Editing Sexagesimal (section)

=== Irrational numbers ===
[[Image:Ybc7289-bw.jpg|right|thumb|200px|Babylonian tablet [[YBC 7289]] showing the sexagesimal number {{nowrap|1;24,51,10}} approximating&nbsp;[[square root of 2|{{sqrt|2}}]]]]
The representations of [[irrational number]]s in any positional number system (including decimal and sexagesimal) neither terminate nor [[Repeating decimal|repeat]].

The [[square root of 2]], the length of the [[diagonal]] of a [[unit square]], was approximated by the Babylonians of the Old Babylonian Period ({{nowrap|1900 BC – 1650 BC}}) as
:<math>1;24,51,10=1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}=\frac{30547}{21600}\approx 1.41421296\ldots</math><ref>{{citation
 | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
 | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
 | doi = 10.1006/hmat.1998.2209
 | issue = 4
 | journal = [[Historia Mathematica]]
 | mr = 1662496
 | pages = 366–378
 | title = Square root approximations in old Babylonian mathematics: YBC 7289 in context
 | volume = 25
 | year = 1998| doi-access = free
 }}</ref>
Because {{sqrt|2}}&nbsp;≈&nbsp;{{val|1.41421356}}... is an [[irrational number]], it cannot be expressed exactly in sexagesimal (or indeed any integer-base system), but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44... ({{OEIS2C|A070197}})

The value of [[pi|{{pi}}]] as used by the [[Ancient Greece|Greek]] mathematician and scientist [[Ptolemy]] was 3;8,30 = {{nowrap|3 + {{sfrac|8|60}} + {{sfrac|30|60<sup>2</sup>}}}} = {{sfrac|377|120}} ≈ {{val|3.141666}}....<ref>{{Citation | editor-last = Toomer | editor-first = G. J. | editor-link = Gerald J. Toomer | year = 1984 | title = Ptolemy's Almagest | publisher = Springer Verlag | place = New York | page = 302 | isbn = 0-387-91220-7 }}</ref> [[Jamshīd al-Kāshī]], a 15th-century [[Persia]]n mathematician, calculated 2{{pi}} as a sexagesimal expression to its correct value when rounded to nine subdigits (thus to {{sfrac|1|60<sup>9</sup>}}); his value for 2{{pi}} was 6;16,59,28,1,34,51,46,14,50.<ref>{{citation|contribution=Al-Kashi|first=Adolf P.|last=Youschkevitch|editor-first=Boris A.|editor-last=Rosenfeld|page=256|title=Dictionary of Scientific Biography|title-link=Dictionary of Scientific Biography}}.</ref><ref>{{harvtxt|Aaboe|1964}}, p. 125</ref> Like {{sqrt|2}} above, 2{{pi}} is an irrational number and cannot be expressed exactly in sexagesimal. Its sexagesimal expansion begins 6;16,59,28,1,34,51,46,14,49,55,12,35... ({{OEIS2C|A091649}})
<!-- == Examples ==
The length of the [[tropical year]] in [[Babylonian astronomy]] (see [[Hipparchus]]), {{val|365.24579}}... days, can be expressed in sexagesimal as 6,5;14,44,51 ({{nowrap|6 × 60 + 5 + {{sfrac|14|60}} + {{sfrac|44|60<sup>2</sup>}} + {{sfrac|51|60<sup>3</sup>}}}}) days. The average length of a year in the [[Gregorian calendar]] is exactly 6,5;14,33 in the same notation because the values 14 and 33 were the first two values for the tropical year from the [[Alfonsine tables]], which were in sexagesimal notation. -->