The Powers of 9 and Related Mathematical Tables from Babylon

TL;DR

This paper expands knowledge of Late-Babylonian mathematics by analyzing 16 new fragments, revealing large factorization tables, reciprocal tables, squares, and multiplications, showcasing advanced number crunching techniques from ancient Mesopotamia.

Contribution

It introduces previously unknown large factorization tables and provides new insights into Babylonian computational methods and number tables from the late-Babylonian period.

Findings

Discovery of large sexagesimal factorization tables

Identification of extensive reciprocal and square tables

Analysis of ancient multiplication techniques

Abstract

Late-Babylonian mathematics (450-100 BC), represented by some 60 cuneiform tablets from Babylon and Uruk, is incompletely known compared to its abundantly preserved, well-studied Old-Babylonian predecessor (1800-1600 BC). With the present paper, 16 fragments from Babylon, probably belonging to 13 different tablets, are added to this corpus. Two remarkable tablets represent a hitherto unknown class of very large factorization tables that can be adequately described as Babylonian examples of number crunching (Section I). In these tables a very large sexagesimal number representing a small factor (9 or 12) raised to a high power, or a product of such numbers, is repeatedly divided by its constituent factors (9 or 12), very likely until 1 is reached. In Text A the initial number is a 25-digit number equivalent to 9 to the power 46; in Text B it is a 30-digit number equivalent to 9 to the power 11 times 12 to the power 39 - the longest number attested in ancient Mesopotamia and, probably, in all antiquity. Most other fragments belong to tables with reciprocals (Section II) and squares (Section III). Finally, two fragments contain multiplications of one kind or another (Section IV).