A decimal separator is a symbol used to separate the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
part from the
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
of a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
written in
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choice of symbol also affects the choice of symbol for the
thousands separator
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
used in digit grouping.
Any such symbol can be called a decimal mark, decimal marker, or decimal sign. Symbol-specific names are also used; decimal point and decimal comma refer to an (either baseline or middle) dot and
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
respectively, when it is used as a decimal separator; these are the usual terms used in English, with the aforementioned generic terms reserved for abstract usage.
In many contexts, when a number is spoken, the function of the separator is assumed by the spoken name of the symbol: ''comma'' or ''point'' in most cases. In some specialized contexts, the word ''decimal'' is instead used for this purpose (such as in
International Civil Aviation Organization
The International Civil Aviation Organization (ICAO, ) is a specialized agency of the United Nations that coordinates the principles and techniques of international air navigation, and fosters the planning and development of international sc ...
-regulated
air traffic control
Air traffic control (ATC) is a service provided by ground-based air traffic controllers who direct aircraft on the ground and through a given section of controlled airspace, and can provide advisory services to aircraft in non-controlled airs ...
communications). In mathematics, the decimal separator is a type of
radix point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
, a term that also applies to number systems with bases other than ten.
History
Hellenistic–Renaissance eras
In the
Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
, from the original Indian decimal writing, before printing, a
bar
Bar or BAR may refer to:
Food and drink
* Bar (establishment), selling alcoholic beverages
* Candy bar
* Chocolate bar
Science and technology
* Bar (river morphology), a deposit of sediment
* Bar (tropical cyclone), a layer of cloud
* Bar (u ...
( ¯ ) over the
units digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...
was used to separate the integral part of a number from its
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
, as in 995 (meaning 99.95 in decimal point format). A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal separator, as in 99. Later, a "separatrix" (i.e., a short, roughly vertical ink stroke) between the units and tenths position became the norm among Arab mathematicians (e.g. 99ˌ95), while an L-shaped or
vertical bar
The vertical bar, , is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others.
Usage
...
() served as the separatrix in England. When this character was
typeset
Typesetting is the composition of text by means of arranging physical ''type'' (or ''sort'') in mechanical systems or ''glyphs'' in digital systems representing '' characters'' (letters and other symbols).Dictionary.com Unabridged. Random H ...
, it was convenient to use the existing
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
(99,95) or
full stop
The full stop (Commonwealth English), period (North American English), or full point , is a punctuation mark. It is used for several purposes, most often to mark the end of a declarative sentence (as distinguished from a question or exclamation ...
(99.95) instead.
Positional
decimal fraction
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic num ...
s appear for the first time in a book by the Arab mathematician
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use ...
written in the 10th century. The practice is ultimately derived from the decimal
Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
used in
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
, and popularized by the
Persian
Persian may refer to:
* People and things from Iran, historically called ''Persia'' in the English language
** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples
** Persian language, an Iranian language of the ...
mathematician
Al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
, when
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
translation of
his work
His or HIS may refer to:
Computing
* Hightech Information System, a Hong Kong graphics card company
* Honeywell Information Systems
* Hybrid intelligent system
* Microsoft Host Integration Server
Education
* Hangzhou International School, in ...
on the
Indian numerals
Indian or Indians may refer to:
Peoples South Asia
* Indian people, people of Indian nationality, or people who have an Indian ancestor
** Non-resident Indian, a citizen of India who has temporarily emigrated to another country
* South Asia ...
introduced the
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
positional number system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
to the Western world. His ''
Compendious Book on Calculation by Completion and Balancing
''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematics, Arabic ...
'' presented the first systematic solution of
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
and
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s in Arabic.
Gerbert of Aurillac
Pope Sylvester II ( – 12 May 1003), originally known as Gerbert of Aurillac, was a French-born scholar and teacher who served as the bishop of Rome and ruled the Papal States from 999 to his death. He endorsed and promoted study of Arab and Gre ...
marked triples of columns with an arc (called a "Pythagorean arc"), when using his Hindu–Arabic numeral-based abacus in the 10th century.
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
followed this convention when writing numbers, such as in his influential work ''
Liber Abaci
''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.
''Liber Abaci'' was among the first Western books to describe ...
'' in the 13th century. Tables of
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s prepared by
John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioann ...
in 1614 and 1619 used the period (full stop) as the decimal separator, which was then adopted by
Henry Briggs Henry Briggs may refer to:
*Henry Briggs (mathematician) (1561–1630), English mathematician
*Henry Perronet Briggs (1793–1844), English painter
*Henry George Briggs (1824–1872), English merchant, traveller, and orientalist
*Henry Shaw Briggs ...
in his influential 17th century work.
In
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
, the full stop was already in use in printing to make
Roman numerals
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...
more readable, so the comma was chosen.
Many other countries, such as Italy, also chose to use the comma to mark the decimal units position. It has been made standard by the
ISO
ISO is the most common abbreviation for the International Organization for Standardization.
ISO or Iso may also refer to: Business and finance
* Iso (supermarket), a chain of Danish supermarkets incorporated into the SuperBest chain in 2007
* Iso ...
for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits. In some countries, a raised dot or dash (') may be used for grouping or decimal separator; this is particularly common in handwriting.
English-speaking countries
In the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
, the full stop or period (.) was used as the standard decimal separator.
In the nations of the
British Empire
The British Empire was composed of the dominions, colonies, protectorates, mandates, and other territories ruled or administered by the United Kingdom and its predecessor states. It began with the overseas possessions and trading posts esta ...
(and, later, the
Commonwealth of Nations
The Commonwealth of Nations, simply referred to as the Commonwealth, is a political association of 56 member states, the vast majority of which are former territories of the British Empire. The chief institutions of the organisation are the Co ...
), the full stop could be used in typewritten material and its use was not banned, although the
interpunct
An interpunct , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin script. (Word-separating spaces did no ...
(a.k.a. decimal point, point or mid dot) was preferred as a decimal separator, in printing technologies that could accommodate it, e.g. However, as the mid dot was already in common use in the mathematics world to indicate multiplication, the SI rejected its use as the decimal separator.
During the beginning of British
metrication
Metrication or metrification is the act or process of converting to the metric system of measurement. All over the world, countries have transitioned from local and traditional units of measurement to the metric system. This process began in F ...
in the late 1960s and with impending currency
decimalisation
Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10.
Most countries have decimalised their currencies, converting them from non-decimal ...
, there was some debate in the United Kingdom as to whether the decimal comma or decimal point should be preferred: the
British Standards Institution
The British Standards Institution (BSI) is the national standards body of the United Kingdom. BSI produces technical standards on a wide range of products and services and also supplies certification and standards-related services to busines ...
and some sectors of industry advocated the comma and the
Decimal Currency Board
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
advocated for the point. In the event, the point was chosen by the
Ministry of Technology
The Ministry of Technology was a department of the government of the United Kingdom, sometimes abbreviated as "MinTech". The Ministry of Technology was established by the incoming government of Harold Wilson in October 1964 as part of Wilson's am ...
in 1968.
When
South Africa
South Africa, officially the Republic of South Africa (RSA), is the southernmost country in Africa. It is bounded to the south by of coastline that stretch along the South Atlantic and Indian Oceans; to the north by the neighbouring countri ...
adopted the metric system, it adopted the comma as its decimal separator, although a number of house styles, including some English-language newspapers such as ''
The Sunday Times
''The Sunday Times'' is a British newspaper whose circulation makes it the largest in Britain's quality press market category. It was founded in 1821 as ''The New Observer''. It is published by Times Newspapers Ltd, a subsidiary of News UK, whi ...
'', continue to use the full stop.
Previously, signs along
California
California is a U.S. state, state in the Western United States, located along the West Coast of the United States, Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the List of states and territori ...
roads expressed distances in decimal numbers with the decimal part in superscript, as in 37, meaning 3.7 . Though California has since transitioned to
mixed number
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s with
common fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s, the older style remains on postmile markers and bridge inventory markers.
Ido
Ido () is a constructed language derived from Reformed Esperanto, and similarly designed with the goal of being a universal second language for people of diverse backgrounds. To function as an effective ''international auxiliary language'', I ...
,
Esperanto
Esperanto ( or ) is the world's most widely spoken constructed international auxiliary language. Created by the Warsaw-based ophthalmologist L. L. Zamenhof in 1887, it was intended to be a universal second language for international communi ...
, and
Interlingua
Interlingua (; ISO 639 language codes ia, ina) is an international auxiliary language (IAL) developed between 1937 and 1951 by the American International Auxiliary Language Association (IALA). It ranks among the most widely used IALs and is t ...
, all use the comma as the decimal separator.
Interlingua has used the comma as its decimal separator since the publication of the Interlingua Grammar in 1951.
Esperanto also uses the comma as its official decimal separator, while thousands are separated by non-breaking spaces; e.g. .
Ido's ''Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido'' (Complete Detailed Grammar of the International Language Ido) officially states that commas are used for the decimal separator while full stops are used to separate thousands, millions, etc. So the number 12,345,678.90123 (in American notation) for instance, would be written ''12.345.678,90123'' in Ido.
The 1931 grammar of
Volapük
Volapük (; , "Language of the World", or lit. "World Speak") is a constructed language created between 1879 and 1880 by Johann Martin Schleyer, a Catholic priest in Baden, Germany, who believed that God had told him in a dream to create an i ...
uses the comma as its decimal separator, and – somewhat unusually – uses the middle dot as the thousands separator (12·345·678,90123).
In 1958, disputes between European and American delegates over the correct representation of the decimal separator nearly stalled the development of the
ALGOL
ALGOL (; short for "Algorithmic Language") is a family of imperative computer programming languages originally developed in 1958. ALGOL heavily influenced many other languages and was the standard method for algorithm description used by the ...
computer programming language. ALGOL ended up allowing different decimal separators, but most computer languages and standard data formats (e.g., C,
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, a radix point or radix character is a symbol used in the display of numbers to separate the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
part of the value from its
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
. In
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
and many other languages (including many that are written right-to-left), the integer part is at the left of the radix point, and the fraction part at the right of it.
A radix point is most often used in
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
(base 10) notation, when it is more commonly called the
decimal point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
(the prefix
deci-
''Deci'' (symbol d) is a decimal unit prefix in the metric system denoting a factor of one tenth. Proposed in 1793, and adopted in 1795, the prefix comes from the Latin , meaning "tenth". Since 1960, the prefix is part of the International System ...
English-speaking countries
The following is a list of English-speaking population by country, including information on both native speakers and second-language speakers.
List
* The European Union is a supranational union composed of 27 member states. The total Engl ...
, the decimal point is usually a small dot (.) placed either on the baseline, or halfway between the baseline and the top of the digits (·)
In many other countries, the radix point is a comma (,) placed on the baseline.
These conventions are generally used both in machine displays (
printing
Printing is a process for mass reproducing text and images using a master form or template. The earliest non-paper products involving printing include cylinder seals and objects such as the Cyrus Cylinder and the Cylinders of Nabonidus. The ea ...
,
computer monitor
A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a visual display, support electronics, power supply, housing, electrical connectors, and external user controls.
The di ...
s) and in
handwriting
Handwriting is the writing done with a writing instrument, such as a pen or pencil, in the hand. Handwriting includes both printing and cursive styles and is separate from formal calligraphy or typeface
A typeface (or font family) is ...
. It is important to know which notation is being used when working in different software programs. The respective
ISO standard
The International Organization for Standardization (ISO ) is an international standard development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Ar ...
defines both the comma and the small dot as decimal markers, but does not explicitly define universal radix marks for bases other than 10.
Fractional numbers are rarely displayed in other number bases, but, when they are, a radix character may be used for the same purpose. When used with the
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
( base 2) representation, it may be called "binary point".
Current standards
The 22nd
General Conference on Weights and Measures
The General Conference on Weights and Measures (GCWM; french: Conférence générale des poids et mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established i ...
declared in 2003 that "the symbol for the decimal marker shall be either the point on the line or the comma on the line". It further reaffirmed that
: "numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups"
( for example). This use has therefore been recommended by technical organizations, such as the United States'
National Institute of Standards and Technology
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
.
Past versions of
ISO 8601
ISO 8601 is an international standard covering the worldwide exchange and communication of date and time-related data. It is maintained by the Geneva-based International Organization for Standardization (ISO) and was first published in 1988, wi ...
, but not the 2019 revision, also stipulated normative notation based on SI conventions, adding that the comma is preferred over the full stop.
ISO 80000-1 stipulates that "The decimal sign is either a comma or a point on the line." The standard does not stipulate any preference, observing that usage will depend on customary usage in the language concerned, but adds a note that as per ISO/IEC directives, all ISO standards should use the comma as the decimal marker.
Digit grouping
For ease of reading, numbers with many digits may be divided into groups using a delimiter, such as comma "," or dot ".", half-space (or
thin space
In typography, a thin space is a space character whose width is usually or of an em. It is used to add a narrow space, such as between nested quotation marks or to separate glyphs that interfere with one another. It is not as narrow as the hai ...
) " ",
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
" ", underbar "_" (as in maritime "21_450") or apostrophe «'». In some countries, these "digit group separators" are only employed to the left of the decimal separator; in others, they are also used to separate numbers with a long
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
. An important reason for grouping is that it allows rapid judgement of the number of digits, via telling at a glance (" subitizing") rather than counting (contrast, for example, with 100000000 for one hundred million).
The use of spaces as separators, not dots or commas (for example: and for "twenty thousand" and "one million"), has been official policy of the
International Bureau of Weights and Measures
The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organisation, through which its 59 member-states act together on measurement standards in four areas: chemistry, ...
since 1948 (and reaffirmed in 2003) stating
: "neither dots nor commas are ever inserted in the spaces between groups",
as well as by the
International Union of Pure and Applied Chemistry
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
(IUPAC), the
American Medical Association
The American Medical Association (AMA) is a professional association and lobbying group of physicians and medical students. Founded in 1847, it is headquartered in Chicago, Illinois. Membership was approximately 240,000 in 2016.
The AMA's state ...
's widely followed ''
AMA Manual of Style
''AMA Manual of Style: A Guide for Authors and Editors'' is the style guide of the American Medical Association. It is written by the editors of ''JAMA'' (''Journal of the American Medical Association'') and the JAMA Network journals and is most ...
'', and the Metrication Board, among others.
The groups created by the delimiters tend to follow the use of the local language, which varies. In European languages, large numbers are read in groups of thousands, and the delimiter – which occurs every three digits when it is used – may be called a "thousands separator". In
East Asian culture
The East Asian cultural sphere, also known as the Sinosphere, the Sinic world, the Sinitic world, the Chinese cultural sphere, the Chinese character sphere encompasses multiple countries in East Asia and Southeast Asia that were historically ...
s, particularly
China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
,
Japan
Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ...
, and
Korea
Korea ( ko, 한국, or , ) is a peninsular region in East Asia. Since 1945, it has been divided at or near the 38th parallel, with North Korea (Democratic People's Republic of Korea) comprising its northern half and South Korea (Republic o ...
, large numbers are read in groups of
myriad
A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinospher ...
s (10,000s) but the delimiter commonly separates every three digits.
The Indian numbering system is somewhat more complex: It groups the rightmost three digits together (until the hundreds place) and thereafter groups by sets of two digits. For example, one trillion would thus be written as 10,00,00,00,00,000 or 10 kharab.
The convention for digit group separators historically varied among countries, but usually seeking to distinguish the delimiter from the decimal separator. Traditionally,
English-speaking countries
The following is a list of English-speaking population by country, including information on both native speakers and second-language speakers.
List
* The European Union is a supranational union composed of 27 member states. The total Engl ...
employed commas as the delimiter – 10,000 – and other European countries employed periods or spaces: 10.000 or . Because of the confusion that could result in international documents, in recent years the use of spaces as separators has been advocated by the superseded SI/ISO 31-0 standard, as well as by the
International Bureau of Weights and Measures
The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organisation, through which its 59 member-states act together on measurement standards in four areas: chemistry, ...
and the
International Union of Pure and Applied Chemistry
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
, which have also begun advocating the use of a "
thin space
In typography, a thin space is a space character whose width is usually or of an em. It is used to add a narrow space, such as between nested quotation marks or to separate glyphs that interfere with one another. It is not as narrow as the hai ...
" in "groups of three".
Within the United States, the
American Medical Association
The American Medical Association (AMA) is a professional association and lobbying group of physicians and medical students. Founded in 1847, it is headquartered in Chicago, Illinois. Membership was approximately 240,000 in 2016.
The AMA's state ...
's widely followed ''
AMA Manual of Style
''AMA Manual of Style: A Guide for Authors and Editors'' is the style guide of the American Medical Association. It is written by the editors of ''JAMA'' (''Journal of the American Medical Association'') and the JAMA Network journals and is most ...
'' also calls for a thin space. In some online
encoding
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
environments (for example,
ASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
-only) a thin space is not practical or available, in which case a regular word space or no delimiter are the alternatives.
Data versus mask
Digit group separators can occur either as part of the data or as a mask through which the data is displayed. This is an example of the
separation of presentation and content
Separation of content and presentation (or separation of content and style) is the separation of concerns design principle as applied to the authoring and presentation of content. Under this principle, visual and design aspects (presentation and s ...
, making it possible to display numbers with spaced digit grouping in a way that does not insert any
whitespace character
In computer programming, whitespace is any character or series of characters that represent horizontal or vertical space in typography. When rendered, a whitespace character does not correspond to a visible mark, but typically does occupy an area ...
s into the string of digits in the
content
Content or contents may refer to:
Media
* Content (media), information or experience provided to audience or end-users by publishers or media producers
** Content industry, an umbrella term that encompasses companies owning and providing mas ...
. In many computing contexts, it is preferred to omit digit group separators from the data and instead overlay them as a mask (an
input mask In computer programming, an input mask refers to a string expression, defined by a developer, that constrains user input. It can be said to be a template, or set format that entered data must conform to, ensuring data integrity by preventing transcr ...
or an output mask).
Common examples include
spreadsheet
A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
s and
database
In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases sp ...
s in which currency values are entered without such marks but are displayed with them inserted. (Similarly, phone numbers can have hyphens, spaces or parentheses as a mask rather than as data.) In
web content
Web content is the wikt:text, text, visual system, visual or audio signal, audio Content (media), content that is made available online and user encountered as part of the online usage and user experience, experience on websites. It may include t ...
, such digit grouping can be done with CSS style. It is useful because the number can be copied and pasted into
calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s (including a web browser's
omnibox
In a web browser, the address bar (also location bar or URL bar) is the element that shows the current URL. The user can type a URL into it to navigate to a chosen website. In most modern browsers, non-URLs are automatically sent to a search eng ...
) and parsed by the computer as-is (i.e., without the user manually purging the extraneous characters). For example,
Wikipedia
Wikipedia is a multilingual free online encyclopedia written and maintained by a community of volunteers, known as Wikipedians, through open collaboration and using a wiki-based editing system. Wikipedia is the largest and most-read refer ...
content can display numbers this way, as in the following examples:
:: metres is 1
astronomical unit
The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits t ...
:: is rounded to 20 decimal places
:: is rounded to 20 decimal places.
In some
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s, it is possible to group the digits in the program's
source code
In computing, source code, or simply code, is any collection of code, with or without comments, written using a human-readable programming language, usually as plain text. The source code of a program is specially designed to facilitate the wo ...
Ada
Ada may refer to:
Places
Africa
* Ada Foah, a town in Ghana
* Ada (Ghana parliament constituency)
* Ada, Osun, a town in Nigeria
Asia
* Ada, Urmia, a village in West Azerbaijan Province, Iran
* Ada, Karaman, a village in Karaman Province, ...
Haskell
Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...
:
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
OCaml
OCaml ( , formerly Objective Caml) is a general-purpose programming language, general-purpose, multi-paradigm programming language which extends the Caml dialect of ML (programming language), ML with object-oriented programming, object-oriented ...
:
Perl
Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...
:
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
:
PHP
PHP is a general-purpose scripting language geared toward web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by The PHP Group. ...
:
Ruby
A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sa ...
:
Rust
Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO(OH ...
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
,
Swift
Swift or SWIFT most commonly refers to:
* SWIFT, an international organization facilitating transactions between banks
** SWIFT code
* Swift (programming language)
* Swift (bird), a family of birds
It may also refer to:
Organizations
* SWIFT, ...
, and free-form Fortran 90 use the underscore (_) character for this purpose; as such, these languages allow seven hundred million to be entered as 700_000_000.
Fixed-form Fortran ignores whitespace (in all contexts), so 700 000 000 has always been accepted. Fortran 90 and its successors allow (ignored) underscores in numbers in free-form.
C++14
C++14 is a version of the ISO/IEC 14882 standard for the C++ programming language. It is intended to be a small extension over C++11, featuring mainly bug fixes and small improvements, and was replaced by C++17. Its approval was announced on Augus ...
Red
Red is the color at the long wavelength end of the visible spectrum of light, next to orange and opposite violet. It has a dominant wavelength of approximately 625–740 nanometres. It is a primary color in the RGB color model and a secondar ...
all allow the use of an
apostrophe
The apostrophe ( or ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes:
* The marking of the omission of one o ...
for digit grouping, so 700'000'000 is permissible.
Below is shown an example of Kotlin code using separators to increase readability:
val exampleNumber = 12_004_953 // Twelve million four thousand nine hundred fifty-three
Exceptions to digit grouping
The
International Bureau of Weights and Measures
The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organisation, through which its 59 member-states act together on measurement standards in four areas: chemistry, ...
states that "when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit". Likewise, some manuals of style state that thousands separators should not be used in normal text for numbers from to inclusive where no decimal fractional part is shown (in other words, for four-digit whole numbers), whereas others use thousands separators and others use both. For example,
APA style
APA style (also known as APA format) is a writing style and format for academic documents such as scholarly journal articles and books. It is commonly used for citing sources within the field of behavioral and social sciences, including sociol ...
stipulates a thousands separator for "most figures of or more" except for page numbers, binary digits, temperatures, etc.
There are always "common-sense" country-specific exceptions to digit grouping, such as year numbers,
postal code
A postal code (also known locally in various English-speaking countries throughout the world as a postcode, post code, PIN or ZIP Code) is a series of letters or digits or both, sometimes including spaces or punctuation, included in a postal a ...
s, and ID numbers of predefined nongrouped format, which style guides usually point out.
In non-base-10 numbering systems
In binary (base-2), a full space can be used between groups of four digits, corresponding to a
nibble
In computing, a nibble (occasionally nybble, nyble, or nybl to match the spelling of byte) is a four-bit aggregation, or half an octet. It is also known as half-byte or tetrade. In a networking or telecommunication context, the nibble is oft ...
, or equivalently to a
hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
digit. For integer numbers, dots are used as well to separate groups of four bits.
Alternatively, binary digits may be grouped by threes, corresponding to an
octal
The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
digit. Similarly, in hexadecimal (base-16), full spaces are usually used to group digits into twos, making each group correspond to a
byte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
. Additionally, groups of eight bytes are often separated by a hyphen.
Influence of calculators and computers
In countries with a decimal comma, the decimal point is also common as the "international" notation because of the influence of devices, such as
electronic calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s, which use the decimal point. Most computer
operating systems
An operating system (OS) is system software that manages computer hardware, software resources, and provides common services for computer programs.
Time-sharing operating systems schedule tasks for efficient use of the system and may also inc ...
allow selection of the decimal separator; programs that have been carefully internationalized will follow this, but some programs ignore it and a few may even fail to operate if the setting has been changed.
Computer interfaces may be set to the Unicode international "Common locale" using as defined at Details of the current (2020) definitions may be found at
Conventions worldwide
Hindu–Arabic numerals
Countries using decimal comma
Countries where a comma "," is used as decimal separator include:
Countries using decimal point
Countries where a dot "." is used as decimal separator include:
Other numeral systems
Unicode defines a ''decimal separator key symbol'' (⎖ in hex U+2396, decimal 9110) which looks similar to the
apostrophe
The apostrophe ( or ) is a punctuation mark, and sometimes a diacritical mark, in languages that use the Latin alphabet and some other alphabets. In English, the apostrophe is used for two basic purposes:
* The marking of the omission of one o ...
. This symbol is from
ISO/IEC 9995
ISO/IEC 9995 ''Information technology — Keyboard layouts for text and office systems'' is an ISO/IEC standard series defining layout principles for computer keyboards. It does not define specific layouts but provides the base for national and ind ...
and is intended for use on a keyboard to indicate a key that performs decimal separation.
In the
Arab world
The Arab world ( ar, اَلْعَالَمُ الْعَرَبِيُّ '), formally the Arab homeland ( '), also known as the Arab nation ( '), the Arabsphere, or the Arab states, refers to a vast group of countries, mainly located in Western A ...
, where
Eastern Arabic numerals
The Eastern Arabic numerals, also called Arabic-Hindu numerals or Indo–Arabic numerals, are the symbols used to represent numerical digits in conjunction with the Arabic alphabet in the countries of the Mashriq (the east of the Arab world) ...
are used for writing numbers, a different character is used to separate the integer and fractional parts of numbers. It is referred to as an Arabic decimal separator (U+066B, rendered: ) in
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
. An Arabic thousands separator (U+066C, rendered: ) also exists. Example: (9,999.99)
In
Persian
Persian may refer to:
* People and things from Iran, historically called ''Persia'' in the English language
** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples
** Persian language, an Iranian language of the ...
, the decimal separator is called ''momayyez''. The Unicode Consortium's investigation concluded that "computer programs should render U+066B as a shortened, lowered, and possibly more slanted
slash
Slash may refer to:
* Slash (punctuation), the "/" character
Arts and entertainment Fictional characters
* Slash (Marvel Comics)
* Slash (''Teenage Mutant Ninja Turtles'')
Music
* Harry Slash & The Slashtones, an American rock band
* Nash ...
(); this should be distinguishable from the slash at the first sight." To separate
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s of three digits, an Arabic thousands separator (rendered as: ), a Latin comma, or a
blank space
"Blank Space" is a song by American singer-songwriter Taylor Swift, taken from her fifth studio album, ''1989''. The song was released to US radio stations as the second single from ''1989'' on November 10, 2014, by Republic Records in partners ...
may be used; however this is not a standard. Example: (9,999.99)
In
English Braille
English Braille, also known as ''Grade 2 Braille'', is the braille alphabet used for English. It consists of around 250 letters ( phonograms), numerals, punctuation, formatting marks, contractions, and abbreviations (logograms). Some English Bra ...
, the decimal point, , is distinct from both the comma, , and the full stop, .
Examples of use
The following examples show the decimal separator and the thousands separator in various countries that use the Arabic numeral system.
*In Albania, Belgium (French), Estonia, Finland, France, Hungary, Poland, Slovakia and much of
Latin Europe
The Latins were originally an Italic tribe in ancient central Italy from Latium. As Roman power and colonization spread Latin culture during the Roman Republic.
Latins culturally "Romanized" or "Latinized" the rest of Italy, and the word Lati ...
as well as French Canada: (In Spain, in handwriting it is also common to use an upper comma: 1.234.567'89)
*In Belgium (Dutch), Brazil, Denmark, Germany, Greece, Indonesia, Italy, Netherlands, Portugal, Romania, Russia, Slovenia, Sweden and much of Europe: or 1.234.567,89. In
handwriting
Handwriting is the writing done with a writing instrument, such as a pen or pencil, in the hand. Handwriting includes both printing and cursive styles and is separate from formal calligraphy or typeface
A typeface (or font family) is ...
, 1˙234˙567,89 is also seen, but never in Belgium, Brazil, Denmark, Estonia, Germany, the Netherlands, Portugal, Romania, Russia, Slovenia or Sweden. In Italy, a straight apostrophe is also used in handwriting: 1'234'567,89. In the Netherlands and Dutch-speaking Belgium, the points thousands separator is used, and is preferred for currency amounts, but the space is recommended by some style guides, mostly in technical writing.
*In Estonia, currency numbers often use a dot "." as the decimal separator, and a space as a thousands separator. This is most visible on shopping receipts and in documents that also use other numbers with decimals, such as measurements. This practice is used to better distinguish between prices and other values with decimals. An older convention uses dots to separate thousands (with commas for decimals) — this older practice makes it easier to avoid word breaks with larger numbers.
*Historically, in Germany and Austria, thousands separators were occasionally denoted by alternating uses of comma and point, e.g. 1.234,567.890,12, entry "Bilanz", sums in last table for "eine Milliarde 234 Millionen ...", but this is never seen in modern days and requires explanation to a contemporary German reader.
*Switzerland: There are two cases: An apostrophe as a thousands separator along with a dot "." as the decimal separator are used for currency values (for example: 1'234'567.89). For other values, the SI-style is used with a comma "," as the decimal separator. The apostrophe is also the most common variety for non-currency values: 1'234'567,89 — though this usage is officially discouraged.
*In Ireland, Israel, Japan,
Korea
Korea ( ko, 한국, or , ) is a peninsular region in East Asia. Since 1945, it has been divided at or near the 38th parallel, with North Korea (Democratic People's Republic of Korea) comprising its northern half and South Korea (Republic o ...
(both), Malaysia, the Philippines, Singapore, Taiwan, Thailand, the United Kingdom, and the United States: 1,234,567.89 or 1,234,567·89; the latter is generally found only in older, and especially handwritten documents.
*English Canada: There are two cases: The preferred method for currency values is $10,000.00 —while for numeric values, it is ; however, commas are also sometimes used, although no longer taught in school or used in official publications.
* SI style: or (in their own publications, the dot "." is used in the English version, and the comma "," in the official French version).
*In China, comma and space are used to mark digit groups, because dot is used as decimal separator. There is no universal convention on digit grouping, so both thousands grouping and no digit grouping can be found. Japan and Taiwan are similar; although when grouping by myriads,
kanji
are the logographic Chinese characters taken from the Chinese family of scripts, Chinese script and used in the writing of Japanese language, Japanese. They were made a major part of the Japanese writing system during the time of Old Japanese ...
or
characters
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
are frequently used as separators: 1億2345万6789 / 1億2345萬6789. Commas are used when grouping by thousands.
*In
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
, due to a numeral system using ''
lakh
A lakh (; abbreviated L; sometimes written lac) is a unit in the Indian numbering system equal to one hundred thousand (100,000; scientific notation: 105). In the Indian 2,2,3 convention of digit grouping, it is written as 1,00,000. For ex ...
s'' (''lacs'') (1,23,456 equal to 123,456) and ''
crore
A crore (; abbreviated cr) denotes ten million (10,000,000 or 107 in scientific notation) and is equal to 100 lakh in the Indian numbering system. It is written as 1,00,00,000 with the local 2,2,3 style of digit group separators (one lakh is e ...
s'' (1,23,45,678 equal to ), a comma is used at levels of thousand, lakh, and crore. For example, 10 million (1 crore) would be written as 1,00,00,000. In
Pakistan
Pakistan ( ur, ), officially the Islamic Republic of Pakistan ( ur, , label=none), is a country in South Asia. It is the world's List of countries and dependencies by population, fifth-most populous country, with a population of almost 24 ...
, there is a greater tendency to use the standard western system, while using the Indian numbering system when conducting business in
Western Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
Eastern Arabic numerals
The Eastern Arabic numerals, also called Arabic-Hindu numerals or Indo–Arabic numerals, are the symbols used to represent numerical digits in conjunction with the Arabic alphabet in the countries of the Mashriq (the east of the Arab world) ...
(٠١٢٣٤٥٦٧٨٩):
*
*
Used with keyboards:
* (resembles an apostrophe)
See also
*
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system h ...
Decimal floating point
Decimal floating-point (DFP) arithmetic refers to both a representation and operations on decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically occur when convert ...
*
Decimal place
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expres ...
*
Decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
*
Decimal section numbering
A paragraph () is a self-contained unit of discourse in writing dealing with a particular point or idea. Though not required by the orthographic conventions of any language with a writing system, paragraphs are a conventional means of organizing e ...
*
Dot-decimal notation
Dot-decimal notation is a presentation format for numerical data. It consists of a string of decimal numbers, using the full stop (''dot'') as a separation character.
A common use of dot-decimal notation is in information technology where it is ...
*
International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
*
ISO 2145
International standard ISO 2145 defines a typographic convention for the "numbering of divisions and subdivisions in written documents". It applies to any kind of document, including manuscripts, books, journal articles, and standards.
Description ...
*
RKM code
The RKM code, also referred to as "letter and numeral code for resistance and capacitance values and tolerances", "letter and digit code for resistance and capacitance values and tolerances", or informally as "R notation" is a notation to specif ...
*
Version numbering
Software versioning is the process of assigning either unique ''version names'' or unique ''version numbers'' to unique states of computer software. Within a given version number category (e.g., major or minor), these numbers are generally assig ...