Editing Positional notation (section)

=== Sexagesimal system ===
The [[sexagesimal]] or base-60 system was used for the integral and fractional portions of [[Babylonian numerals]] and other Mesopotamian systems, by [[Hellenistic]] astronomers using [[Greek numerals]] for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.

Modern time separates each position by a colon or a [[Prime (symbol)|prime symbol]]. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be {{nowrap|10°{{px2}}25′{{px2}}59″}} (10 [[degree (angle)|degree]]s 25 [[minute (angle)|minute]]s 59 [[second (angle)|second]]s). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second.{{citation needed|date=January 2019}} This contrasts with the numbers used by Hellenistic and [[Renaissance]] astronomers, who used [[third (angle)|third]]s, [[fourth (angle)|fourth]]s, etc. for finer increments. Where we might write {{nowrap|10°{{px2}}25′{{px2}}59.392″}}, they would have written {{nowrap|10°{{px2}}25′{{px2}}59′′{{px2}}23′′′{{px2}}31′′′′{{px2}}12′′′′′}} or {{nowrap|10°{{px2}}25<sup>{{smallcaps|i}}</sup>{{px2}}59<sup>{{smallcaps|ii}}</sup>{{px2}}23<sup>{{smallcaps|iii}}</sup>{{px2}}31<sup>{{smallcaps|iv}}</sup>{{px2}}12<sup>{{smallcaps|v}}</sup>}}.

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.

In the 1930s, [[Otto Neugebauer]] introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion.<ref>{{Citation | last1 = Neugebauer | first1 = Otto | author-link = Otto Neugebauer | last2 = Sachs | first2 = Abraham Joseph | author2-link = Abraham Sachs | last3 = Götze | first3 = Albrecht | title = Mathematical Cuneiform Texts | place = New Haven | publisher = American Oriental Society and the American Schools of Oriental Research | series = American Oriental Series | volume = 29 | year = 1945 | page = 2 | isbn = 9780940490291 | url = https://books.google.com/books?id=i-juAAAAMAAJ&q=%22sexagesimal+notation%22&pg=PA1 | access-date = 18 September 2019 | archive-url = https://web.archive.org/web/20161001222131/https://books.google.com/books?hl=en&lr=&id=i-juAAAAMAAJ&oi=fnd&pg=PA1&dq=%22sexagesimal+notation%22&ots=nkmW8KTnQ_&sig=U_K4iOoKy5Xf70UrbjoTyS3hN2A#v=onepage&q=%22sexagesimal%20notation%22&f=false | archive-date = 1 October 2016 | url-status = live }}</ref> For example, the mean [[synodic month]] used by both Babylonian and Hellenistic astronomers and still used in the [[Hebrew calendar]] is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.