Babylonian mathematics Eleanor Robson University of Cambridge.
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Transcript of Babylonian mathematics Eleanor Robson University of Cambridge.
Babylonian mathematics
Eleanor RobsonUniversity of Cambridge
Outline
• Introducing ourselves• Going to school in ancient Babylonia• Learning about Babylonian numbers• Learning about Babylonian shapes• Question time
Who were the Babylonians?
• Where did they live?• When did they live?• What were their lives like?
We live here
The Babylonians lived here, 5000-2000 years ago
• 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
Babylonian men and women
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
Cuneiform objects
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
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
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?
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
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
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
The ancient city of Nippur
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
19 tablets48 tablets29 tablets348tablets
3 tablets11 tablets967 tablets+ 46 tablets?tanour
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
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
1 52 30
1 52 30 Base 10 equivalent
1 x 3600 52 x 60 30 6750
1 x 60 52 30/60 112 1/2
1 52/60 30/3600 1 7/8
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?
Multiplication tables • 1 30• 2 1• 3 1 30• 4 2• 5 2 30• 6 3• 7 3 30• 8 4• 9 4 30• 10 5• 11 5 30• [12] 6• 13 6 30 …
… continued• [14 7]• [15 7 30]• 16 [8]• 17 [8 30]• 18 9• 20-1 9 30• 20 10• 30 15• 40 20• 50 25
Practicing calculations
5 155 1527 33 45
5.25x 5.25 27.5625
or 325
x 325= 105,625
Was Babylonian maths so different from ours?
• Draw or imagine a triangle
Two Babylonian triangles
Cultural preferences
• Horizontal base• Vertical axis of symmetry• Equilateral
• Left-hand vertical edge• Hanging right-angled triangle or
horizontal axis of symmetry• Elongated
A Babylonian maths book
front back
What are these shapes?
• The side of the square is 60 rods. Inside it are: o 4 triangles, o 16 barges, o 5 cow's noses.
• What are their areas?
"Triangle" is actually santakkum "cuneiform wedge" — and doesn't have
to have straight edges
Barge and cow’s nose
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.”