Babylonian cuneiform numerals

Babylonian cuneiform numerals, also used in Assyria and Chaldea, were written in cuneiform, using a wedge-tipped 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 numeral system inherited from either the Sumerian or 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

This system first appeared around 2000 BC;^[1] its structure reflects the decimal lexical numerals of Semitic languages rather 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

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 digits. 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 2×60²+23×60+3 = 8583.

A space was left to indicate a place without value, similar to the modern-day 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 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 degrees (360° in a circle or 60° in an angle of an equilateral triangle), arcminutes, and arcseconds in trigonometry and the measurement of time, although both of these systems are actually mixed radix.^[3]

A common theory is that 60, a superior highly composite number (the previous and next in the series being 12 and 120), was chosen due to its prime factorization: 2×2×3×5, which makes it divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Integers and fractions 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 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 () to represent zero, but only in the medial positions, and not on the right-hand side of the number, as we do in numbers like 100.^[4]

References

^ ^a ^b ^c Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 247. ISBN 978-0-521-87818-0.
^ ^a ^b Stephen 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

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

Babylonian numerals Archived 2017-05-20 at the Wayback Machine
Cuneiform numbers Archived 2020-06-27 at the Wayback Machine
Babylonian Mathematics
High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine
Babylonian Numerals by 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

[1]