# What were the causes of decline of Babylonian numeric system?

Babylonia developed a numeric system advanced far ahead of its neighbors, and some might say even superior to our contemporary decimal. It was a 60-base system with convenient composition of factors for lower numbers and positional for larger numbers, vastly superior to horribly unwieldy, non-positional Roman and Egyptian.

All signs point to it being widely spread; it survives in more or less niche applications nowadays - our time system (2x12,60,60), 'dozen', 'gross', is quite prevalent in US cooking units and so on.

Still, currently the Arabic system (0..9) is prevalent throughout the world, after a period when other systems (like Roman) were dominant.

What were the causes for Babylonian system to die out?

• Is it merely place value? Did the Babylonians have place value?
– MCW
Sep 3, 2014 at 14:53
• The unpopularity of writing on clay tablets. Sep 3, 2014 at 14:58
• Maybe it was because all the Babylonians died? During the Babylonian wars (319-310 BC) Babylon and the surrounding countryside was completely annihilated and all the Babylonians were killed or died by starvation or drowning. Sep 3, 2014 at 17:10
• @fdb No! Don't . . .
– Mike
Sep 4, 2014 at 0:53
• @MarkC.Wallace: The Babylonians had place value (at least of a sort), but without a true zero placeholder. It gave them an arithmetical power greater than many contemporary systems, but less than our modern system. Sep 4, 2014 at 1:18

Simply put, the decimal system is more convenient for most types of calculations. As you point out, there are systems that still use base 60. And there are others such as binary and hexadecimal which are applied in other areas where they are applicable.

But the main reason for its decline is the unwieldiness. 60 as a base is difficult to use because you have to remember at least 59 unique names ( as opposed to decimal where twenty-one, thirty-one, forty-one etc follow the same pattern). To mitigate these difficulties, arithmeticians looked for a smaller number. This led to the decline of the 60 base. See this page 97.

It is also important to note that various geographies and cultures have used various systems over time, and the sexagesimal system is predated by various other systems:

"Historically, finger counting, or the practice of counting by fives and tens, seems to have come later than counter-casting by twos and threes, yet the quinary and decimal systems almost invariably displaced the binary and ternary schemes. A study of several hundred tribes among the American Indians, for example, showed that almost one-third used a decimal base, and about another third had adopted a quinary or a quinary-decimal system; fewer than a third had a binary scheme, and those using a ternary system constituted less than 1 percent of the group. The vigesimal system, with the number 20 as a base, occurred in about 10 percent of the tribes."

"An interesting example of a vigesimal system is that used by the Maya of Yucatan and Central America. This was deciphered some time before the rest of the Maya languages could be translated. In their representation of time intervals between dates in their calendar, the Maya used a place value numeration, generally with 20 as the primary base and with 5 as an auxiliary."

See this

So it seems a balance between a "small" number - already known- such as 5, and a slightly larger 12, as base was ultimately arrived at- the number being 10.

"The Mesopotamian civilizations of antiquity are often referred to as Babylonian, although such a designation is not strictly correct. The city of Babylon was not at first, nor was it always at later periods, the center of the culture associated with the two rivers, but convention has sanctioned the informal use of the name “Babylonian” for the region during the interval from about 2000 to roughly 600 BCE.When in 538 BCE Babylon fell to Cyrus of Persia, the city was spared, but the Babylonian Empire had come to an end. “Babylonian” mathematics, however, continued through the Seleucid period in Syria almost to the dawn of Christianity."

Edit: Additional Information on "Babylonian" Mathematics

Much of Babylonian Math tablets comes from "House F", a scribal school:

House F was excavated in the first months of 1952 by a team of archaeologists from the universities of Chicago and Pennsylvania. It was their third field season in the ancient southern Iraqi city of Nippur and one of their express aims was to find large numbers of cuneiform tablets (McCown and Haines 1967, viii). For this reason they had chosen two sites on the mound known as Tablet Hill, because of the large number of tablets that had been found there in the late nineteenth century.

Several types of tablet were used for elementary schooling in Nippur, as classified by a scheme devised by Miguel Civil (e.g., 1995, 2308) to describe lexical lists—standardized lists of signs and words. But, as Niek Veldhuis (1997, 28–39) showed, this tablet typology applies equally to all elementary school exercises, including mathematical ones. It happens that mathematics has survived on just three types of tablet of from House F: the small Type IIIs and the larger Type I and IIs, of which it will be important to distinguish between the flat obverse (Type II/1) and the slightly convex reverse (II/2)...

On the multiplication tables:

The standard list of multiplications was described long ago by Neugebauer (1935–7, I 32–67; Neugebauer and Sachs 1945, 19–33) and is very well known.

...

The series starts with a list of one- and two-place reciprocal pairs, encompassing all the regular integers from 2 to 81. It is followed by multiplication ‘tables’ for sexagesimally regular head numbers from 50 down to 1 15, with multiplicands 1–20, 30, 40, and 50.

....

Returning to the standard series of multiplications as attested in House F, nine of the 40 known head numbers—namely 48, 44 26 40, 20, 7 12, 7, 5, 3 20, 2 24, and 2 15—do not survive on known tablets. Should we attribute these omissions to the accidents of recovery or to deliberate exclusion from the series? The patterns of attestation make it easier to make de[ nitive statements about the higher head numbers than the lower. The head number 48, for instance, is included in just five of the 71 ‘combined’ tables catalogued by Neugebauer (two of those five are from Nippur), compared to 23 certain omissions. He lists no ‘single’ tables for 48. Similarly, 2 15 occurs in two out of nine possible ‘combined’ tables, neither of them from Nippur, and in no ‘singles’. It is not surprising, therefore, that the 48 and 2 15 times tables were apparently not taught in House F. The exclusion of 44 26 40, is rather more surprising: given its place near the start of the standard series it is presumably not simply missing by archaeological accident. On the other hand none of Neugebauer’s ‘combined’ tables appear to omit it, while he lists three ‘single’ tables for 44 26 40. 9 is is a deliberate but idiosyncratic omission then, particular to House F—though perhaps a judicious one; none of the other head numbers are three sexagesimal places long.

Unsuccessful methods and mistakes by Students:

The other two calculations identified so far on House F tablets are also attempts to find reciprocals, but conspicuously less successful than the first.

...

As in our first example, the student has split 4 37;46 40 into 4 37; 40 and 0;06 40. He has appropriately taken the reciprocal of the latter—9—and multiplied it by the former, adding 1 to the result. However, instead of arriving at 41 39 + 1 = 41 40, our student has lost a sexagesimal place and found 41;39 + 1 = 42;39. Unable to go further with his calculation (for the next stage is to find the reciprocal of the number just found, but his is coprime to 60) he has abandoned the exercise there. The correct answer would have been 0;00 12 57 36. the last calculation of the three is the most pitiful ... the student has got no further than... On the other hand, 4 37 46 40 does not, as far as I can ascertain, fit the pattern; presumably it was chosen because, like the other two, it terminates in the string 6 40. One possible interpretation of this commonality is that three students were set similar problems at the same time, using a common method and a common starting point but requiring different numerical solutions.

It is clear from these that the techniques were not perfected yet- even if we today believe that a base 60 math was a viable system.

• The system wasn't quite as unwieldy as it used multiples of its many fractions, a'la 'two dozen'.
– SF.
Sep 3, 2014 at 15:41
• The commonest base in history and today as well is base 10. And the reason cited by Ifrah is its manageability wrt other bases such as 60. You may very well disagree. But my source is simply books on math history. Sep 3, 2014 at 17:10
• Not really a historical answer. Sep 3, 2014 at 17:11
• What is a historical answer? I quoted from math history books. Sep 3, 2014 at 17:15
• "60 as a base is difficult to use because you have to remember at least 59 unique names " Sorry, this is nonsense. In Babylonian (Akkadian), as in other languages, the numbers above twenty and formed in the same way as inEnglish: tens and units.
– fdb
Sep 3, 2014 at 18:01

The Babylonian sexagesimal system is used by Ptolemy in his Almagest (2nd century AD) and by Arabic astronomers throughout the Middle Ages. The decimal numerals were introduced from India to the Muslim World in the 9th century AD, and later from the Near East to Europe. It took a long time for the “Indian” numbers to be accepted, but eventually people realised that it is much easier to calculate using the decimal rather than the sexagesimal system, especially if you are using an abacus. That is why the latter died out (except for time-keeping, degrees of an arc, degrees of the zodiac).

• so... only because 60 bead per row abacus would be unwieldy? (because in day-to-day calculations - in memory etc, it's significantly easier than decimal).
– SF.
Sep 3, 2014 at 12:32
• @sf: An abacus doesn't have 10 beads per column, it has 5. These five bear a striking resemblance to a thumb and four fingers, and if one had 18 slaves standing inline with their hands out that would be functionally equivalent to an abacus, if perhaps not quite as convenient to carry around or calculate on. Sep 4, 2014 at 0:14
• @PieterGeerkens : Well, Chinese abaci have 5 beads/columns, russians have 10. There is no reason to think the abacus you’re refering to is the only possible design. Given the sub-bases 6 and 10, it is not difficult to imagine a sexagesimal abacus using pebbles. I cannot find the reference, but I’m sure I read something about it, by an assyriologist analysing the computing errors and the vocabulary used in tablets to deduce than the standard device had 5 sexagesimal places Sep 15, 2014 at 10:08
• I found the article I referred to above (in French), by Christine Proust. Arelated paper by Jen Høyrup is here. Sep 15, 2014 at 10:22