: Matematika (ISO 80000-2:2019,
popravljena verzija 2021-11), ima status slovenskega standarda in je istoveten evropskemu standardu
verzija 2021-11).
Ta standard nadomešča SIST EN ISO 80000-2:2013 in SIST ISO 80000-2:2013.

NACIONALNI PREDGOVOR
Evropski standard EN ISO 80000-2:2019 je pripravil tehnični odbor ISO/TC 12 "Veličine in enote" v
sodelovanju s tehničnim odborom CEN/SS F02 "Enote in simboli", katerega sekretariat vodi CCMC.

Slovenski standard SIST EN ISO 80000-2:2019 je prevod evropskega standarda EN ISO 80000-2:2019.
V primeru spora glede besedila slovenskega prevoda v tem standardu je odločilen izvirni evropski
standard v angleškem jeziku. Slovensko izdajo standarda je pripravil tehnični odbor SIST/TC TRS
Tehnično risanje, veličine, enote, simboli in grafični simboli.

Odočitev za izdajo tega standarda je 16. 10. 2019 sprejel SIST/TC TRS Tehnično risanje, veličine,
enote, simboli in grafični simboli.

ZVEZA Z NACIONALNIMI STANDARDI

S privzemom tega evropskega standarda, ki je istoveten mednarodnemu standardu, veljajo za omejeni
namen referenčnih standardov vsi standardi, navedeni v izvirniku, razen standardov, ki so že sprejeti v
nacionalno standardizacijo:
SIST EN ISO 80000-1:2023 (en) Veličine in enote – 1. del: Splošno (ISO 80000-1:2022)

PREDHODNA IZDAJA
– SIST ISO 80000-2:2013 (sl), Veličine in enote – 2. del: Matematični znaki in simboli za uporabo v
naravoslovnih vedah in tehniki

OPOMBE
– Povsod, kjer se v besedilu standarda uporablja izraz "evropski standard" ali "mednarodni standard",
v SIST EN ISO 80000-2:2019 to pomeni "slovenski standard".

– Nacionalni uvod in nacionalni predgovor nista sestavni del evropskega standarda.

II
EVROPSKI STANDARD
EN ISO 80000‐2
EUROPEAN STANDARD
NORME EUROPÉENNE
oktober 2019
EUROPÄISCHE NORM
ICS 01.060; 01.075 Nadomešča EN ISO 80000-2:2013

Slovenska izdaja
Veličine in enote – 2. del: Matematika
(ISO 80000‐2:2019, popravljena verzija 2021‐11)

Quantities and units – Part 2: Grandeurs et unités – Partie 2: Größen und Einheiten – Teil 2:
Mathematics (ISO 80000- Mathématiques (ISO 80000- Mathematik (ISO 80000-2:2019,
2:2019, corrected version 2:2019, version corrigée korrigierte Fassung 2021-11)
2021-11) 2021-11)
Ta evropski standard je CEN sprejel 5. maja 2019.

Upravni center CEN-CENELEC je ta evropski standard popravil in ponovno izdal 8. decembra
2021.
Člani CEN morajo izpolnjevati določila notranjih predpisov CEN/CENELEC, s katerimi je
predpisano, da mora biti ta evropski standard brez kakršnihkoli sprememb sprejet kot
nacionalni standard. Seznami najnovejših izdaj teh nacionalnih standardov in njihovi
bibliografski podatki so na zahtevo na voljo pri Upravnem centru CEN-CENELEC ali
kateremkoli članu CEN.
Ta evropski standard obstaja v treh uradnih izdajah (angleški, francoski in nemški). Izdaje v
drugih jezikih, ki jih člani CEN na lastno odgovornost prevedejo in izdajo ter prijavijo pri
Upravnem centru CEN-CENELEC, veljajo kot uradne izdaje.

Člani CEN so nacionalni organi za standarde Avstrije, Belgije, Bolgarije, Cipra, Češke republike,
Danske, Estonije, Finske, Francije, Grčije, Hrvaške, Irske, Islandije, Italije, Latvije, Litve,
Luksemburga, Madžarske, Malte, Nemčije, Nizozemske, Norveške, Poljske, Portugalske,
Republike Severne Makedonije, Romunije, Slovaške, Slovenije, Srbije, Španije, Švedske, Švice,
Turčije in Združenega kraljestva.

CEN
EVROPSKI ODBOR ZA STANDARDIZACIJO
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG

Upravni center CEN‐CENELEC: Rue de la Science 23, B‐1040 Bruselj

© 2019 CEN Vse pravice do izkoriščanja v kakršnikoli obliki in na Ref. oz. EN ISO 80000-2:2019 E
kakršenkoli način imajo nacionalni člani CEN.

SIST EN ISO 80000-2 : 2019
Evropski predgovor
Ta dokument (EN ISO 80000-2:2019) je pripravil tehnični odbor ISO/TC 12 "Veličine in enote" v
sodelovanju s tehničnim odborom CEN/SS F02 "Enote in simboli", katerega sekretariat vodi
CCMC.
Ta evropski standard mora dobiti status nacionalnega standarda bodisi z objavo istovetnega
besedila ali z razglasitvijo najpozneje do aprila 2020, nacionalne standarde, ki so v nasprotju s
tem standardom, pa je treba razveljaviti najpozneje do aprila 2020.

Opozoriti je treba na možnost, da je lahko nekaj elementov tega dokumenta predmet patentnih
pravic. CEN ne prevzema odgovornosti za identifikacijo katerihkoli ali vseh takih patentnih pravic.

Ta dokument nadomešča standard EN ISO 80000-2:2013.

V skladu z notranjimi pravili CEN/CENELEC morajo ta evropski standard obvezno uvesti
nacionalne organizacije za standarde naslednjih držav: Avstrije, Belgije, Bolgarije, Cipra, Češke
republike, Danske, Estonije, Finske, Francije, Grčije, Hrvaške, Irske, Islandije, Italije, Latvije, Litve,
Luksemburga, Madžarske, Malte, Nemčije, Nizozemske, Norveške, Poljske, Portugalske, Republike
Severne Makedonije, Romunije, Slovaške, Slovenije, Srbije, Španije, Švedske, Švice, Turčije in
Združenega kraljestva.
Razglasitvena objava
Besedilo mednarodnega standarda ISO 80000-2:2019, popravljena verzija 2021-11, je CEN
odobril kot evropski standard EN ISO 80000-2:2019 brez kakršnihkoli sprememb.

SIST EN ISO 80000-2 : 2019
VSEBINA Stran
Evropski predgovor . 2
Predgovor . 4
Uvod . 6
1 Področje uporabe . 7
2 Zveze s standardi . 7
3 Izrazi in definicije . 7
4 Spremenljivke, funkcije in operatorji . 7
5 Matematična logika . 8
6 Množice . 9
7 Standardne številske množice in intervali . 11
8 Razni simboli . 12
9 Elementarna geometrija . 14
10 Operacije . 14
11 Kombinatorika . 17
12 Funkcije . 19
13 Eksponentne in logaritemske funkcije . 23
14 Krožne in hiperbolične funkcije . 24
15 Kompleksna števila . 26
16 Matrike . 26
17 Koordinatni sistemi . 28
18 Skalarji, vektorji in tenzorji . 30
19 Transformi . 33
20 Posebne funkcije . 35
Literatura . 41
Abecedno kazalo . 42

SIST EN ISO 80000-2 : 2019
Predgovor
ISO (Mednarodna organizacija za standardizacijo) je svetovna zveza nacionalnih organov za
standarde (članov ISO). Mednarodne standarde praviloma 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).

Postopki, uporabljeni pri razvoju tega dokumenta, in postopki, predvideni za njegovo nadaljnje
vzdrževanje, so opisani v Direktivah ISO/IEC, 1. del. Posebna pozornost naj se nameni različnim
kriterijem odobritve, potrebnim za različne vrste dokumentov ISO. Ta dokument je bil pripravljen
v skladu z uredniškimi pravili Direktiv ISO/IEC, 2. del (glej www.iso.org/directives).

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.
Podrobnosti o morebitnih patentnih pravicah, identificiranih med pripravo tega dokumenta, bodo
navedene v uvodu in/ali na seznamu patentnih izjav, ki jih je prejela organizacija ISO (glej
www.iso.org/patents).
Morebitna trgovska imena, uporabljena v tem dokumentu, so informacije za uporabnike in ne
pomenijo podpore blagovni znamki.

Za razlago prostovoljne narave standardov, pomena specifičnih pojmov in izrazov ISO, povezanih
z ugotavljanjem skladnosti, ter informacij o tem, kako ISO spoštuje načela Mednarodne trgovinske
organizacije (WTO) v Tehničnih ovirah pri trgovanju (TBT), glej www.iso.org/iso/foreword.html.

Ta dokument je pripravil tehnični odbor ISO/TC 12, Veličine in enote, v sodelovanju s tehničnim
odborom IEC/TC 25, Veličine in enote.

Ta druga izdaja razveljavlja in nadomešča prvo izdajo (ISO 80000-2:2009), ki je tehnično
spremenjena.
V primerjavi s prejšnjo izdajo so glavne tehnične spremembe naslednje:

‒ točka 4 je popravljena z dodanim pojasnilom glede pisanja vrst pisav; popravljeno je pravilo
za delitev enačb čez dve ali več vrstic;

‒ točka 18 je popravljena in vključuje pojasnilo glede skalarjev, vektorjev in tenzorjev;

‒ v drugem stolpcu preglednic "Simbol, izraz" so dodani manjkajoči simboli in izrazi, v četrtem
stolpcu "Opombe in primeri" so po potrebi podana dodatna pojasnila;

‒ dodatek A je izbrisan.
OPOMBA: Čeprav so bili v to drugo izdajo ISO 80000-1 dodani manjkajočo simboli in izrazi, pa je dokument še
vedno nepopoln.
Na spletnih straneh ISO in IEC je seznam vseh delov skupine standardov ISO 80000 in IEC 80000.

Za morebitne povratne informacije ali vprašanja glede tega dokumenta naj se uporabniki obrnejo
na svoj nacionalni organ za standarde. Popoln seznam teh organov je mogoče najti na povezavi
www.iso.org/members.html.
SIST EN ISO 80000-2 : 2019
Ta popravljena verzija ISO 80000-2:2019 vsebuje naslednje popravke:

‒ v 2-12.20, "Opombe in primeri", zadnja vrstica, f je zamenjan s f ;

‒ v 2-13.1, "Opombe in primeri", vrednost 2,718 81 28 . je zamenjana z 2,718 281 828 .;

‒ v 2-20.20, "Opombe in primeri", je prva enačba popravljena in se sedaj glasi:
m
d
m
m
L1zz L ; t.j. dodan je +m v podpisu pri L ;
  
nnm
m
dz
‒ v 2-20.21, "Opombe in primeri", druga vrstica, zveza v oklepajih je popravljena in se sedaj
glasi:
(za n ∈ N , z ≤ 1); t.j. doda se |z| ≤ 1.

SIST EN ISO 80000-2 : 2019
Uvod
Ureditev preglednic
V vsaki preglednici simbolov in izrazov (razen preglednice 13) je nakazan (v tretjem stolpcu)
pomen izraza oziroma kako se ta lahko prebere pri vsaki točki (v prvem stolpcu ošt
...