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|Numeral systems by culture|
|Positional systems by base|
|Non-standard positional numeral systems|
|List of numeral systems|
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 Sumerian and also Akkadian 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 ( to count units and 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 way similar to that of Roman numerals; for example, the combination 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 : 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.
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. In fact, it is the smallest integer divisible by all integers from 1 to 6. Integers and fractions were represented identically — a radix point was not written but rather made clear by context.
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.
- 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.
|Wikimedia Commons has media related to Babylonian numerals.|
- 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", MathWorld.
- CESCNC - a handy and easy-to use numeral converter