Mathematics in ancient Mesopotamia

The Sumerian civilization flourished about 3000 BC

• Mesopotamia is the region lying between the Tigris and Euphrates rivers, which corresponds roughly to modern Iraq. The inhabitants of this regions were called Sumerians.   

Outside of Mosul, Iraq

• The Mesopotamia region was fertile, and the agriculture was the main resource of this region, but not as fertile as the Nile river valley is. The floods of the Tigris and Euphrates river were not as benign as the floods of the Nile river: they were irregular and often ruinous. Not all the territory that was close to the rivers was equally fertile.

• By 3000 BC a system of city state had developed in the lower Mesopotamia. To the best of our knowledge, the city of Baghdad was  founded much later, (around 8th century AD). These city state were in a perennial state of war. But there were also times of imperial stability, when centralized control was established. The greatest empires of the region were Babylonia  and Assyria.

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• The ancient Nimrud (a major Assyrian city, approx. 1250 BC) was among them. Until a few months ago, it was beautifully preserved

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• Nimrud was blown up by Isis on March 6, 2015• The ancient city of Hatra (approx. 300 BC) was bulldozed

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• Over the centuries, many different people lived in this area creating a collection of independent states

• Sumerians- southern part (3500-2000 BC)• Akkadians- northern part (2340 – 2180 BC)• Assyrians- North-Eastern part (1100 -612 BC)• These regions were unified (1830-1500 BC and

650-500 BC) forming the so-called Babylonia)

King Hammurabi was a Sumerian king who conquered Akkad and Assyria around 1830 BC. 1830 BC. He build new walls to protect the cities and new canals and dikes to improve crops.

Under his reign, individuals could own land around cities, and artisans and merchants could keep most profits and even formed guilds / associations

Grain was used as the medium of exchange, but they also used money (and the Egyptian didn’t)

• 1 shekel = 180 grains of barley; • 1 mina = 60 shekels

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•Under King Hammurabi, Under King Hammurabi, Mesopotamia was unified, but not Mesopotamia was unified, but not for long…for long…

Code of Hammurabi• To enforce his rule, Hammurabi collected all the

laws of Babylon in a code that would apply everywhere in the land.

• It is the most extensive law code from the ancient world (c. 1800 BC). It consists of 282 laws inscribed on a stone pillar placed in the public hall for all to see. The punishments were designed to fit the crimes (eye for an eye… )

• But the consequences for crimes depended on rank in society (i.e. only fines for nobility, no amputations of body parts …)

Religion

• The position of King was enhanced and supported by religion. Kingship was believed to be created by gods and the king’s power was divinely ordained. Kings and priests acted as interpreters of the gods as they told the people what the god wanted them to do (ie., by examining the liver or lungs of a slain sheep)

Like the Egyptians, the Like the Egyptians, the Babylonians had a Babylonians had a polytheistic religion polytheistic religion consisting of over 3600 gods consisting of over 3600 gods and demigods. and demigods. Gods were worshipped at huge temples called ziggurats. The Babylonians believed that the gods lived on the distant mountaintops, and each god had control of certain things and each city was ruled by a different god

Writing

• The Greatest contribution of Mesopotamia to western civilization was the invention of an alphabet (3000 BCE) Their words were made of a limited number of wedge-shaped symbols called cuneiform.

• The Scribes were only ones who could read and write and served as priests, record keepers and accountants

• Sumerians wrote on wet clay tablets with the point of a reed, then dried in the sun, or baked, to make a tablet. Thousands of these tablets were found buried in tombs.

• Many tablets kept in the Baghdad museum where stolen during the looting that followed the 2002 invasion.

Early cuneiform

• This tablet (87 BC) reports the • passage of the Halley’s comet

• The Babylonians used a positional base 60 system, but to represent numbers they used only 2 characters:  a 'unit' symbol and a 'ten' symbol. Since they did not have a symbol for zero (sometimes they left an empty space, but in some cases they used the same symbol that they used to indicate 20), then the symbol for 1 was used also to indicate 60, 3600, 60^3 and ... 1/60, 1/3600 ...

• There is evidence that they knew how to multiply numbers  and some evidence of the fact that they knew how to do the long division. More often, they reduced division to the multiplication by the  "base 60"   equivalent of decimals.

• Example. Divide [7, 12] by  5.

This is equivalent to multiplying [7,12] (  7x60 + 12 in base 10), by 1/5, i.e, 12/60= .12. Thus, the problem is equivalent to multiplying [7, 12] by [.12] (the multiplication is in base 60, of course), and then shifting the decimal point.

The quadratic equation

• To solve a quadratic equation the Babylonians essentially used the standard formula. They considered two types of quadratic equation, namely

• x^2 + bx = c and x^2 - bx = c• where here b, c were positive

but not necessarily integers. The form that their solutions took was, respectively

• x = √[(b/2)^2 + c] - (b/2) and x = √[(b/2)^2 + c] + (b/2).

• Problem: the area of a rectangle is [1, 0 ] and its length exceeds its breadth by 7. Find the sides of the rectangle.

• Problem: the area of a rectangle is [1, 0 ] and its length exceeds its breadth by 7. Find the sides of the rectangle.

• The breadth of the rectangle is a solution of the equation

• X(X + 7) = [1, 0] • of course, the scribe did not

write the equation in this modern form…. But the solution to the problem is calculated as in the formula for the quadratic equation !

• He computed half of 7, namely [3: 30]. Squared it to get [12: 15]. Then, added [1, 0] to get [1,12:15]. Take its square root (from a table of squares like the one in the picture) to get [8: 30] From this subtract [3: 30] to get 5 for the breadth of the rectangle.

Geometry• Babylonians knew the common rules for measuring

volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. Also, there was a recent discovery in which a tablet used π as 3 and 1/8.

• The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases.

• … and the Pythagorean theorem was also known to the Babylonians.  

• This clay tablet (circa 1800–

1600 BC) from the Yale Babylonian Collection it appears to be a practice school exercise undertaken by a novice scribe. It contains constructed illustration of a geometric square with intersecting diagonals, but also a numerical estimate of  √ 2 correct to three sexagesimal, or six decimal places. It is believed that the tablet’s author copied the results from an existing table of values and did not compute them himself. 

• The Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins. Assuming that the first number is [1: 24,51,10] = 1+24/60+ 51/60^2+ 10/60^3 = 1.414212956 while  √ 2 = 1.414213562.

• Calculating 30 x  [1: 24,51,10] gives [42: 25,35] which is the second number. In other worlds, the diagonal of a square of side 30 is found by multiplying 30 by the approximation to √ 2.