|Numeral systems by culture|
|Positional systems by base|
|Non-standard positional numeral systems|
|List of numeral systems|
Different cultures use different symbols for the decimal mark. The choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article.
In mathematics the decimal mark is a type of radix point, a term that also applies to number systems with bases other than ten.
In the Middle Ages, before printing, a bar ( ¯ ) over the units digit was used to separate the integral part of a number from its fractional part, a tradition derived from the decimal system used in Indian mathematics. Its regular usage and classification can be attributed to the Persian mathematician Al-Khwarizmi, when Latin translation of his work on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal point, e.g. 99.
Later, a separator (ˌ) (a short, roughly vertical, ink stroke) between the units and tenths position became the norm among Arabic mathematicians. When this character was typeset, it was convenient to use the existing comma (,) or full stop (.) instead.
Gerbert of Aurillac marked triples of columns with an arc (called a "Pythagorean arc") when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century.
In France, the full stop was already in use in printing to make Roman numerals more readable,[how?] 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 for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits.
In the United States, the full stop or period (.) was used as the standard decimal mark. In the nations of the British Empire, although the full stop could be used in typewritten material and its use was not banned, the point or mid-dot (·), which can also be called an interpunct (often referred to as the decimal point) was preferred for the decimal mark in printing technologies that could accommodate it. This had the advantage of reducing confusion in the countries that used the full stop to separate groups of digits and it was generally clearer in handwriting (particularly when writing on a dotted baseline as on many forms). 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 mark. British aviation magazines thus switched to the US form in the late 20th century.
When South Africa adopted the metric system, it adopted the comma as its decimal mark, although a number of house styles, including leading newspapers like The Star and The Sunday Times continue to use the decimal point.
The three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua all use the comma as the official radix point, or decimal point. Interlingua has used the comma as its decimal mark since the publication of the Interlingua Grammar in 1951. Esperanto also uses the comma as its official decimal mark, while thousands are separated by non-breaking spaces: 12 345 678,9. 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 point while full stops are used to separate thousands, millions, etc. So the number 12,345,678.90123 for instance, would be written 12.345.678,90123 in Ido. The 1931 grammar of Volapük by Arie de Jong uses the comma as its decimal mark, 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 mark nearly stalled the development of the ALGOL computer programming language. ALGOL ended up allowing different decimal marks, but most computer languages and standard data formats (e.g. C, Java, Fortran, Cascading Style Sheets (CSS)) specify a dot.
The 22nd General Conference on Weights and Measures 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".
Digit grouping 
For ease of reading, numbers with many digits before or after the decimal mark may be divided into groups using a delimiter, with the counting of groups starting from the decimal mark in both directions. This delimiter is usually called a "thousands separator", because the digits are usually in groups of three, that is, thousands. The most general name for this delimiter is "digit group separator", because thousands are not always the relevant group. For example, in various countries (e.g., China, India, and Japan), there have been traditional conventions of grouping by 2 or 4 digits. These conventions are still observed in some contexts, although the 3-digit group convention is also well-known and often used. Making groups of three digits also emphasizes that there is a base-1000 of the numeral system that is being used, which is not always a given (for example, in computer science).
As with the decimal mark, there have been several common conventions for which character to use for the digit group separator. If the decimal mark is a point, the digit group separator is often a comma or a space. If the decimal mark is a comma, the digit group separator is often a point or a space. The problem with the point and the comma as either decimal mark or digit group separator is that, internationally, they have both often been used for both meanings, and their meaning is context-dependent (one must know which notational system is being used in order to interpret them). Therefore the space is recommended in the SI/ISO 31-0 standard, and the International Bureau of Weights and Measures states that "for numbers with many digits the digits may be divided into groups of three by a thin space, in order to facilitate reading. Neither dots nor commas are inserted in the spaces between groups of three". Other style-defining bodies are also moving toward this clearer notation. For example, the International Union of Pure and Applied Chemistry (IUPAC) suggest never using a comma or a point as thousands separator: "For numbers with many digits, the digits may be separated in groups of three, counting from the decimal sign toward the left and the right. The groups should be separated by a thin space (half space), and never by a comma or a point, or by any other means." The American Medical Association, whose AMA Manual of Style is widely followed in health professions literature, also endorses a space for the digit group separator. (Specifically, a thin space is endorsed for typesetting; a regular word space is sufficient wherever typographical nicety is impractical.) The United Kingdom Metrication Board also proposed this system for Metrication in the United Kingdom and, while not universally adopted, it is the standard in the UK construction industry.
Exceptions to digit grouping 
The International Bureau of Weights and Measures 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". Some manuals of style state that thousands separators should not be used in normal text for numbers from 1000 to 9999 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, the Publication Manual of the American Psychological Association stipulates a thousands separator for "most figures of 1,000 or more" except for page numbers, binary digits, temperatures, etc.
There are always common-sense exceptions to digit grouping, such as postal codes 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.
Similarly, in hexadecimal (base-16), full spaces are usually used to group digits into twos.
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 calculators, which use the decimal point. Most computer operating systems allow selection of the decimal mark and programs that have been carefully internationalized will follow this, but some programs ignore it and a few are even broken by it.
Hindu–Arabic numeral system 
Countries using Arabic numerals with decimal point 
Countries where a dot "." is used to mark the radix point include:
- Australia, Botswana, British West Indies, Brunei, Canada (except Province of Quebec), Dominican Republic, Guatemala, Hong Kong, India, Ireland, Israel, Japan, Kenya, Korea (both North and South), Lebanon, Luxembourg (uses both marks officially), Macau (in Chinese and English text), Malaysia, Malta, Mexico, Mongolia, Nepal, New Zealand, Nicaragua, Nigeria, Pakistan, Panama, Peru, People's Republic of China, Philippines, Singapore, Sri Lanka, Switzerland, Taiwan, Tanzania, Thailand, Uganda, United Kingdom, United States (including insular areas), Zimbabwe.
Countries using Arabic numerals with decimal comma 
Countries where a comma "," is used to mark the radix point include:
- Albania, Andorra, Argentina, Armenia, Austria, Azerbaijan, Belarus, Belgium, Bolivia, Bosnia and Herzegovina, Brazil, Bulgaria, Cameroon, Canada (only Province of Quebec), Chile, Colombia, Costa Rica, Croatia (comma used officially, but both forms are in use elsewhere) Cuba, Cyprus, Czech Republic, Denmark, Ecuador, Estonia, Faroes, Finland, France, Germany, Georgia, Greece, Greenland, Honduras, Hungary, Iceland, Indonesia, Italy, Kazakhstan, Kirgistan, Latvia, Lebanon, Lithuania, Luxembourg (uses both marks officially), Macau (in Portuguese text), Macedonia, Moldova, Mongolia, Morocco, Netherlands, Norway, Paraguay, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, South Africa (officially), Spain, Switzerland, Sweden, Tunisia, Turkey, Ukraine, Uruguay, Uzbekistan, Venezuela, Vietnam.
Other numeral systems 
In the Arab world, where Eastern Arabic numerals 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 (٫) in Unicode. An Arabic Thousands Separator (٬) also exists.
In Persian, the decimal mark is called Momayyez, which is written like a forward slash—there is a small difference between the "comma" character used in sentences and the Momayyez (٫) used to separate sequences of three digits. To separate sequences of three digits, a comma or blank space may be used; however this is not a standard.
In English Braille, the decimal point, ⠨, is distinct from both the comma, ⠂, and the period / full stop ⠲.
Examples of use 
The following examples show the decimal mark and the thousands separator; the detailed lists below are ordered chronologically, by when each country adopted the use:
|1 234 567,89||SI style (French version), Brazil, Germany, Netherlands, Denmark, Italy, Portugal, Romania, Sweden, Slovenia, Greece, Switzerland, Albania, Belgium, Bosnia, Estonia, France, Finland, Hungary, Poland, Czech Republic, Slovakia, Latin Europe, French Canada|
|1 234 567.89||SI style (English version), China|
|1,234,567.89||English Canada, Ireland, Israel, Japan, Korea, Malaysia, New Zealand, Singapore, Taiwan, Thailand, Philippines, United Kingdom, United States, China|
|1,234,567·89||English Canada, Ireland, Israel, Japan, Korea, Malaysia, New Zealand, Singapore, Taiwan, Thailand, Philippines, United Kingdom, United States (older, typically hand written)|
|1.234.567,89||Brazil, Germany, Netherlands, Denmark, Italy, Portugal, Romania, Spain, Sweden, Slovenia, Greece,|
|1˙234˙567,89||Germany, Italy, Romania, Greece|
- In Albania, Belgium, Bosnia, Estonia, France, Finland, Hungary, Poland, Czech Republic, Slovakia and much of Latin Europe as well as French Canada: 1 234 567,89 (In Spain, in handwriting it is also common to use an upper comma: 1.234.567'89)
- In Brazil, Germany, Netherlands, Denmark, Italy, Portugal, Romania, Sweden, Slovenia, Greece and much of Europe: 1 234 567,89 or 1.234.567,89. In handwriting, 1˙234˙567,89 is also seen, but never in Denmark, the Netherlands, Portugal, Sweden or Slovenia. In Italy a straight apostrophe is also used in handwriting: 1'234'567,89.
- In Switzerland: There are two cases. 1'234'567.89 is used for currency values. An apostrophe as thousands separator along with a "." as decimal symbol. For other values the SI style 1 234 567,89 is used with a "," as decimal symbol. When handwriting, a straight apostrophe is often used as the thousands separator for non-currency values: 1'234'567,89.
- In English Canada, Ireland, Israel, Japan, Korea (both), Malaysia, New Zealand, Singapore, Taiwan, Thailand, the Philippines, 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. Australia used this style up until the 1970s; now it uses the SI style.
- SI style: 1 234 567.89 or 1 234 567,89 (in their own publications the dot is used in the English version and the comma in the French version).
- In China, comma and space are used to mark digit groups because dot is used as decimal mark. There is no universal convention on digit grouping, so both thousands grouping and no digit grouping can be found. However, grouping can also be done every four digits: 123,4567.89, since names for large numbers in Chinese are based on powers of 10,000 (e.g. the next new word is for 108). Japan is similar.
- In India, due to a numeral system using lakhs (lacs) (1,23,456 equal to 123 456) and crores (1,23,45,678 equal to 12 345 678), comma is used at levels of thousand, lakh and crore, for example, 10 million (1 crore) would be written as 1,00,00,000.
|सहस्त्र (Sahastr) / हजार (Hazaar)||1,000|
See also 
- Reimer, L., and Reimer, W. Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Vol. 2. 1995. pp. 22-22. Parsippany, NJ: Pearson Education, Inc. as Dale Seymor Publications. ISBN 0-86651-823-1.
- Khwarizmi, Abu Jafar Muhammad ibn Musa al-, Oxford Islamic Studies Online
- Devlin, Keith (2011). The Man of Numbers: Fibonacci's Arithmetic Revolution. New York: Walker & Company. pp. 44–45. ISBN 9780802779083.
- Enciclopedia Universal Santillana, 1996 by SANTILLANA S.A., Barcelona, Spain. ISBN 84-294-5129-3. Comma, def.2: "coma: MAT. Signo utilizado en los números no enteros para separar la parte entera de la parte decimal o fraccionaria; p.ej., 2,123."
- Reimer, L., and Reimer, W. Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Vol. 1. 1990 p. 41. Parsippany, NJ: Pearson Education, Inc. as Dale Seymor Publications. ISBN 0-86651-509-7.
- Government Notice R. 1144, Government Gazette 4326, 5 July 1974
- Grammar of Interlingua: Parts of Speech – Numerals
- Gramat Volapüka. Cathair na Mart: Evertype. ISBN 978-1-904808-94-7
- Perlis, Alan, The American Side of the Development of ALGOL, ACM SIGPLAN Notices, August 1978.
- , Resolution 10.
- Iverson, Cheryl, et al. (eds) (2007). AMA Manual of Style (10th ed.). Oxford, Oxfordshire: Oxford University Press. p. 793. ISBN 978-0-19-517633-9.
- "Decimals Score a Point on International Standards". 2006-11-22. Retrieved 2008-11-27.
- International Bureau of Weights and Measures, Rules and style conventions for expressing values of quantities, Formatting numbers, and the decimal marker.
- "Guidelines for drafting IUPAC technical reports and recommendations". 2007. Retrieved 2008-11-27.
- GCIS (2011). "'Editorial Style Guide', Government Communications & Information System, Pretoria, page-# 28". Retrieved 2013-02-13.
- Pournader, Roozbeh (2000-10-15). "Persian decimal separator". Unicode Mail List Archive. Unicode Consortium. Retrieved 2008-06-21.
- "The Decimal Numeral". Academic Grammar of New Persian. Archived from the original on 2006-06-20. Retrieved 2006-06-19.
- Descriptive Grammar of New Persian (archived)
- Steinle, Vicki. "Teaching and Learning about Decimals". Retrieved 20 April 2012.