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BABYLONIANMATHEMATICS

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Map of the region

The Babylonians lived in Mesopotamia,a fertile plain between the Tigris and

Euphrates river 

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• Babylonian mathematics refers to any mathematics of the peoples

of Mesopotamia (Iraq), from the days of the early Sumerians to the

fall of Babylon in 539 BC. • In contrast to the scarcity of sources in Egyptian mathematics, our 

knowledge of Babylonian mathematics is derived from some 400

clay tablets unearthed since the 1850s. •

Written in Cuneiform script, tablets were inscribed whilst the claywas moist, and baked hard in an oven or by the heat of the sun. 

• The majority of recovered clay tablets date from 1800 to 1600 BC,

and cover topics which include fractions, algebra, quadratic and

cubic equations, the Pythagorean theorem, and the calculation of 

Pythagorean triples and possibly trigonometric functions .

• The Babylonian tablet YBC 7289 gives an approximation to accurate

to nearly six

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The Babylonian Tablets

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Babylonian Numerals

• Certainly in terms of their number system the Babylonians inherited

ideas from the Sumerians and from the Akkadians.

• From the number systems of these earlier peoples came the base of 

60, that is the sexagesimal system. 

• Babylonian system was a positional base 60 system, it had some

vestiges of a base 10 system within it. This is because the 59

numbers, which go into one of the places of the system, were built

from a 'unit' symbol and a 'ten' symbol.

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Here are the 59 symbols built from these two symbols 

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• From numeral system we derive the modern day usage of 60seconds in a minute, 60 minutes in an hour, and 360 (60×6) degrees

in a circle.

• The Babylonians were able to make great advances in mathematics

for two reasons. Firstly, the number 60 is a Highly compositenumber , having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30,

facilitating calculations with fractions.

•  Additionally, unlike the Egyptians, Romans, the Babylonians and

Indians had a true place-value system, where digits written in the left

column represented larger values (much as in our base ten system:

734 = 7×100 + 3×10 + 4×1).

Number System

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• The Babylonians divided the day into 24 hours, each

hour into 60 minutes, each minute into 60 seconds .

• To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds,

is just to write the sexagesimal fraction, 5 25/60 30/3600

• We adopt the notation 5; 25, 30 for this sexagesimal

number.

•  As a base 10 fraction the sexagesimal number 5; 25, 30

is 5 4/10 2/100 5/1000 which is written as 5.425 in

decimal notation.

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Two tablets found at Senkerah on the Euphrates in 1854 date from

2000 BC. They give squares of the numbers up to 59 and cubes of 

the numbers up to 32. The table gives 82 = 1,4 which stands for 

82 = 1, 4 = 1 60 + 4 = 64

and so on up to 592 = 58, 1 (= 58 60 +1 = 3481).

The Babylonians used the formula

ab = [(a + b)2 - a2 - b2]/2

To make multiplication easier. Even better is their formula

ab = [(a + b)2 - (a - b)2]/4

which shows that a table of squares is all that is necessary to multiplynumbers, simply taking the difference of the two squares that were

looked up in the table then taking a quarter of the answer.

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• Division is a harder process. The Babylonians did not have analgorithm for long division. Instead they based their method on thefact that

a/b = a (1/b)

• so all that was necessary was a table of reciprocals. We still havetheir reciprocal tables going up to the reciprocals of numbers up toseveral billion. Of course these tables are written in their numerals,but using the sexagesimal notation we introduced above, thebeginning of one of their tables would look like

2 0; 30

3 0; 204 0; 155 0; 126 0; 108 0; 7, 309 0; 6, 4010 0; 6

12 0; 515 0; 416 0; 3, 4518 0; 3, 2020 0; 324 0; 2, 30

25 0; 2, 2427 0; 2, 13, 20

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• Now the table had gaps in it since 1/7, 1/11, 1/13, etc.are not finite base 60 fractions. This did not mean that

the Babylonians could not compute 1/13, say. Theywould write

1/13 = 7/91 = 7 (1/91) = (approx) 7 (1/90)

• and these values, for example 1/90, were given in their tables. In fact there are fascinating glimpses of theBabylonians coming to terms with the fact that divisionby 7 would lead to an infinite sexagesimal fraction.

... an approximation is given since 7 does not divide. 

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 Arithmetic

• The Babylonians made extensive use of pre-calculated tables to

assist with arithmetic. For example, two tablets found at Senkerah

on the Euphrates in 1854, dating from 2000 BC, give lists of the

squares of numbers up to 59 and the cubes of numbers up to 32.

The Babylonians used the lists of squares together with the formulas

to simplify multiplication. 

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• The Babylonians did not have an algorithm for long division. Instead

they based their method on the fact that

• together with a table of reciprocals. Numbers whose only prime

factors are 2, 3 or 5 have finite reciprocals in sexagesimal notation,and tables with extensive lists of these reciprocals have been found.

• Reciprocals such as 1/7, 1/11, 1/13, etc. do not have finite

representations in sexagesimal notation. To compute 1/13 or to

divide a number by 13 the Babylonians would use an approximationsuch as

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 Algebra

•  As well as arithmetical calculations, Babylonian mathematicians also

developed algebraic methods of solving equations. Once again,

these were based on pre-calculated tables.

• To solve a quadratic equation the Babylonians essentially used the

standard quadratic formula. They considered quadratic equations of 

the form

 –  

• where here b and c were not necessarily integers, but c was always

positive. They knew that a solution to this form of equation is

 –  • .

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• They would use their tables of squares in reverse to find square

roots. They always used the positive root because this made

sense when solving "real" problems. Problems of this type

included finding the dimensions of a rectangle given its area and

the amount by which the length exceeds the width.

• Tables of values of n3+n2 were used to solve certain cubic 

equations. For example, consider the equation

• '''Multiplying the equation by''' a2 and dividing by b3 gives

 –  

• Substituting y = ax /b gives

 –  

• which could now be solved by looking up the n3+n2 table to find

the value closest to the right hand side. The Babylonians

accomplished this without algebraic notation, showing a

remarkable depth of understanding. However, they did not have a

method for solving the general cubic equation.

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