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'@@ -1,69 +1,2 @@ -{{also|Babylonian mathematics}} -[[Image:Babylonian numerals.svg|450px|thumb|Babylonian numerals]] -'''Babylonian numerals''' were written in [[cuneiform (script)|cuneiform]], using a wedge-tipped [[Phragmites|reed]] [[stylus]] to make 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 and calculations (aided by their invention of the [[abacus]]), used a [[sexagesimal]] (base-60) positional [[numeral system]] inherited from either the [[Sumer]]ian or the [[Ebla]]ite civilizations.<ref name="Chrisomalis">{{cite book|url=https://books.google.nl/books?id=ux--OWgWvBQC&pg=PA247#v=onepage&q&f=false|title= Numerical Notation: A Comparative History|author=Stephen Chrisomalis|page= 247|year= 2010}}</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|url=https://books.google.nl/books?id=ux--OWgWvBQC&pg=PA248#v=onepage&q&f=false|title= Numerical Notation: A Comparative History|author=Stephen Chrisomalis|page= 248|year= 2010}}</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}} - -==Characters== -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), making calculations difficult. - -Only two symbols ([[File:Babylonian 1.svg|20px]] to count units and [[File:Babylonian 10.svg|20px]] 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 [[File:Babylonian 20.svg|20px]][[File:Babylonian 3.svg|20px]] represented the digit for 23 (see table of digits below). 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 : [[File:Babylonian 20.svg|20px]][[File:Babylonian 3.svg|20px]] could have represented 23 or 23&times;60 or 23&times;60&times;60 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]]), [[minute]]s, and [[second]]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/</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&times;2&times;3&times;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]], and [[30 (number)|30]]. [[Integer]]s and [[fraction (mathematics)|fraction]]s were represented identically &mdash; a radix point was not written but rather made clear by context. - -==Numerals== -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. What the Babylonians had instead was a space (and later a disambiguating placeholder symbol [[File:Babylonian digit 0.svg|40 px]]) to mark the nonexistence of a digit in a certain place value.{{Citation needed|date=May 2015}} - -== See also == -{{portal|Mathematics|Ancient Near East}} -* [[Babylon]] -* [[Babylonia]] -* [[0_%28number%29#History|History of zero]] -* [[Numeral system]] - -==Notes== -{{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 -}} - -== External links == -{{Commons category|Babylonian numerals}} -* [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html Babylonian numerals] -* [http://it.stlawu.edu/%7Edmelvill/mesomath/Numbers.html Cuneiform numbers] -* [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] -* [http://demonstrations.wolfram.com/BabylonianNumerals/ Babylonian Numerals] by Michael Schreiber, [[Wolfram Demonstrations Project]]. -* {{MathWorld | urlname=Sexagesimal | title=Sexagesimal}} -* [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]] +9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 '