layout: topic categories: history, ancient-world nodeid: bookofproofs$1637 orderid: 5600 parentid: bookofproofs$610 title: Sumerian Mathematics description: ABOUT 2050 BC: PLACE-VALUE SEXAGESIMAL SYSTEM ★ history of mathematics ✚ science ➜ visit BookOfProofs now! references: bookofproofs$641 keywords: mathematics,sumerian contributors: bookofproofs
In comparison to the [Egyptian mathematics][bookofproofs$8246], 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 ($60$ seconds).