Editing Decimal (section)

{{Short description|Number in base-10 numeral system}}
{{Other uses}}
[[File:Decimal digit.png|thumb|upright=1.2|Place value of number in decimal system]]
The '''decimal''' [[numeral system]] (also called the '''base-ten''' [[positional numeral system]] and '''denary''' {{IPAc-en|ˈ|d|iː|n|ər|i}}<ref>{{OED|denary}}</ref> or '''decanary''') is the standard system for denoting [[integer]] and non-integer [[number]]s. It is the extension to non-integer numbers (''decimal fractions'') of the [[Hindu–Arabic numeral system]]. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''.<ref>{{Cite book |last1=Yong |first1=Lam Lay |url=http://dx.doi.org/10.1142/5425 |title=Fleeting Footsteps |last2=Se |first2=Ang Tian |date=April 2004 |publisher=[[World Scientific]] |isbn=978-981-238-696-0 |at=268 |doi=10.1142/5425 |access-date=March 17, 2022 |archive-date=April 1, 2023 |archive-url=https://web.archive.org/web/20230401132256/https://www.worldscientific.com/worldscibooks/10.1142/5425 |url-status=live }}</ref>

A '''decimal numeral''' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a [[decimal separator]] (usually "." or "," as in {{math|25.9703}} or {{math|3,1415}}).<ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |date=March 10, 2022 |title=Decimal Point |url=https://mathworld.wolfram.com/DecimalPoint.html |url-status=live |access-date=March 17, 2022 |website=Wolfram MathWorld |language=en |archive-date=March 21, 2022 |archive-url=https://web.archive.org/web/20220321195047/https://mathworld.wolfram.com/DecimalPoint.html }}</ref>
''Decimal'' may also refer specifically to the digits after the decimal separator, such as in "{{math|3.14}} is the approximation of {{pi}} to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the [[#Decimal fractions|'''decimal fractions''']]. That is, [[fraction (mathematics)|fractions]] of the form {{math|''a''/10<sup>''n''</sup>}}, where {{math|''a''}} is an integer, and {{math|''n''}} is a [[non-negative integer]]. Decimal fractions also result from the addition of an integer and a ''[[fractional part]]''; the resulting sum sometimes is called a ''fractional number''.

Decimals are commonly used to [[approximation (mathematics)|approximate]] real numbers. By increasing the number of digits after the decimal separator, one can make the [[approximation error]]s as small as one wants, when one has a method for computing the new digits.

{{anchor|terminating decimal}}
Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. 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.