Decimal mark

A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form.

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.

History

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, e.g. 9995. This practice derived from the decimal system used in Indian mathematics^[1] and was popularized by the Persian mathematician Al-Khwarizmi,^[2] 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. 99⁹⁵. Later, a separator (a short, roughly vertical, ink stroke) between the units and tenths position became the norm among Arab mathematicians, e.g. 99ˌ95. When this character was typeset, it was convenient to use the existing comma (99,95) or full stop (99.95) 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.^[3]

In France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen.^{[citation needed]} Many other countries, such as Italy, also chose to use the comma to mark the decimal units position.^[4] 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 interpunct (also known as the decimal point, point, or mid-dot) was preferred for the decimal mark in printing technologies that could accommodate it, e.g. 99·95.^[5] 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,^[6] although a number of house styles, including leading newspapers like The Star and The Sunday Times continue to use the full stop.

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.^[7] 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 (in American notation) 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).^[8]

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.^[9] 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".^[10] This usage has therefore been recommended by technical organizations, such as the United States' National Institute of Standards and Technology.

Digit grouping

For ease of reading, numbers with many digits may be divided into groups using a delimiter.^[11] In some countries, these "digit group separators" are only employed to the left of the decimal mark; in others, they are also used to separate long decimal numbers as well.

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 3 digits when it is used) may be called a "thousands separator". In East Asian cultures, particularly China and Japan, large numbers are read in groups of myriads (10000s) and the delimiter accordingly separates every four digits. The Indian numbering system is somewhat more complex: it groups the first three digits in a similar manner to European languages but then groups every two digits thereafter: 1.5 million would accordingly be written 15,00,000 and read as "15 lakh".

The convention for digit group separators varies but usually seeks to distinguish the delimiter from the decimal mark. Typically, English-speaking countries employ commas as the delimiter and other European countries employ periods or spaces. Because of the confusion that can result in international documents, the withdrawn SI/ISO 31-0 standard advocates the use of spaces^[12] and the International Bureau of Weights and Measures and International Union of Pure and Applied Chemistry advocate the use of a "thin space" in "groups of three".^[13][14] Within the United States, the American Medical Association's widely-followed AMA Manual of Style also calls for a thin space.^[11] In some online encoding environments (for example, ASCII-only) a thin space is not practical/available, in which case a regular word space or no delimiter are the alternatives. The Metrication Board has proposed this system for the United Kingdom and, while it is not universally adopted, it is standard within the country's construction industry.

In some programming languages, it is possible to group the digits in the program's source code to make it easier to read. Ada, Perl and Ruby uses the underscore (_) character for this purpose. All these languages allow writing seven hundred thousand as 700_000_000.

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".^[13] Likewise, 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, APA style 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

Decimal marks:^{[citation needed]}
Full Stop "." – Blue
Comma "," – Green
Eastern Arabic numerals – Red
Data unavailable – Grey
This map is unsourced and many of the country may be wrong.

Countries using Arabic numerals with decimal point

Countries where a dot "." is used to mark the radix point comprise roughly 60% of the world's population.^{[citation needed]} They include:^{[citation needed]}

Australia, Botswana, British West Indies, Brunei, Canada (when using English), Dominican Republic, Guatemala, Honduras, 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, 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 comprise roughly 24% of the world's population.^{[citation needed]} They include:^{[citation needed]}

Albania, Andorra, Argentina, Armenia, Austria, Azerbaijan, Belarus, Belgium, Bolivia, Bosnia and Herzegovina, Brazil, Bulgaria, Cameroon, Canada (when using French), Chile, Colombia, Costa Rica, Croatia (comma used officially, but both forms are in use), Cuba, Cyprus, Czech Republic, Denmark, Ecuador, Estonia, Faroes, Finland, France, Germany, Georgia, Greece, Greenland, 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, Peru, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, South Africa (officially^[15]), Spain, Switzerland, Sweden, Tunisia, Turkey, Ukraine, Uruguay, Uzbekistan, Venezuela, Vietnam.

Other numeral systems

Unicode defines a decimal separator key symbol (⎖ in hex U+2396, decimal 9110) which looks similar to the apostrophe.

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.^[16][17][18]

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:

• In Albania, Belgium, Bosnia, Czech Republic, Estonia, Finland, France, Hungary, Poland, 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)^{[citation needed]}

• In Brazil, Denmark, Germany, Greece, Indonesia, Italy, Netherlands, Portugal, Romania, Slovenia, Sweden 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 Brazil, Denmark, the Netherlands, Portugal, Slovenia or Sweden. In Italy a straight apostrophe is also used in handwriting: 1'234'567,89. In the Netherlands, the points thousands separator is preferred in currency amounts, and the space for other large numbers^{[citation needed]}.

• 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. 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 (both), Malaysia, New Zealand, 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. Australia used this style up until the 1970s; now it uses the SI style.^[20]

• In English Canada: There are two cases. The preferred method for currency values is $4,000.00 while for numeric values it is 1 234 567.89; however, commas are also sometimes used although no longer taught in school or used in official publications.^[21]

• 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 South Asia, 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.

See also

• Algorism
• International System of Units

References