Babylonian cuneiform numerals

Babylonian cuneiform numerals,also used in Assyria and Chaldea, were written incuneiform,using a wedge-tippedreedstylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

TheBabylonians,who were famous for their astronomical observations, as well as their calculations (aided by their invention of theabacus), used asexagesimal(base-60)positional numeral systeminherited from either theSumerianor the Akkadian civilizations.^[1]Neither of the predecessors was a positional system (having a convention for which 'end' of the numeral represented the units).

Origin[edit]

This system first appeared around 2000 BC;^[1]its structure reflects the decimal lexical numerals ofSemitic languagesrather than Sumerian lexical numbers.^[2]However, the use of a special Sumerian sign for 60 (beside two Semitic signs for the same number)^[1]attests to a relation with the Sumerian system.^[2]

Symbols[edit]

The Babylonian system is credited as being the first knownpositional 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 (to count units andto count tens) were used to notate the 59 non-zerodigits.These symbols and their values were combined to form a digit in asign-value notationquite similar to that ofRoman numerals;for example, the combinationrepresented 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,would represent 2×60²+23×60+3 = 8583.

A space was left to indicate a place without value, similar to the modern-dayzero.Babylonians later devised a sign to represent this empty place. They lacked a symbol to serve the function ofradix point,so the place of the units had to be inferred from context:could have represented 23 or 23×60 or 23×60×60 or 23/60, etc.

Their system clearly used internaldecimalto represent digits, but it was not really amixed-radixsystem 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 thearithmeticneeded to work with these digit strings was correspondingly sexagesimal.

The legacy of sexagesimal still survives to this day, in the form ofdegrees(360° in acircleor 60° in anangleof anequilateral triangle),arcminutes,andarcsecondsintrigonometryand the measurement oftime,although both of these systems are actually mixed radix.^[3]

A common theory is that60,asuperior highly composite number(the previous and next in the series being12and120), was chosen due to itsprime factorization:2×2×3×5, which makes it divisible by1,2,3,4,5,6,10,12,15,20,30,and60.Integersandfractionswere represented identically—a radix point was not written but rather made clear by context.

Zero[edit]

The Babylonians did not technically have a digit for, nor a concept of, the numberzero.Although they understood the idea ofnothingness,it was not seen as a number—merely the lack of a number. Later Babylonian texts used a placeholder () to represent zero, but only in the medial positions, and not on the right-hand side of the number, as we do in numbers like100.^[4]

References[edit]

^^a ^b ^cStephen Chrisomalis (2010).Numerical Notation: A Comparative History.Cambridge University Press. p. 247.ISBN 978-0-521-87818-0.
^^a ^bStephen Chrisomalis (2010).Numerical Notation: A Comparative History.Cambridge University Press. p. 248.ISBN 978-0-521-87818-0.
^Scientific American – Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?
^Lamb, Evelyn (August 31, 2014),"Look, Ma, No Zero!",Scientific American,Roots of Unity

Bibliography[edit]

Menninger, Karl W.(1969).Number Words and Number Symbols: A Cultural History of Numbers.MIT Press.ISBN 0-262-13040-8.
McLeish, John (1991).Number: From Ancient Civilisations to the Computer.HarperCollins.ISBN 0-00-654484-3.

External links[edit]

Babylonian numerals Archived2017-05-20 at theWayback Machine
Cuneiform numbers Archived2020-06-27 at theWayback Machine
Babylonian Mathematics
High resolution photographs, descriptions, and analysis of theroot(2)tablet (YBC 7289) from the Yale Babylonian Collection
Photograph, illustration, and description of theroot(2)tablet from the Yale Babylonian Collection Archived2012-08-13 at theWayback Machine
Babylonian Numeralsby Michael Schreiber,Wolfram Demonstrations Project.
Weisstein, Eric W."Sexagesimal".MathWorld.
CESCNC – a handy and easy-to use numeral converter

[Chrisomalis-1] Stephen Chrisomalis (2010).Numerical Notation: A Comparative History.Cambridge University Press. p. 247.ISBN 978-0-521-87818-0.

[Chrisomalis2-2] Stephen Chrisomalis (2010).Numerical Notation: A Comparative History.Cambridge University Press. p. 248.ISBN 978-0-521-87818-0.

[3] Scientific American – Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?

[lmnz-4] Lamb, Evelyn (August 31, 2014),"Look, Ma, No Zero!",Scientific American,Roots of Unity

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