Babylonian numerals and problems

In Mesopotamia the scribes of Babylon and the other big cities were impressing on clay tablets economic and administrative records, literary, religious and scientific works, word-lists, and mathematical problems and tables.

Nearly all of the texts that give us our fullest understanding of Babylonian mathematics date from the Old Babylonian Period about 1800-1600 BCE.

There is also a second corpus of later evidence, from around 650 BCE to perhaps as late as the first century AD, but until recently this has been largely ignored by historians and is only now undergoing serious study.

As a result of the extensive excavations of the nineteenth century there are many more tablets available in museums and universities throughout the world than have yet been translated or even catalogued.

However, of those that have been translated, only a relatively small proportion have been shown to have mathematical content, perhaps five hundred or so, compared with upwards of 500,000 extant tablets. Nevertheless, this is a significant number when compared with the paucity of Egyptian mathematical texts.

Most of the tablets are rectangular but there are some that are round in shape. They usually fit comfortably into the palm of a hand and are about an inch thick, although some are as small a postage stamp and others the size of a large book.

Creative Commons Image Gordontour via Flickr
Cuneform tablet in the British Museum [Image: gordontour under CC-BY-NC-ND licence]

The writing is in cuneiform (‘wedge-shaped’) script and it is usually found on the front and the back of the tablets, and sometimes on the side as well. All of the Babylonian tablets are written in Akkadian, a Semitic language, although some mathematical tablets do use a few Sumerian words.

For their numeral system, the Babylonians used the sexagesimal (base 60) place-value system. Why they chose a sexagesimal system is not known but it may have been related to their astronomy, with its 360 day year. They wrote their numerals from left to right using just two symbols:

Open2 team for the unit and

Creative Commons Image Gordontour via Flickr for ten.

If there was no value in a place (which is what our zero symbol signifies) a space was sometimes left but otherwise meant 1, 60, or 3600 (or indeed 1/60, 1/3600 etc) according to context. In much later sources, mainly astronomical texts dating from 300 BCE onwards, a special symbol is introduced to mark empty places within numerals; but not at the end of a numeral, so the absolute value of the whole is still left floating.

For example (assuming we know from the context where the integer part ends and the fractional part begins):

Open2 team

denotes 2 x 60 + 31 + 4/60 + 13/3600.

Like the Egyptian texts, the mathematical tablets from the Old Babylonian period fall broadly into two categories, table texts and problem texts. Several hundred table texts, tablets consisting solely of tables of numbers, have been found, and many types of calculations appear to have been carried using them. There are tables of squares, multiplication tables, tables of reciprocals (used for division), tables of square and cube roots, combined tables where several of these are present, tables for working out compound interest, tables of weights and measures, and others. Numerical tables seem to have been a staple constituent of Babylonian life, as ubiquitous for them as is the pocket calculator for us today.

Problem texts, by contrast, are rarer, only a hundred or so tablets featuring these have been found and they seem to relate to an educational context or advanced scribal training. Some merely give the problem and the answer; others are more forthcoming on what to do to reach the answer.

They are generally written in the context of everyday life and activities, such as weighing and measuring, paying wages, and digging ditches; although they rarely appear to be using real-world examples. Typical examples involve the flooding of a field to a depth of one finger for irrigation, and finding the length of a broken reed used for measuring a field!

Almost invariably the central purpose of a Babylonian problem is the computation of a specific number. The solution is then generally given through a series of instructions. Often these instructions include a step, such as calculating a square root, which is sufficiently difficult to imply the off-stage use of a table text.

Many of the problems involve linear and quadratic equations, and there are even some involving cubic and biquadratic equations. There are also problems involving geometrical constructions but these too require the computation of a number, such as the length of a side, or an area, or a volume.

Some problems even include diagrams of basic shapes such as triangles, squares and circles. Although all the problems are formulated using specific numbers, it is evident from the methods used to solve them that the Babylonians were in possession of some general rules.