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'{{nofootnotes|date=April 2011}} {{numeral systems}} '''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 the [[Sumer]]ian and also [[Akkad]]ian civilizations. Neither of the predecessors was a positional system (having a convention for which ‘end’ of the numeral represented the units). This system first appeared around 3100 B.C. It is also 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|{{{width|20px}}}|]] to count units and [[File:Babylonian_10.svg|{{{width|20px}}}|]] to count tens) were used to notate the 59 non-zero [[digit]]s. These symbols and their values were combined to form a digit in a [[sign-value notation]] way similar to that of [[Roman numerals]]; for example, the combination [[File:Babylonian_20.svg|{{{width|20px}}}|]][[File:Babylonian_3.svg|{{{width|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|{{{width|20px}}}|]][[File:Babylonian_3.svg|{{{width|20px}}}|]] could have represented 23 or 23×60 or 23×60×60 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. 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]], and [[30 (number)|30]]. In fact, it is the smallest integer divisible by all integers from 1 to 6. [[Integer]]s and [[fraction (mathematics)|fraction]]s were represented identically — a radix point was not written but rather made clear by context. [[Image:Babylonian numerals.svg|450px|thumb|Babylonian numerals]] ==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:Chiffre-babylonien-0.png]]) to mark the nonexistence of a digit in a certain place value. ==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 }} == See also == {{portalbox|Mathematics|Ancient Near East}} *[[Babylonia]] *[[Babylon]] *[[0_%28number%29#History|History of zero]] *[[Numeral system]] == External links == {{Commonscat|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}} [[Category:Numeration]] [[Category:Numerals]] [[Category:Babylonian mathematics]] [[ar:أرقام بابلية]] [[ca:Numeració babilònica]] [[da:Babyloniske tal]] [[es:Numeración babilónica]] [[fr:Numération mésopotamienne]] [[gl:Numeración babilónica]] [[id:Angka-angka Babilonia]] [[it:Sistema di numerazione babilonese]] [[he:ספרות בבליות]] [[ms:Angka Babylon]] [[nl:Babylonische cijfers]] [[nds:Babyloonsche Tallen]] [[sh:Vavilonski brojevi]] [[fi:Babylonialaiset numerot]] [[sv:Babyloniska talsystemet]] [[ta:பபிலோனிய எண்ணுருக்கள்]] [[zh:巴比伦数字]]'