Decimal: Difference between revisions

Originally and in most uses, a decimal has only a finite number of digits after the decimal seperatorseparator. However, the decimal system has been extended to ''infinite decimals'' for representing any [[real number]], by using an [[sequence (mathematics)|infinite sequence]] of digits after the decimal separator (see [[decimal representation]]). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called '''terminating decimals'''. A ''[[repeating decimal]]'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., {{math|1=5.123144144144144... = 5.123{{overline|144}}}}).<ref>The [[Vinculum (symbol)|vinculum (overline)]] in 5.123<span style="text-decoration: overline;">144</span> indicates that the '144' sequence repeats indefinitely, i.e. {{val|5.123144144144144|s=...}}.</ref> An infinite decimal represents a [[rational number]], the [[quotient]] of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

Many [[numeral system]]s of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the [[Egyptian numerals]], then the [[Brahmi numerals]], [[Greek numerals]], [[Hebrew numerals]], [[Roman numerals]], and [[Chinese numerals]].<ref name=":0">{{Cite book |last=Lockhart |first=Paul |title=Arithmetic |date=2017 |publisher=The Belknap Press of Harvard University Press |isbn=978-0-674-97223-0 |location=Cambridge, Massachusetts London, England}}</ref> Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the [[Hindu–Arabic numeral system]] for representing [[integer]]s. This system has been extended to represent some non-integer numbers, called ''[[#Decimal fractions|decimal fractions]]'' or ''decimal numbers'', for forming the ''decimal numeral system''.<ref name=":0" />

For writing numbers, the decimal system uses ten [[decimal digit]]s, a [[decimal mark]], and, for [[negative number]]s, a [[minus sign]] "−". The decimal digits are [[0]], [[1]], [[2]], [[3]], [[4]], [[5]], [[6]], [[7]], [[8]], [[9]];<ref>In some countries, such as [[Arabic]]-speaking ones, other [[glyph]]s are used for the digits</ref> the [[decimal separator]] is the dot "{{math|.}}" in many countries (mostly English-speaking),<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Decimal|url=https://mathworld.wolfram.com/Decimal.html|access-date=2020-08-22|website=mathworld.wolfram.com|language=en|archive-date=2020-03-18|archive-url=https://web.archive.org/web/20200318204545/https://mathworld.wolfram.com/Decimal.html|url-status=live}}</ref> and a comma "{{math|,}}" in other countries.<ref name=":1" />

| volume = 86}}.</ref> Where this counting system is known, it is based on the "[[long hundred]]" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's [https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon Introduction to Old Norse] {{Webarchive|url=https://web.archive.org/web/20160415205641/https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon |date=2016-04-15 }} p.&nbsp;293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. [http://ads.ahds.ac.uk/catalogue/adsdata/arch-352-1/dissemination/pdf/vol_123/123_395_418.pdf Goodare]{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }} details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.<ref>{{Cite journal|last=Stevenson|first=W.H.|date=1890|title=The Long Hundred and its uses in England|journal=Archaeological Review|volume=December 1889|pages=313–22}}</ref><ref>{{Cite book|last=Poole, Reginald Lane|title=The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911|date=2006|publisher=Lawbook Exchange|isbn=1-58477-658-7|location=Clark, NJ|oclc=76960942}}</ref>