Grouping of digits

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Digit grouping refers to the grouping of digits in longer numbers . In most languages ​​today, larger numbers are spoken in a thousand base place system . Thousands separators make it easier to read large numbers, with the help of which the digits of a number are grouped into groups of three, starting with the lowest digit. Historically, the Arabic separators were used when using Arabic numerals : an apostrophe for thousands and a comma for decimal. Today, however, depending on the region, the type of writing (e.g. letterpress vs. handwriting), special spaces, periods and commas in different arrangements are also common.

Methods of digit grouping

With the thousand separator

The grouping of digits in a place value system of numbers in blocks of three, in the decimal system in powers of thousands (thousands separation) is widespread .

In addition to the digits to the left of the decimal separator , the digits to the right of the decimal separator, i.e. the decimal places, can also be grouped. Examples:

  • 1 987 654 321 , 123 456 7
  • 1 987 654 321 , 123 456 78

The division of numbers into groups of digits serves for better readability. It also simplifies in German and z. B. in English speaking, since here too numbers are divided into groups of three: One hundred and twenty-three million, four hundred and fifty-six thousand, seven hundred and eighty-nine for the number 123 456 789.

The division into groups of three also makes it easier to work with the prefixes for units of measurement , i.e. the prefixes micro, milli, kilo, mega, etc. Each group of three of a number corresponds to a prefix: 123 456 789 watts = 123.456 789 megawatts = 123 456.789 kilowatts.

The conversion into the exponential notation is also easy: 10,000 m = 10 km = 10e3 m

Exceptions to the division into groups of three are, for example, years, postcodes or phone numbers .

Relevant German standards are DIN 1333 , DIN 5008 , international ISO 31 and its successor ISO 80000 . ÖNORM A 1080 and the Austrian dictionary apply to Austria .

Digit grouping in numerical mathematics

Deviating from the blocks of thousands, long rows of digits are also grouped in blocks of five: = 2.71828 18284 59045 23536 02874 71352 7 ...

In addition to numbers with non-terminating decimal notation (e.g. irrational numbers ), the long series of numbers for the orbital elements or ephemeris ( VSOP , ELP , JPL ) are also given in astronomy .

Other number systems

In the Chinese number system and in East Asian cultures that have adopted it, ten thousand ( Chinese    /  , Pinyin wàn ) is the largest elementary number word and the base number for indicating larger numerical values. Large numbers are therefore read in blocks of four digits each. A structure in blocks of three does not help reading in such languages. The consequence of this is that the simultaneous translation of large numbers, e.g. B. between Chinese and English is not easy.

In the Indian number system, it is customary to group the three lowest digits first and then two digits each and to give the numbers their own elementary names. For example, the spelling 10'00'000 and the designation 10 lakh are used for one million .

Characters for grouping numbers

Different characters have been and are used as grouping characters in different countries and languages:

On the problem of points and commas for thousands and decimal separators

According to German and international standards, the narrow space should be used as a thousand separator (e.g. 123 456 789 ). In addition to conforming to standards, the space has the advantage that it cannot be confused with the decimal separator in international communication . A disadvantage is the unwieldiness, since (protected) narrow spaces - e.g. B. on keyboards with German key assignment - cannot be entered by a single (combined) keystroke. Even with handwritten writing, it is difficult to achieve an optimal space width.

Traditionally, in Germany , Austria and France the point was used as a thousand separator and the comma as a decimal separator. This is the default setting in various programs, e.g. B. LibreOffice or Microsoft Office . In contrast, in England , for example, the two characters are used in exactly the opposite way:

  • Germany, Austria, France: 123,456,789.123
  • England, USA: 123,456,789,123

A number such as 12.345 cannot easily be interpreted correctly. For this reason, standards provide for the use of a space as a thousand separator ( DIN 1333 , DIN 5008 and ISO 80000 ). A narrow space is recommended if this is technically available. An exception are amounts of money, which for security reasons can be separated with a space, which is at least the width of one of the digits, or a separator (such as a point).

Sometimes (e.g. in Switzerland) the digits are separated from each other with a prime to exclude the mentioned problems with commas and periods:

  • Example: 123'456'789.123 or 123'456'789.123

Also on the LCDs of most pocket and desk calculators, electronic measuring devices with digital displays , such as B. multimeters , frequency counters , etc., or on scales with digital displays, hyphens are used as thousands and dots as decimal separators. Here, point and hyphen each have a clear meaning.

On typography and computer typesetting problems

Typographical white space , which should be narrower than normal word spacing, is used for grouping digits . Often one is sixth square ( six-per-em ) is recommended. If the computer system used correctly converts Unicode, corresponding spaces from the Unicode block General Punctuation can be used.

If groups are grouped with spaces, a protected space is required to avoid a line break within the column of digits. The same applies to the narrow spaces; the narrow break-protected space is to be preferred here. Alternatively, breaking white space can be protected from breaking by assigning an appropriate formatting property. The width of the white space (with normal spaces) can also be changed by scaling , spacing , negative locking or reducing the word spacing.

Under TeX and LaTeX , large (5/18 em ) and small spaces (3/18 em) are available in the form a\;bor a\,b.

Representation in programming

The grouping of digits in programming languages ​​is problematic. In a few languages ​​(for example in Perl , Ruby , Java , C # and Verilog ) it is possible to group numbers with the underscore (_) if necessary , in Algol 68 the use of spaces () is possible in most other programming languages however, no equivalent can be found. In C ++ 14, the single apostrophe is introduced as a grouping character.

See also

Individual evidence

  1. See, for example, chapter 5.3.4 of the main topic The International System of Units (SI) . In: Physikalisch-Technische Bundesanstalt: PTB-Mitteilungen. 117 (2007), issue 2, ISSN  0030-834X , p. 175 (= PDF-p. 34); ptb.de (PDF; 1.0 MB) accessed on October 30, 2018.
  2. Notation of numbers. In: quality.de. January 28, 2015, accessed January 8, 2018.
  3. See e.g. B. Jens Weller: Looking at C ++ 14. In: meetingcpp.com. March 14, 2014, accessed August 26, 2014.