Topic: Sumerian Mathematics

In comparison to the Egyptian mathematics, the Sumerian period (3rd dynasty of Ur in Mesopotamia) was much more sophisticated. The Sumers developed a placed-valued sexagesimal system, which differs from the currently world-wide used decimal system only by the base of $60$ instead of $10$. Thus, for instance, the number $59$ would be written as a single Sumerian digit, while the number $61$ would be written as two Sumerian digits used for $1,$ i.e. as $11=1\cdot 60^1+1\cdot 60^0.$

Sumerians used clay tablets with numerical notation. Their unique symbols are shown below.


(from Wikimedia, uploaded by Josell7)

The earliest texts contain tables with symbols for $1$, $60$ and $60^2$ as well as $\frac {1}{60}$ and $\frac {1}{60^2}$ and also contain examples of calculations connected with administrative tasks like cattle breeding, or taxes.

The sexagesimal placed-valued system was superior in comparison to the Egyptian number system, however, it still had some ambiguities. In particular, for a long time, the Sumers did not have a symbol for $0$. Moreover, the same symbols of numbers could mean different numbers, depending on the context. For instance, $11$ could mean $1\cdot 60^1+1\cdot 60^0$ or $1\cdot 60^0+1\cdot 60^{-1}.$

The remnants of the sexagesimal system can be found in the modern measurements of angles ($360^\circ$), or of time ($60$ minutes, $60$ seconds).

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  1. Govers, Timothy: "The Princeton Companion to Mathematics", Princeton University Press, 2008,