Floating calculation in Mesopotamia

Sophisticated computation methods were developed 4000 years ago in Mesopotamia in the context of scribal schools. The basics of the computation can be detected in clay tablets written by young students educated in these scribal schools. At first glance, the mathematical exercises contained in school tablets seem to be very simple and quite familiar, and therefore, they have little attracted the attention of historians of mathematics. Yet if we look more closely at these modest writings, their simplicity proves deceptive. The feeling of familiarity primarily results from a projection on to the past of thought patterns ingrained from our own early learning. Careful observation of school tablets reveals subtle differences between ancient and modern notions of number, quantity, measurement unit, order, divisibility, algorithm, etc. In fact, we find a completely original mathematical world with rules of its own. This article explores the computation practices developed in scholarly milieus in the light of the mathematical basics that was taught in elementary education. Table of content 1Cuneiform mathematics ......................................................................................................3 2The problem of the orders of magnitude .............................................................................3 3Mathematics taught at Nippur around the 18th century BCE ...............................................6 The sources.........................................................................................................................6 The elementary curriculum at Nippur..................................................................................6 Metrological tables .............................................................................................................7 Reciprocal tables ................................................................................................................8 4Computing and quantifying .............................................................................................. 10 Computing surfaces and volumes ...................................................................................... 11 5Floating calculation in action: some examples .................................................................. 12 Finding the reciprocal of a number.................................................................................... 12 Floating numbers and fixed values: the example of the surface of a disk ........................... 15 6Linear and quadratic problems in YBC 4663 .................................................................... 17 Multiplications and reciprocals ......................................................................................... 17 The intervention of addition .............................................................................................. 19 7Orders of magnitude and place of the unit in the number .................................................. 22 Appendix: numbers and measurement units .......................................................................... 23 1Numbers ....................................................................................................................... 23 2Measurement units ....................................................................................................... 24 3MesoCalc, a Mesopotamian calculator .......................................................................... 24 Sources ................................................................................................................................. 25 Bibliography......................................................................................................................... 25 1 The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement n. 269804. This article is the English translation of Proust, “Du calcul flottant en Mésopotamie”, published in La Gazette des Mathématiciens 138, p. 23-48 (2014). This English version was presented at the conference "Arith 22", Ecole Normale de Lyon, June 22, 2015. General information on cuneiform mathematics, scribal schools, and place value notation is well known to specialists. However, some of the arguments developed here, especially those concerning the duality between measurement values and numbers in sexagesimal place value notation, are personal and are not the subject of consensus (see more details in [11]).