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.”