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# The Sexagesimal Place Value Notation

The Sexagesimal Place Value Notation (SPVN thereafter) is a notation of numbers using a positional principle and the base 60. It was used mainly in mathematical texts from the end of 3^{rd} millennium to the end of 1^{st}millennium.

To understand this notation, the best is to look at school exercises intended to teach it in Old Babylonian scribal schools, for example a multiplication table. Table 2 shows the photographs and a copy of a multiplication table by 9 from Nippur (HS 217a). Looking the left hand column of the table, it is easy to guess that the vertical wedge () represents the number one, and the angle wedge () represents the number ten. The right hand column provides the products by 9 of the numbers written in the left hand column. For example, in front of (3), we see (27).

Obverse

` 1 9`

2 18

3 27

4 36

5 45

6 54

7 1.3

8 1.12

9 1.21

10 1.30

11 1.39

12 1.48

13 1.57

14 2.6

Reverse

` 15 2.15`

16 2.24

17 2.33

18 2.42

20-1 2.51

20 3

30 4.30

40 6

50 7.30

8.20 a-ra_{2} 1 8.20

**transliteration of HS 217a, school tablet from Nippur, multiplication table by 9**

Thus, in front of 7, we expect the number 63. Instead, we see the notation , where the sixty is represented by a wedge in the second place (left position), which means that this number is expressed in sexagesimal place value notation. We can transcribe our number 1.3, where the dot separates sexagesimal digits. The system works as our sexagesimal system for time expressed in hours, minutes and second, by the way inherited from Mesopotamia.

However, there is a difference with the modern time system. If we continue the reading of the multiplication table, in front of 20, we expect 180, that is, 3×60, or 3.0 in our modern sexagesimal system, but we see . This means that 3×60 is noted with exactly the same sign as the “3” in the third row of the left column. In other words, at the same time the sign denotes 3 and 3×60. When we see the sign in a multiplication table, we do not know if the wedges represent units, sixties, sixtieth, etc. The floating character of the cuneiform SPVN is a general feature in mathematical texts.

The reader can facilitate his/her calculations with SPVN by using the online tool developed by Baptiste Mélès, MesoCalc, and the attached tutorial.

## Bibliography

- Neugebauer, Otto. 1957.
*The exact sciences in antiquity*. 2ième ed. Providence: Brown University Press. - Proust, Christine. 2013. Du calcul flottant en Mésopotamie.
*La Gazette des Mathématiciens*138:23-48. - Thureau-Dangin, François. 1932.
*Esquisse d'une histoire du système sexagésimal*. Paris: Geuthner.

CP

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