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Babylonian mathematics. Eleanor Robson University of Cambridge. Outline. Introducing ourselves Going to school in ancient Babylonia Learning about Babylonian numbers Learning about Babylonian measurement and fairness Question time. Who were the Babylonians?. Where did they live?

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Babylonian mathematics l.jpg

Babylonian mathematics

Eleanor Robson

University of Cambridge

Outline l.jpg

  • Introducing ourselves

  • Going to school in ancient Babylonia

  • Learning about Babylonian numbers

  • Learning about Babylonian measurement and fairness

  • Question time

Who were the babylonians l.jpg
Who were the Babylonians?

  • Where did they live?

  • When did they live?

  • What were their lives like?

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We livehere

The Babylonians lived here, 5000-2000 years ago

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Babylonia, 1900–1650 BC

  • 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

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Babylonian men and women

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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 Middle Eastern languages

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Employed by:



Courts of law

Rich families



Senior officials


In order to write:

Receipts and lists

Monthly and annual accounts

Loans, rentals, and sales

Marriage contracts, dowries, and wills

Royal inscriptions

Records of legal disputes


Professional scribes

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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

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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?

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Some mathematical things in modern schools the night …

  • 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

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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 archaeology of Babylonian maths

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Maths at school: single objects House F

  • A small house in Nippur, 10m x 5m

  • Dug by archaeologists in 1951

  • Used as a school in the 1740s BC — nearly 4000 years ago!

  • 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

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The House F curriculum single objects

  • Wedges and signs

  • People’s names

  • Words for things (wood, reed, stone, metal, …)

  • How cuneiform writing works

  • Weights, measures, and multiplications

  • Whole sentences

  • Sayings and proverbs

  • Stories, myths, hymns to gods

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The way it looks: single objects



Clay, paper, screen

The way they thought about it:




Was Babylonian maths so different from ours?Isn’t maths just maths?

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Babylonian numbers single objects

  • Different: cuneiform signs pressed into clay

    • Vertical wedges 1–9

    • Arrow wedges 10–50

  • Different: no zero and no decimal point

  • Different/same: in base 60

    • What do we still count in base 60?

  • Same: order matters

    • Place value systems

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1 30 single objects

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Babylonian numbers single objects

  • Try to write:

    • 32

    • 23

    • 18

    • 81

    • 167

    • 4 1/2

  • Think of a number for your friend to write. Did they do it right?

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  • 32 single objects

  • 23

  • 18

  • 81 = 60 + 21

  • 167 = 120 + 47

  • 4 1/2 = 4 + 30/60

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• The Babylonians didn’t use symbols like + or = single objects

• What do you think they wrote instead?

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A Babylonian maths book single objects



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What are these shapes? single objects

  • The side of the square is 60 rods. Inside it are:

    • 4 triangles,

    • 16 barges,

    • 5 cow's noses.

  • What are their areas?

"Triangle" is actually santakkum "cuneiform wedge" — and doesn't have to have straight edges

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Barge and cow’s nose single objects

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Babylonian measurements single objects

  • 1 finger

  • 1 cubit = 30 fingers

  • 1 rod = 12 cubits = 360 fingers

  • 1 cubit is about 50 cm

  • How long is 1 (Babylonian) finger?

  • How long is 1 rod?

  • How many m2 is 1 rod x 1 rod?

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Why did the Babylonians need to measure things? single objects

  • Why do we need to know how long or tall or big things are?

  • Can you think of any times when you settled an argument or made things fairer using maths? (Arguments about maths homework don’t count!)

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Two students quarrel: single objects

  • (Girini-isag speaks): “You wrote a tablet, but you cannot grasp its meaning. […] Go to apportion a field, and you cannot even hold the tape and rod properly; you are unable to place the field pegs; you cannotfigure out its shape, so that when wronged men have a quarrel you are not able to bring peace but you allow brother to attack brother. Among the scribes you (alone) are unfit for the clay. What are you fit for? Can anybody tell us?”

  • (Enki-manšum replies): “Why should I be good for nothing? When I go to divide a plot, I can divide it; when I go to apportion a field, I can apportion the pieces, so that when wronged men have a quarrel I soothe their hearts and […]. Brother will be at peace with brother, their hearts [soothed].”

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King Zimri-Lim (c.1780 BC) receiving measuring measurement to the kingequipment from the great goddess Inana

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The goddesses’ most prized possessions measurement to the king

  • When she entered the sixth gate, the lapis-lazuli measuring reed and measuring rope were removed from her hand.

  • "What is this?"

  • "Be satisfied, Inana, a divine power of the underworld has been fulfilled. Inana, you must not open your mouth against the rites of the underworld."

Inana’s Descent to the Underworld, ETCSL 1.7.1

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Mathematical measurement to the kingjustice in action

If a merchant gives grain or silver as an interest-bearing loan, he shall take 100 sila per gur (= 33%) as interest; if he gives silver as an interest-bearing loan, he shall take 36 grains of silver per shekel (= 20%) as interest.

King Hammurabi and the sun-god Shamash, c.1760 BC(on Hammurabi’s law code)

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Mathematical measurement to the kingjustice in action

If a (female) innkeeper refuses to accept grain for the price of beer but accepts only silver measured by the large weight, thus reducing the value of beer in relation to the value of grain, they shall establish the guilt of that (female) innkeeper and they shall throw her into the river.

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Questions measurement to the king