ANCIENT MATHEMATICS*ANCIENT CIVILIZATION*

BASIC FACTS:

MESOPOTAMIAFirst and oldest world’s civilization took

place.A valley between the two rivers, Tigris

and Euphrates.Thought to be the (or at least a)“cradle of

civilization.” Delta region extremely fertile –

The“Fertile Crescent” Semi-arid climate required extensive

irrigation projects

MESOPOTAMIAN

The Sumerian Mathematical SystemSumer (a region of Mesopotamia, modern-day Iraq) was the

birthplace of writing, the wheel, agriculture, the arch, the plow, irrigation and many other innovations, and is often referred to as the Cradle of Civilization.

The Sumerians developed the earliest known writing system - a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets - and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early Egyptian mathematics. Indeed, we even have what appear to school exercises in arithmetic and geometric problems.

Contributions:A. Sumerian

A. SumerianThe Sumerian System, called "sexagesimal", combined a mundane 10... with a "celestial" 6, to obtain the base figure 60. This system is in some ways superior to our present one, and much superior to later Greek and Roman systems.

It enabled Sumerians to divide into fractions and multiply into the million, to calculate roots or raise numbers several powers.

CUNEIFORM WRITING

B. Akkadian

C. BabylonianHammurabi, founderof the Old Babylonian Empire• Code of Hammurabi -232 laws, lex talionus, an eye for an eye– If anyone strikes the body of a man higher in rank than he, he shall receive sixty blows with an ox-whip in public.

C. BabylonianMost of what we know about

Mesopotamian mathematics comes from several hundred clay tablets belonging to the Old Babylonian kingdom, around roughly 1800-1600 BCE.

•Tablets are of two kinds:– Table texts– Problem texts

Babylonian Numbers In Mathematics, the Babylonians were somewhat more

advanced than the Egyptians: Their mathematical notation was positional but

sexagesimal. They used no zero. More general fractions, though not all fractions, were

admitted. They could extract square roots. They could solve linear systems. They worked with Pythagorean triples. They solved cubic equations with the help of tables. They studied circular measurement. Their geometry was sometimes incorrect.

Babylonian Number SystemA base-60 positional system with individual numbers

formed by two different wedge-shaped marks: A horizontal wedge ‹ worth 10 and a vertical wedge ˅ worth 1.

Numbers less than 60 were written using these two symbols in a purely additive fashion.

There was no “0” or placeholder so we really can’t be sure which power of 60 is being used.

Even though the Babylonians used this system to write fractions as sexagecimals, there was no “sexagecimal point” or other way of marking where the fractional part began.

Babylonian Number System

Babylonian Number SystemThese 59 symbols would be written in place

value system based on powers of 60. Powers of 60increased from right to left, just as powers of 10 increase from right to left in our system.

Thus, writing ‹˅ ‹‹‹˅˅˅˅ ‹‹˅˅˅ would most likely represent

11 × 60^2 + 34 × 60 + 23or 41,663.