 # Babylonian Mathematics

## Introduction to Babylonian Mathematics

Mathematics was an active discipline in Mesopotamia since the earliest periods of writing in the mid-fourth millennium BC. However, if we are to judge by the number and age of the cuneiform mathematical texts we possess, it appears that mathematics reached its apex during the Old Babylonian period.

At the outset, it should be mentioned that Babylonian mathematics was largely an applied science. Unlike classical Greek mathematics, Babylonian mathematics was not centered around geometric figures or abstract proofs of mathematical theorems. Instead, it focused on practical problems that could be solved algorithmically. Many texts from that era are simply multiplication tables of various sizes, designed to aid scribes in performing complicated arithmetic. Other texts describe fairly realistic situations that might arise in economic exchanges or engineering endeavors, such as constructing dams, canals, and brick walls. There are many examples of texts dealing with more general geometric or algebraic problems, but they are usually stated in terms of concrete situations such as the division of a field or distribution of rations, and are always discussed using specific values.

Thus there are very few instances of Babylonian ‘proofs,’ or even instruction tablets that outline a general method for solving a class of problems. This is not to say that Babylonian mathematicians did not understand the abstract principles behind their solutions; they simply thought of them in the context of concrete examples, similar to the way a mechanic might think of physics principles in terms of an automobile.

An example of such a concrete mathematical text comes from YBC 4669, which states : ‘2/3 of 1/3 of my ration I have eaten. 7 remains. What was the initial ration?’ In modern notation, the problem asks to solve for s in the equation (1-2/9)s = 7.

Contrary to the preceeding example, Babylonian math problems can be quite complicated, involving large numbers and multiple unknowns (sometimes as many as fifteen - see Friberg, pg. 559). Often, though, the problems are pedagogical in construction, and so the input values are skillfully chosen to provide a clean answer.

## The Babylonian Number System

Perhaps the most widely known fact about Babylonian mathematics is that it used an artificial sexagesimal number system. That is, the place values corresponded to postive and negative powers of 60. For example, the number 4000 would be expressed as 1,6,40 = 1 x 60^2 + 6 x 60 + 40 x 1 in sexagesimal notation. The Babylonians had cuneiform symbols for 1 and 10, which they combined like tally marks to express numbers between 1 and 59. They used place value notation for larger numbers and fractional numbers just as we do now, except that they did not have a symbol for zero to indicate empty place values.

The Babylonians also understood the concept of fractions, or ratios. They did not have a special symbol corresponding to the decimal point, nor did they have anything like our modern fraction notation. Instead, they based their computations on the concept of reciprocal numbers. The reciprocal of a number n is just n’ := 1/n. If n’ has a finite sexagesimal expansion, then we say that n is regular. Otherwise we call n irregular. Likewise, an irregular fraction is a fractional number with non-terminating sexagesimal expansion. The Babylonians usually restricted their computations to regular numbers, and constructed large tables of these numbers and their reciprocals. Such tables would often consist of pairs of columns of numbers of the form igi-n-gal-bi n’ in accent on gal or just n n’, where n’ is the reciprocal of n. For instance, 2 30 would mean that 30 x 60^-1 = .5 is the reciprocal of 2. The Babylonians avoided working with irregular numbers and irregular fractions when possible, which sometimes meant rounding or figuring out ways to express irregular fractions in terms of sums and products of regular fractions.

The Babylonians did not as a rule write mathematical equations as precisely as we do. Until late Old Babylonian times they did not have symbols for + and =, and they did not use symbols to stand for variables. Still, they understood algebra well enough to perform most of the manipulations we do today, such as factorizing expressions, adding, subtracting, multiplying, or dividing both sides of an equation by the same quantity, and taking roots. They demonstrated remarkable ingenuity in solving equations, given their lack of suggestive notation.

For instance, the Babylonians might consider the problem: given a rectangle whose area and sum of its length and width are known, determine the length and width. Suppose the length plus width is l + w = 50, and the area is lw = 600. Moreover, suppose the length is greater than the width. Then, working only with the specific numbers given in the problem, the Babylonians would first compute half the sum of the length and width, i.e. h := (l+w)/2 = 25. Then they would compute h^2 = 625, and subtract the area from this, getting 625-600=25. Finally, they would take the square root of the result and add this to the half sum to get l = 30, and subtract it from the half sum to get w = 20. When the above procedure is worked out using the variables l and w, the solution method becomes clear.

The Babylonians had other methods for solving many types of problems involving quadratic equations, sometimes based on reducing to a problem like the one above. There are even instances of solutions for cubic equations. Yet while the Babylonians were aware of the existence of irrational numbers like sqrt(2), they did not formalize the concept to the extent the Greeks did, and were content to compute rational approximations. Similarly, their estimate of pi usually ended up being simply 3.

## The Legacy of Babylonian Mathematics

Because of the great antiquity of Babylonian mathematics, scholars have often conjectured about its influence on the development of mathematics in other ancient cultures. Although the paucity of cuneiform texts from the 3rd millennium makes conclusions about Mesopotamia’s mathematical influence highly speculative, in the 2nd and 1st millennia there is strong evidence for cross-cultural mathematical exchange. During the Old Babylonian period it is evident that Babylonian mathematical techniques were known in Susa, Mari, and Egypt. Later, in the great cultural movements of the 1st millennium the influence of Babylonian mathematics can be seen in numerous mathematical texts and numerical tables from Assyria, Ugarit, and Palestine.

Babylon’s mathematical influence beyond the Ancient Near East is harder to identify, although conjectures have been made about its relation to Greek, Indian, and Chinese mathematics. In the case of the Greeks arguments for a relation are slightly more cogent, as ancient Greek commentators unanimously claimed that their geometrical and astronomical sciences came from Egypt and Babylon. Indeed, the famous ‘Pythagorean theorem’ was known in Babylonia well over a thousand years before Pythagoras. Thus while the Greeks radically reformulated mathematics along the lines of abstract geometry, many of the concepts and theorems they used certainly were known to the Babylonians long before.

## Bibliographical References

### Web Sites

Black, Jeremy A., Graham G. Cunningham, Esther Flückiger-Hawker, Eleanor Robson, and Gabor Zólyomi (1998- ) The Electronic Text Corpus of Sumerian Literature. http://www-etcsl.orient.ox.ac.uk/

Duncan Melville's site "Mesopotamian Mathematics" http://it.stlawu.edu/%7Edmelvill/mesomath/index.html

### Books and Articles

Damerow, Peter (2001) "Kannten die Babylonier den Satz des Pythagoras? Epistemologische Anmerkungen zur Natur der babylonischen Mathematik" in Changing Views on Ancient Near Eastern Mathematics (Jens Høyrup and Peter Damerow, eds.), Berlin: 219-310.

Englund, Robert K. (1988) "Administrative timekeeping in Ancient Mesopotamia" JESHO 31: 121-185.

and Jean-Pierre Grégoire (1991) The Proto-Cuneiform Texts from Jemdet Nasr (MSVO 1), Berlin.

Friberg, Jöran (1978) "Early Roots of Babylonian Mathematics 1. A method for the decipherment … of proto- Sumerian and proto-Elamite semi-pictographic inscriptions" Department of Mathematics CTH-GU 1978- 9, Gothenburg.

Friberg, Jöran (1990) "Mathematik" (in English) in RlA 7 (Dietz Otto Edzard, ed.), Berlin, New York: 531-585.

(1992) "Numbers and counting in the Ancient Near East" in The Anchor Bible Dictionary (D. N. Freedman, ed.) New York, etc.: 1139-1146.

(2000) "Mesopotamian mathematics" in The History of Mathematics from Antiquity to Present. A Selective Annotated Bibliography. Revised edition on CD-ROM (J. W. Dauben, ed.), American Mathematical Society: 128-152.

Gurzadyan, V. G. (2000) "Astronomy and the fall of Babylon" Sky & Telescope 100(1): 40-45. http://arxiv.org/abs./physics/0311114

Høyrup, Jens (1996) "Changing trends in the historiography of Mesopotamian mathematics: An insider's view" HistSci 34: 1-32.

(1998) "Pythagorean 'rule' and 'theorem'- mirror of the relation between Babylonian and Greek mathematics" RUCPS 1998(3): 1-15.

Nissen, Hans J., Peter Damerow, and Robert K. Englund (1993) Archaic Bookkeeping, Early Writing and Techniques of Economic Administration in the Ancient Near East, Chicago, London Nougayrol, Jean, et al. (1968) Ugaritica V, Paris.

Powell, M. A., Jr. (1976a) "The antecedents of Old Babylonian place notation and the early history of Babylonian mathematics" HM 3: 417-439. 