1. Babylonian mathematics Eleanor Robson University of Cambridge

2. Outline Introducing ourselves Going to school in ancient Babylonia Learning about Babylonian numbers Learning about Babylonian shapes Question time

3. Who were the Babylonians? Where did they live? When did they live? What were their lives like?

4. We live here The Babylonians lived here, 5000- 2000 years ago

5. Cities and writing for 1500 years already Brick-built cities on rivers and canals Wealth through farming: barley and sheep Central temples, to worship many gods King Hammurabi (1792–1750 BC) Most children didn’t go to school Babylonia, 1900–1650 BC

6. Babylonian men and women

7. Cuneiform writing Wedges on clay –Whole words –Syllables –Word types –600 different signs Sumerian language –No known relatives Akkadian language –Related to Hebrew, Arabic, and other modern Middle Eastern languages

8. Cuneiform objects

9. Professional scribes Employed by: –Temples –Palaces –Courts of law –Wealthy families Status: –Slaves –Senior officials –Nobility In order to write: –Receipts and lists –Monthly and annual accounts –Loans, leases, rentals, and sales –Marriage contracts, dowries, and wills –Royal inscriptions –Records of legal disputes –Letters

10. I’m an archaeologist of maths Archaeology is the study of rubbish –To discover how people lived and died –To discover how people made and used objects to work with and think with Doing maths leaves a trail of rubbish behind I study the mathematical rubbish of the ancient Babylonians

11. Imagine an earthquake destroys your school in the middle of the night … An archaeologist comes to your school 500 years from now … What mathematical things might she find in your school? What would they tell her about the maths you do?

12. Some mathematical things in modern schools Text books and exercise books Scrap paper and doodles Mathematical instruments from rulers to calculators Mathematical displays from models to posters Computer files and hardware

13. But isn’t maths the same everywhere? Two different ways of thinking about maths: Maths is discovered, like fossils –Its history is just about who discovered what, and when Maths is created by people, like language –Its history is about who thought and used what, and why

14. The archaeology of Babylonian maths Looking at things in context tells us far more than studying single objects What sort of people wrote those tablets and why? Tablets don’t rot like paper or papyrus do They got lost, thrown away, or re-used Archaeologists dig them up just like pots, bones or buildings

15. The ancient city of Nippur

16. Maths at school: House F A small house in Nippur, 10m x 5m Excavated in 1951 From the 1740s BC 1400 fragments of tablets with school exercises –Tablets now in Chicago, Philadelphia, and Baghdad Tablet recycling bin Kitchen with oven Room for a few students

17. The House F curriculum Wedges and signs People’s names Words for things (wood, reed, stone, metal, …) How cuneiform works Weights, measures, and multiplications Sumerian sentences Sumerian proverbs Sumerian literature

18. Babylonian numbers Different: cuneiform signs pressed into clay –Vertical wedges 1–9 –Arrow wedges 10–50 Different/same: in base 60 –What do we still count in base 60? Same: order matters –Place value systems Different: no zero – and no boundary between whole numbers & fractions

19. 1 52 30 15230 Base 10 equivalent 1 x 360052 x 60306750 1 x 605230/60112 1/2 152/6030/36001 7/8

20. Playing with Babylonian numbers Try to write: –32 –23 –18 –81 –107 –4 1/2 Think of a number for your friend to write. Did they do it right?

21. Multiplication tables 130 21 31 30 42 52 30 63 73 30 84 94 30 105 115 30 [12]6 136 30 …

22. … continued [147] [157 30] 16[8] 17[8 30] 189 20-19 30 2010 3015 4020 5025

23. Practicing calculations 5 15 5 15 27 33 45 5.25 x 5.25 27.5625 or 325 x 325 = 105,625

24. Was Babylonian maths so different from ours? Draw or imagine a triangle

26. Two Babylonian triangles

27. Cultural preferences Horizontal base Vertical axis of symmetry Equilateral Left-hand vertical edge Hanging right-angled triangle or horizontal axis of symmetry Elongated

28. A Babylonian maths book front back

29. What are these shapes? The side of the square is 60 rods. Inside it are: o4 triangles, o16 barges, o5 cow's noses. What are their areas? "Triangle" is actually santakkum "cuneiform wedge" — and doesn't have to have straight edges

30. Barge and cow’s nose

31. A father praises his son’s teacher: “My little fellow has opened wide his hand, and you made wisdom enter there. You showed him all the fine points of the scribal art; you even made him see the solutions of mathematical and arithmetical problems.”