Why base 10 ?Why base 10 ?

The most common The most common base in historybase in history

Conflicting forcesConflicting forces

Theoretician Pragmatist

Tigris-Euphrates River Tigris-Euphrates River Valley Valley

Mesolithic Mesolithic and Neolithic and Neolithic tokens from tokens from the Middle the Middle

EastEastAs early as 9th millennium

Pictograms Pictograms

Development of Development of Pictograms into Pictograms into

Cuneiform Cuneiform

Archaic Sumerian numbersArchaic Sumerian numbers

1 10 60 600 3 600 36 000

Ancient Sumerian WritingAncient Sumerian Writing

Bulla, Calculus, and Bulla, Calculus, and SealsSeals

Bulla: binding Bulla: binding contracts contracts

in an illiterate in an illiterate societysociety

the enigma of sexagesimal the enigma of sexagesimal basebase

the enigma of the sexagesimal the enigma of the sexagesimal systemsystem

► origin unknown but…vestiges origin unknown but…vestiges remain todayremain today

► numerous hypothesesnumerous hypotheses smallest number with greatest smallest number with greatest number of dividers number of dividers mixture of pre-existent base 10 andmixture of pre-existent base 10 and the new base 60 of emigrants the new base 60 of emigrants mystical reasons- cult of the mystical reasons- cult of the celestialcelestial god god

Long division Sumerian-Long division Sumerian-stylestyle

►Example from 2650 B.C. in Example from 2650 B.C. in ŠŠuruppakuruppak

►1,152,000 s1,152,000 sìla of barleyìla of barley

►Each person must get 7 Each person must get 7 ssìlaìla

►How many people will get barley?How many people will get barley?

Šàrgal-iá Šàr-u-minŠàrgal-iá Šàr-u-min1,152,0001,152,000

Literally means Literally means

216,000 x 5 + 36,000 x 2216,000 x 5 + 36,000 x 2

oror

5 x 605 x 6033 + 2 x (10 x 60 + 2 x (10 x 6022))

how the Sumerians did their how the Sumerians did their sumssums

Calculation with pellets, cones, and Calculation with pellets, cones, and spheresspheres

After the eclipse of After the eclipse of SumerSumer

BabyloniaBabylonia

►Everyday Babylonia strictly decimalEveryday Babylonia strictly decimal

►Stopped using calculiStopped using calculi

►Most probably had a counting deviceMost probably had a counting device

►Developed the rule of positionDeveloped the rule of position

►Developed the idea of zeroDeveloped the idea of zero

Babylonian positional Babylonian positional numberingnumbering