# Babylonian numerals

**Babylonian numerals** were written in cuneiform, using a wedge-tipped 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 Sumerian or the Eblaite 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]}

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Hindu–Arabic numeral system |

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Non-standard positional numeral systems |

List of 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), which can make calculations more difficult.

Only two symbols ( to count units and 20px 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 20px20px 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 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 : 20px20px 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 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), minutes, and seconds 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, and 30. 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. What the Babylonians had instead was a space (and later a disambiguating placeholder symbol ) to mark the nonexistence of a digit in a certain place value.

## See also

- Babylon
- Babylonia
- History of zero
- Numeral system

## Notes

- ↑
^{1.0}^{1.1}^{1.2}Stephen Chrisomalis (2010).*Numerical Notation: A Comparative History*. p. 247. https://books.google.nl/books?id=ux--OWgWvBQC&pg=PA247#v=onepage&q&f=false. - ↑
^{2.0}^{2.1}Stephen Chrisomalis (2010).*Numerical Notation: A Comparative History*. p. 248. https://books.google.nl/books?id=ux--OWgWvBQC&pg=PA248#v=onepage&q&f=false. - ↑ http://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/

## 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
- Cuneiform numbers
- 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 - Babylonian Numerals by Michael Schreiber, Wolfram Demonstrations Project.
- Weisstein, Eric W.. "Sexagesimal". http://mathworld.wolfram.com/Sexagesimal.html.
- CESCNC - a handy and easy-to use numeral converter