Editing Babylonian cuneiform numerals

{{short description|Numeral system}}
[[Image:Babylonian numerals.svg|upright=1.8|thumb|Babylonian cuneiform numerals]]
'''Babylonian cuneiform numerals''', also used in [[Assyria]] and [[Chaldea]], were written in [[cuneiform (script)|cuneiform]], using a wedge-tipped [[Phragmites|reed]] stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

The [[Babylonians]], who were famous for their astronomical observations, as well as their calculations (aided by their invention of the [[abacus]]),  used a [[sexagesimal]] (base-60) [[Positional notation|positional numeral system]] inherited from either the [[Sumer]]ian or the Akkadian civilizations.<ref name="Chrisomalis">{{cite book|title= Numerical Notation: A Comparative History |title-link=Numerical Notation: A Comparative History |author=Stephen Chrisomalis|page=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA247 247]|year= 2010|publisher=Cambridge University Press|isbn= 978-0-521-87818-0}}</ref> Neither of the predecessors was a positional system (having a convention for which 'end' of the numeral represented the units).

==Origin==
This system first appeared around 2000 BC;<ref name="Chrisomalis" /> its structure reflects the decimal lexical numerals of [[Semitic languages]] rather than Sumerian lexical numbers.<ref name="Chrisomalis2">{{cite book|title= Numerical Notation: A Comparative History |title-link=Numerical Notation: A Comparative History |author=Stephen Chrisomalis|page=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA248 248]|year= 2010|publisher= Cambridge University Press|isbn= 978-0-521-87818-0}}</ref>  However, the use of a special Sumerian sign for 60 (beside two Semitic signs for the same number)<ref name="Chrisomalis" /> attests to a relation with the Sumerian system.<ref name="Chrisomalis2" />
{{numeral systems}}

==Symbols==
The Babylonian system is credited as being the first known [[positional numeral system]], in which the value of a particular digit depends both on the digit itself and its position within the number. This was an extremely important development because non-place-value systems require unique symbols to represent each power of a base (ten, one hundred, one thousand, and so forth), which can make calculations more difficult.

Only two symbols (𒁹 to count units and 𒌋 to count tens) were used to notate the 59 non-zero [[Numerical digit|digit]]s. These symbols and their values  were combined to form a digit in a [[sign-value notation]] quite similar to that of [[Roman numerals]]; for example, the combination 𒌋𒌋𒁹𒁹𒁹 represented the digit for 23 (see table of digits above). 

These digits were used to represent larger numbers in the base 60 (sexagesimal) positional system. For example, 𒁹𒁹 𒌋𒌋𒁹𒁹𒁹 𒁹𒁹𒁹 would represent 2×60<sup>2</sup>+23×60+3 = 8583.

A space was left to indicate a place without value, similar to the modern-day [[0 (number)|zero]]. Babylonians later devised a sign to represent this empty place. They lacked a symbol to serve the function of [[radix point]], so the place of the units had to be inferred from context: 𒌋𒌋𒁹𒁹𒁹 could have represented 23, 23×60 (𒌋𒌋𒁹𒁹𒁹␣), 23×60×60 (𒌋𒌋𒁹𒁹𒁹␣␣), or 23/60, etc.

Their system clearly used internal [[decimal]] to represent digits, but it was not really a [[mixed radix|mixed-radix]] system of bases 10 and 6, since the ten sub-base was used merely to facilitate the representation of the large set of digits needed, while the place-values in a digit string were consistently 60-based and the [[arithmetic]] needed to work with these digit strings was correspondingly sexagesimal.

The legacy of sexagesimal still survives to this day, in the form of [[degree (angle)|degree]]s (360° in a [[circle]] or 60° in an [[angle]] of an [[equilateral triangle]]), [[arcminute]]s, and [[arcsecond]]s in [[trigonometry]] and the measurement of [[time]], although both of these systems are actually mixed radix.<ref>[http://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/ Scientific American – Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?]</ref>

A common theory is that [[60 (number)|60]], a [[superior highly composite number]] (the previous and next in the series being [[12 (number)|12]] and [[120 (number)|120]]), was chosen due to its [[prime factorization]]: 2×2×3×5, which makes it divisible by [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[5 (number)|5]], [[6 (number)|6]], [[10 (number)|10]], [[12 (number)|12]], [[15 (number)|15]], [[20 (number)|20]], [[30 (number)|30]], and [[60 (number)|60]]. [[Integer]]s and [[fraction (mathematics)|fraction]]s were represented identically—a radix point was not written but rather made clear by context.

===Zero===
The Babylonians did not technically have a digit for, nor a concept of, the number [[0 (number)|zero]]. Although they understood the idea of [[nothingness]], it was not seen as a number—merely the lack of a number. 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 is done in numbers like {{val|100}}.<ref>{{Cite journal |last=Boyer |first=Carl B. |date=1944 |title=Zero: The Symbol, the Concept, the Number |url=https://www.jstor.org/stable/3030083 |journal=National Mathematics Magazine |volume=18 |issue=8 |pages=323–330 |doi=10.2307/3030083 |issn=1539-5588|url-access=subscription }}</ref>

== See also ==
{{Portal|Mathematics}}
* {{format link|Akkadian language#Numerals}}
* [[Babylon]]
* [[Babylonia]]
* [[Babylonian mathematics]]
* [[Cuneiform (Unicode block)]]
* [[0 (number)#History|History of zero]]
* [[Numeral system]]
* {{format link|Sumerian language#Numerals}}

==References==
{{reflist}}

===Bibliography===
*{{cite book
 | last = Menninger
 | first = Karl W.
 | author-link = Karl Menninger (mathematics)
 | year = 1969
 | title = Number Words and Number Symbols: A Cultural History of Numbers
 | publisher = MIT Press
 | isbn = 0-262-13040-8
}}
*{{cite book
 | last = McLeish
 | first = John
 | year = 1991
 | title = Number: From Ancient Civilisations to the Computer
 | publisher = HarperCollins
 | isbn = 0-00-654484-3
 | url-access = registration
 | url = https://archive.org/details/number00john
 }}

== External links ==
{{Commons category|Babylonian numerals}}
* [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals] {{Webarchive|url=https://web.archive.org/web/20170520152528/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html |date=2017-05-20 }} 
* [http://it.stlawu.edu/%7Edmelvill/mesomath/Numbers.html Cuneiform numbers] {{Webarchive|url=https://web.archive.org/web/20200627030234/http://it.stlawu.edu/~dmelvill/mesomath/Numbers.html |date=2020-06-27 }}
* [http://mathforum.org/alejandre/numerals.html Babylonian Mathematics]
* [http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection]
* [http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection] {{Webarchive|url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |date=2012-08-13 }}
* [http://demonstrations.wolfram.com/BabylonianNumerals/ Babylonian Numerals] by Michael Schreiber, [[Wolfram Demonstrations Project]].
* {{MathWorld | urlname=Sexagesimal | title=Sexagesimal}}
* [https://archive.today/20130410174251/http://cutedgesoft.com/our-products/cescnc-numerical-converter/ CESCNC – a handy and easy-to use numeral converter]

[[Category:Babylonian mathematics]]
[[Category:Non-standard positional numeral systems]]
[[Category:Numeral systems]]
[[Category:Numerals]]