View source for Sexagesimal

{{Short description|Base sixty numeral system}}
{{numeral systems}}
'''Sexagesimal''', also known as '''base 60''',<ref>Pronounced {{IPAc-en|s|ɛ|k|s|ə|ˈ|dʒ|ɛ|s|ᵻ|m|əl}} and {{IPAc-en|s|ɛ|k|ˈ|s|æ|dʒ|ᵻ|n|ər|i}}; see {{OED|sexagesimal|mode=cs2}}</ref> is a [[numeral system]] with [[60 (number)|sixty]] as its [[radix|base]]. It originated with the ancient [[Sumer]]ians in the 3rd millennium BC, was passed down to the ancient [[Babylonia]]ns, and is still used—in a modified form—for measuring [[time]], [[angle]]s, and [[geographic coordinate system|geographic coordinates]].

The number 60, a [[superior highly composite number]], has twelve [[divisor]]s, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are [[prime number]]s. With so many factors, many [[fraction]]s involving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. 60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the [[lowest common multiple]] of 1, 2, 3, 4, 5, and 6.

''In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. For example, the largest sexagesimal digit is "59".''

==Origin==

According to [[Otto Neugebauer]], the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. Their use has also always included (and continues to include) inconsistencies in where and how various bases are used to represent numbers even within a single text.<ref name=Neugebauer>{{citation|title=The Exact Sciences In Antiquity|last=Neugebauer|first=O.|journal=Acta Historica Scientiarum Naturalium et Medicinalium|volume=9|publisher=Dover|year=1969|isbn=0-486-22332-9|pages=17–19|pmid=14884919}}</ref>

[[File:Proto-cuneiform sexagesimal type Sa.svg|thumb|upright=1.2|Early [[Proto-cuneiform]] (4th millennium BCE) and [[cuneiform]] signs for the sexa&shy;gesimal system (60, 600, 3600, etc.)]]
The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data.<ref name=Neugebauer/> Another practical factor that helped expand the use of sexagesimal in the past, even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. In the late 3rd millennium BC, [[Sumer|Sumerian]]/[[Akkadian Empire|Akkadian]] units of weight included the ''kakkaru'' ([[Talent (measurement)|talent]], approximately 30 kg) divided into 60 ''manû'' ([[Mina (unit)|mina]]), which was further subdivided into 60 ''šiqlu'' ([[shekel]]); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10–compatible ratio of a ''shekel'' being one 50th of a ''mina''.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basic [[cuneiform]] symbols used to represent numeric quantities.<ref name=Neugebauer/> For example, the cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The details and even the magnitudes implied (since [[0#History|zero was not used consistently]]) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. In modern times there is the recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.<ref name=Neugebauer/>

==Usage==

===Babylonian mathematics===
{{main|Babylonian cuneiform numerals}}
The sexagesimal system as used in ancient [[Mesopotamia]] was not a pure base-60 system, in the sense that it did not use 60 distinct symbols for its [[numerical digit|digits]]. Instead, the [[cuneiform (script)|cuneiform]] digits used [[10 (number)|ten]] as a sub-base in the fashion of a [[sign-value notation]]: a sexagesimal digit was composed of a group of narrow, wedge-shaped marks representing units up to nine ([[File:Babylonian 1.svg|20px]], [[File:Babylonian 2.svg|20px]], [[File:Babylonian 3.svg|20px]], [[File:Babylonian 4.svg|20px]], ..., [[File:Babylonian 9.svg|20px]]) and a group of wide, wedge-shaped marks representing up to five tens ([[File:Babylonian 10.svg|20px]], [[File:Babylonian 20.svg|20px]], [[File:Babylonian 30.svg|20px]], [[File:Babylonian 40.svg|20px]], [[File:Babylonian 50.svg|20px]]). The value of the digit was the sum of the values of its component parts:

[[Image:Babylonian numerals.svg|frameless|upright=2.5]]

Numbers larger than 59 were indicated by multiple symbol blocks of this form in [[positional notation|place value notation]]. Because there was no symbol for [[0 (number)|zero]] it is not always immediately obvious how a number should be interpreted, and its true value must sometimes have been determined by its context. For example, the symbols for 1 and 60 are identical.<ref>{{citation|title=Topics in Contemporary Mathematics |first1=Ignacio |last1=Bello |first2=Jack R. |last2=Britton |first3=Anton |last3=Kaul |edition=9th |publisher=Cengage Learning |year=2009 |isbn=9780538737791 |page=182 |url=https://books.google.com/books?id=Aji2yzbopxgC&pg=PA182}}.</ref><ref name="lmnz"/> Later Babylonian texts used a placeholder ([[File:Babylonian digit 0.svg]]) to represent zero, but only in the medial positions, and not on the right-hand side of the number, as in numbers like {{val|13200}}.<ref name="lmnz">{{citation |url=http://blogs.scientificamerican.com/roots-of-unity/2014/08/31/look-ma-no-zero/ |first=Evelyn |last=Lamb |journal=[[Scientific American]] |series=Roots of Unity |date=August 31, 2014 |title=Look, Ma, No Zero!}}</ref>

===Other historical usages===
[[File:sexagenary_cycle_years.svg|thumb|Combinations of the five elements and twelve animals of the [[Chinese zodiac]] form the {{nowrap|60-year}} [[sexagenary cycle]].]]
In the [[Chinese calendar]], a system is commonly used in which days or years are named by positions in a sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle.

Book VIII of [[Plato]]'s ''[[The Republic (Plato)|Republic]]'' involves an allegory of marriage centered on the number 60<sup>4</sup> = {{val|12,960,000}} and its divisors. This number has the particularly simple sexagesimal representation 1,0,0,0,0. Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.<ref>{{citation
 | last = Barton | first = George A.
 | title = On the Babylonian origin of Plato's nuptial number
 | year = 1908
 | journal = Journal of the American Oriental Society
 | volume = 29
 | pages = 210–219
 | doi = 10.2307/592627
 | jstor = 592627}}. {{citation
 | last = McClain |first = Ernest G.
 | author-link = Ernest G. McClain
 | year = 1974
 | title = Musical "Marriages" in Plato's "Republic"
 | journal = Journal of Music Theory
 | volume = 18
 | issue = 2
 | pages = 242–272
 | author2 = Plato
 | jstor = 843638}}</ref>

[[Ptolemy]]'s ''[[Almagest]]'', a treatise on [[mathematical astronomy]] written in the second century AD, uses base 60 to express the fractional parts of numbers. In particular, his [[table of chords]], which was essentially the only extensive [[trigonometric table]] for more than a millennium, has fractional parts of a degree in base&nbsp;60, and was practically equivalent to a modern-day table of values of the [[Sine and cosine|sine]] function.

Medieval astronomers also used sexagesimal numbers to note time. [[Al-Biruni]] first subdivided the hour sexagesimally into [[minute]]s, [[second]]s, [[third (angle)|third]]s and [[fourth (angle)|fourth]]s in 1000 while discussing Jewish months.<ref name="al-Biruni">{{ citation | author=Al-Biruni | year=1879 | orig-year=1000 | title=The Chronology of Ancient Nations | url=https://books.google.com/books?id=pFIEAAAAIAAJ&pg=PA147 | pages=147–149 | translator-last=Sachau | translator-first=C. Edward | author-link=Al-Biruni}}</ref> Around 1235 [[John of Sacrobosco]] continued this tradition, although Nothaft thought Sacrobosco was the first to do so.<ref>{{ Citation | last = Nothaft | first = C. Philipp E. | date = 2018 | title = Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe | publisher = Oxford University Press | place = Oxford | page = 126 | isbn = 9780198799559 | quote = Sacrobosco switched to sexagesimal fractions, but rendered them more congenial to computistical use by applying them not to the day but to the hour, thereby inaugurating the use of hours, minutes, and seconds that still prevails in the twenty-first century.}}</ref> The Parisian version of the [[Alfonsine tables]] (ca. 1320) used the day as the basic unit of time, recording multiples and fractions of a day in base-60 notation.<ref>{{ Citation | last = Nothaft | first = C. Philipp E. | date = 2018 | title = Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe | publisher = Oxford University Press | place = Oxford | page = 196 | isbn = 9780198799559 | quote = One noteworthy feature of the Alfonsine Tables in their Latin-Parisian incarnation is the strict 'sexagesimalization' of all tabulated parameters, as … motions and time intervals were consistently dissolved into base-60 multiples and fractions of days or degrees.}}</ref>

The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as 1671.<ref>{{citation|last= Newton |first= Isaac |author-link= Isaac Newton |title= The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. |year= 1671 |publication-date= 1736 |publisher= [[Henry Sampson Woodfall|Henry Woodfall]] |location= [[London]] |page= 146 |url= https://books.google.com/books?id=WyQOAAAAQAAJ&pg=PA146 |quote= The most remarkable of these is the Sexagenary or Sexagesimal Scale of Arithmetick, of frequent use among Astronomers, which expresses all possible Numbers, Integers or Fractions, Rational or Surd, by the Powers of ''Sixty'', and certain numeral Coefficients not exceeding fifty-nine.}}</ref> For instance, [[Jost Bürgi]] in ''[[Fundamentum Astronomiae]]'' (presented to [[Emperor Rudolf II]] in 1592), his colleague Ursus in ''Fundamentum Astronomicum'', and possibly also [[Henry Briggs (mathematician)|Henry Briggs]], used multiplication tables based on the sexagesimal system in the late 16th century, to calculate sines.<ref name="folkerts">{{citation
 | last1 = Folkerts | first1 = Menso
 | last2 = Launert | first2 = Dieter
 | last3 = Thom | first3 = Andreas
 | arxiv = 1510.03180
 | doi = 10.1016/j.hm.2016.03.001
 | issue = 2
 | journal = Historia Mathematica
 | mr = 3489006
 | pages = 133–147
 | title = Jost Bürgi's method for calculating sines
 | volume = 43
 | year = 2016| s2cid = 119326088
 }}</ref>

In the late 18th and early 19th centuries, [[Tamil people|Tamil]] astronomers were found to make astronomical calculations, reckoning with shells using a mixture of decimal and sexagesimal notations developed by [[Hellenistic#Indian reference|Hellenistic]] astronomers.<ref>{{citation
| title=Tamil Astronomy: A Study in the History of Astronomy in India
| first=Otto
| last=Neugebauer
| author-link= Otto Neugebauer 
| journal=Osiris
| volume=10
| pages=252–276
| year=1952
| doi=10.1086/368555
| s2cid=143591575
}}; reprinted in {{Citation
| first=Otto
| last=Neugebauer
| author-link= Otto Neugebauer 
| title = Astronomy and History: Selected Essays
| place = New York
| publisher = [[Springer-Verlag]]
| year = 1983
| bibcode=1983ahse.book.....N
| isbn = 0-387-90844-7}}</ref>

Base-60 number systems have also been used in some other cultures that are unrelated to the Sumerians, for example by the [[Ekari people]] of [[Western New Guinea]].<ref>{{citation
 |title       = Kapauku numeration: Reckoning, racism, scholarship, and Melanesian counting systems
 |first       = Nancy
 |last        = Bowers
 |journal     = Journal of the Polynesian Society
 |volume      = 86
 |issue       = 1
 |pages       = 105–116
 |year        = 1977
 |url         = http://www.ethnomath.org/resources/bowers1977.pdf
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20090305190121/http://www.ethnomath.org/resources/bowers1977.pdf
 |archive-date = 2009-03-05
}}</ref><ref>{{citation
 |first       = Glendon Angove
 |last        = Lean
 |title       = Counting Systems of Papua New Guinea and Oceania
 |publisher   = Ph.D. thesis, [[Papua New Guinea University of Technology]]
 |year        = 1992
 |url         = http://www.uog.ac.pg/glec/thesis/thesis.htm
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20070905170848/http://www.uog.ac.pg/glec/thesis/thesis.htm
 |archive-date = 2007-09-05
}}. See especially [http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm chapter 4] {{webarchive|url=https://web.archive.org/web/20070928061154/http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm |date=2007-09-28 }}.</ref>

===Modern usage===
{{further|Degree (angle)#Subdivisions|Minute and second of arc|Hour|Minute|Second}}

Modern uses for the sexagesimal system include measuring [[angle]]s, [[geographic coordinates]], electronic navigation, and [[time]].<ref>{{cite book | doi=10.1007/978-1-4020-4425-0_9055 | chapter=Sexagesimal System | title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | date=2008 | last1=Powell | first1=Marvin A. | pages=1998–1999 | isbn=978-1-4020-4559-2 }}</ref>

One [[hour]] of time is divided into 60 [[minute]]s, and one minute is divided into 60 seconds. Thus, a measurement of time such as 3:23:17 {{nowrap|(3 hours, 23 minutes, and 17 seconds)}} can be interpreted as a whole sexagesimal number (no sexagesimal point), meaning {{nowrap|3 × 60{{sup|2}} + 23 × 60{{sup|1}} + 17 × 60{{sup|0}} seconds}}. However, each of the three sexagesimal digits in this number (3,&nbsp;23, and&nbsp;17) is written using the [[decimal]] system.

Similarly, the practical unit of angular measure is the [[degree (angle)|degree]], of which there are [[360 (number)|360]] (six sixties) in a circle. There are 60 [[minutes of arc]] in a degree, and 60 [[arcseconds]] in a minute.

==== YAML ====
In version 1.1<ref>{{Cite web|url=http://yaml.org/spec/1.1/|title = YAML Ain't Markup Language (YAML™) Version 1.1}}</ref> of the [[YAML]] data storage format, sexagesimals are supported for plain scalars, and formally specified both for integers<ref>{{Cite web|url=http://yaml.org/type/int.html|title = Integer Language-Independent Type for YAML Version 1.1}}</ref> and floating point numbers.<ref>{{Cite web|url=http://yaml.org/type/float.html|title = Floating-Point Language-Independent Type for YAML™ Version 1.1}}</ref> This has led to confusion, as e.g. some [[MAC address]]es would be recognised as sexagesimals and loaded as integers, where others were not and loaded as strings. In YAML 1.2 support for sexagesimals was dropped.<ref>{{citation|url=https://yaml.org/spec/1.2/spec.html#id2805071|title=YAML Ain't Markup Language (YAML™) Version 1.2 (3rd Edition, Patched at 2009-10-01) §10.3.2 Tag Resolution|author1=Oren Ben-Kiki|author2=Clark Evans|author3=Brian Ingerson|date=2009-10-01|website=The Official YAML Web Site|access-date=2019-01-30}}</ref>

==Notations==
{{see also|Positional notation#Sexagesimal system}}
In [[Hellenistic Greece|Hellenistic Greek]] astronomical texts, such as the writings of [[Ptolemy]], sexagesimal numbers were written using [[Greek numerals#Hellenistic zero|Greek alphabetic numerals]], with each sexagesimal digit being treated as a distinct number. Hellenistic astronomers adopted a new symbol for zero, {{overset|—|°}}, which morphed over the centuries into other forms, including the Greek letter omicron, ο, normally meaning 70, but permissible in a sexagesimal system where the maximum value in any position is 59.<ref>{{citation | edition = 2 | publisher = [[Dover Publications]] | last = Neugebauer | first = Otto | author-link = Otto E. Neugebauer | title = The Exact Sciences in Antiquity | journal = Acta Historica Scientiarum Naturalium et Medicinalium | orig-year = 1957 | year = 1969 | volume = 9 | isbn = 978-0-486-22332-2 | pages = 13–14, plate 2| pmid = 14884919 }}</ref><ref name=Mercier>{{citation |last=Mercier |first=Raymond |title=Consideration of the Greek symbol 'zero' |url=http://www.raymondm.co.uk/prog/GreekZeroSign.pdf |website=Home of Kairos}}</ref> The Greeks limited their use of sexagesimal numbers to the fractional part of a number.<ref>{{Citation |last= Aaboe |first= Asger |author-link = Asger Aaboe |year= 1964 |title= Episodes from the Early History of Mathematics |series = New Mathematical Library |volume = 13 |publisher= Random House |location= New York |pages= 103–104 }}</ref>

In medieval Latin texts, sexagesimal numbers were written using [[Arabic numerals]]; the different levels of fractions were denoted ''minuta'' (i.e., fraction), ''minuta secunda'', ''minuta tertia'', etc. By the 17th century it became common to denote the integer part of sexagesimal numbers by a superscripted zero, and the various fractional parts by one or more accent marks. [[John Wallis]], in his ''Mathesis universalis'', generalized this notation to include higher multiples of 60; giving as an example the number {{nowrap|49‵‵‵‵36‵‵‵25‵‵15‵1°15′25″36‴49⁗}}; where the numbers to the left are multiplied by higher powers of 60, the numbers to the right are divided by powers of 60, and the number marked with the superscripted zero is multiplied by 1.<ref>{{citation | last = Cajori | first = Florian | author-link = Florian Cajori | title = A History of Mathematical Notations | publisher = Cosimo, Inc. | volume = 1 | orig-year = 1928 | year = 2007 | location = New York | page = 216 | url = https://books.google.com/books?id=OQZxHpG2y3UC | isbn = 9781602066854 }}</ref> This notation leads to the modern signs for degrees, minutes, and seconds. The same minute and second nomenclature is also used for units of time, and the modern notation for time with hours, minutes, and seconds written in decimal and separated from each other by colons may be interpreted as a form of sexagesimal notation.

In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: ''[[prime (symbol)|prime]]'' or ''primus'', ''seconde'' or ''secundus'', ''tierce'', ''quatre'', ''quinte'', etc. To this day we call the second-order part [[second|of an hour]] or [[Minute and second of arc|of a degree]] a "second". Until at least the 18th century, {{sfrac|1|60}} of a second was called a "tierce" or "third".<ref>{{citation|title=A natural history of vision|last=Wade|first=Nicholas|publisher=MIT Press|year=1998|isbn=978-0-262-73129-4|page=193}}</ref><ref>{{citation|title=Middle English Dictionary|first=Robert E.|last=Lewis|publisher=University of Michigan Press|year=1952|isbn=978-0-472-01212-1|page=231}}</ref>

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 integer 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 | url = https://books.google.com/books?id=i-juAAAAMAAJ&pg=PA2}}</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. This notation is used in this article.

== Fractions and irrational numbers ==
=== Fractions ===
In the sexagesimal system, any [[fraction (mathematics)|fraction]] in which the [[denominator]] is a [[regular number]] (having only 2, 3, and 5 in its [[prime factorization]]) may be expressed exactly.<ref>{{citation
| title=Astronomical Cuneiform Texts
| last=Neugebauer
| first=Otto E.
| author-link=Otto E. Neugebauer
| year=1955
| publisher=Lund Humphries
| place=London
}}</ref> Shown here are all fractions of this type in which the denominator is less than or equal to 60:

{{div col |colwidth=8em |style=max-width:40em }}
:{{frac|2}} = 0;30
:{{frac|3}} = 0;20
:{{frac|4}} = 0;15
:{{frac|5}} = 0;12
:{{frac|6}} = 0;10
:{{frac|8}} = 0;7,30
:{{frac|9}} = 0;6,40
:{{frac|10}} = 0;6
:{{frac|12}} = 0;5
:{{frac|15}} = 0;4
:{{frac|16}} = 0;3,45
:{{frac|18}} = 0;3,20
:{{frac|20}} = 0;3
:{{frac|24}} = 0;2,30
:{{frac|25}} = 0;2,24
:{{frac|27}} = 0;2,13,20
:{{frac|30}} = 0;2
:{{frac|32}} = 0;1,52,30
:{{frac|36}} = 0;1,40
:{{frac|40}} = 0;1,30
:{{frac|45}} = 0;1,20
:{{frac|48}} = 0;1,15
:{{frac|50}} = 0;1,12
:{{frac|54}} = 0;1,6,40
:{{frac|60}} = 0;1
{{div col end}}

However numbers that are not regular form more complicated [[repeating fraction]]s. For example:

:{{frac|7}} = 0;{{overline|8,34,17}} (the bar indicates the sequence of sexagesimal digits 8,34,17 repeats infinitely many times)
:{{frac|11}} = 0;{{overline|5,27,16,21,49}}
:{{frac|13}} = 0;{{overline|4,36,55,23}}
:{{frac|14}} = 0;4,{{overline|17,8,34}}
:{{frac|17}} = 0;{{overline|3,31,45,52,56,28,14,7}}
:{{frac|19}} = 0;{{overline|3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41}}
:{{frac|59}} = 0;{{overline|1}}
:{{frac|61}} = 0;{{overline|0,59}}

The fact that the two numbers that are adjacent to sixty, 59 and 61, are both prime numbers implies that fractions that repeat with a period of one or two sexagesimal digits can only have regular number multiples of 59 or 61 as their denominators, and that other non-regular numbers have fractions that repeat with a longer period.

=== Irrational numbers ===
[[Image:Ybc7289-bw.jpg|thumb|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. -->

==See also==
* [[Clock]]
* [[Latitude]]
* [[Trigonometry]]

==References==
{{reflist|colwidth=30em}}

==Further reading==
*{{citation
 | first = Georges
 | last = Ifrah
 | title = The Universal History of Numbers: From Prehistory to the Invention of the Computer
 | publisher = Wiley
 | year = 1999
 | isbn = 0-471-37568-3}}.
*{{citation
 | first1 = Hans J.
 | last1 = Nissen
 | first2 = P.
 | last2 = Damerow
 | first3 = R.
 | last3 = Englund
 | title = Archaic Bookkeeping
 | publisher = University of Chicago Press
 | year = 1993
 | isbn = 0-226-58659-6}}

==External links==
*[http://www.wdl.org/en/item/4259 "Facts on the Calculation of Degrees and Minutes"] is an Arabic language book by [[Sibt al-Maridini|Sibṭ al-Māridīnī, Badr al-Dīn Muḥammad ibn Muḥammad]] (b. 1423). This work offers a very detailed treatment of sexagesimal mathematics and includes what appears to be the first mention of the periodicity of sexagesimal fractions.

[[Category:Positional numeral systems]]
[[Category:Babylonian mathematics]]