History of Babylonians Mathematics

Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics

developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall

of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time

they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other

mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any

difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character

and content, for nearly two millenniums.

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 while the clay was 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 BCE, and cover topics that

include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian

tablet YBC 7289 gives an approximation to accurate to three significant sexagesimal digits (seven

significant decimal digits).

Babylonian mathematics is a range of numeric and more advanced mathematical practices in

the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian

period in the early second millennium BC due to the wealth of data available. There has been debate over

the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between

the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform

script in the Akkadian or Sumerian languages. "Babylonian mathematics" is perhaps an unhelpful term

since the earliest suggested origins date to the use of accounting devices, such as  bullae and tokens, in the

5th millennium BC.

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Babylonian Numbers

In mathematics, the Babylonians (Sumerians) were somewhat more advanced than the Egyptians.

Their mathematical notation was positional but sexagesimal (base 60).

They used no zero.

More general fractions, though not all fractions, were admitted.

They could extract square roots.

They could solve linear systems.

They worked with Pythagorean triples.

They solved cubic equations with the help of tables.

They studied circular measurement.

Their geometry was sometimes incorrect.

For enumeration the Babylonians used symbols for 1, 10, 60, 600, 3,600, 36,000, and 216,000, similar to

the earlier period. Below are four of the symbols. They did arithmetic in base 60, sexagesimal.

When they wrote "60", they would put a single wedge mark in the second place of the numeral.

When they wrote "120", they would put two wedge marks in the second place.

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Here are the 59 symbols built from these two symbols

There is no clear reason why the Babylonians selected the sexagesimal system 6. It was possibly

selected in the interest of metrology, this according to Theon of Alexandria, a commentator of the fourth

century A.D.: i.e. the values 2,3,5,10,12,15,20, and 30 all divide 60. Remnants still exist today with time

and angular measurement. However, a number of theories have been posited for the Babylonians

choosing the base of 60. For example :

1. The number of days, 360, in a year gave rise to the subdivision of the circle into 360 degrees, and that

the chord of one sixth of a circle is equal to the radius gave rise to a natural division of the circle into six

equal parts. This in turn made 60 a natural unit of counting. (Moritz Cantor, 1880)

2. The Babylonians used a 12 hour clock, with 60 minute hours. That is, two of our minutes is one minute

for the Babylonians. (Lehmann-Haupt, 1889) Moreover, the (Mesopotamian) zodiac was divided into

twelve equal sectors of 30 degrees each.

3. The base 60 provided a convenient way to express fractions from a variety of systems as may be

needed in conversion of weights and measures. In the Egyptian system, we have seen the values 1/1,1/2,

2/3, 1, 2, . . . , 10. Combining we see the factor of 6 needed in the denominator of fractions. This with the

base 10 gives 60 as the base of the new system. (Neugebauer, 1927)

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4. The number 60 is the product of the number of planets (5 known at the time) by the number of months

in the year, 12. (D. J. Boorstin, 1986)

Old Babylonian Mathematics (2000 – 1600bc)

Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, which is why the

mathematics of Mesopotamia is commonly known as Babylonian mathematics. Some clay tablets contain

mathematical lists and tables, others contain problems and worked solutions.

i) Arithmetic

The Babylonians used 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 formulae

to simplify multiplication.

ii) 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.

iii) Growth

Babylonians modeled exponential growth, constrained growth (via a form of sigmoid functions),

and doubling time, the latter in the context of interest on loans. Clay tablets from c. 2000 BCE include

the exercise "Given an interest rate of 1/60 per month (no compounding), compute the doubling time."

This yields an annual interest rate of 12/60 = 20%, and hence a doubling time of 100% growth/20%

growth per year = 5 years.

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iv) Plimpton 322

As we have seen there is solid evidence that the ancient Chinese were aware of the Pythagorean theorem,

even though they may not have had anything near to a proof. The Babylonians, too, had such an

awareness. Indeed, the evidence here is very much stronger, for an entire tablet of Pythagorean triples has

been discovered. The events surrounding them reads much like a modern detective story, with the sleuth

being archaeologist Otto Neugebauer. We begin in about 1945 with the Plimpton 322 tablet, which is now

the Babylonian collection at Yale University and dates from about 1700 BCE. It appears to have the left

section broken away. Indeed, the presence of glue on the broken edge indicates that it was broken after

excavation. What the tablet contains is fifteen rows of numbers, numbered from 1 to 15.

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v) 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. 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. The Pythagorean theorem was also known to the

Babylonians.

Babylonian texts usually approximated π≈3, sufficient for the architectural projects of the time (notably

also reflected in the description of Solomon's Temple in the Hebrew Bible). The Babylonians were aware

that this was an approximation, and one Old Babylonian mathematical tablet excavated near  Susa in 1936

(dated to between the 19th and 17th centuries BCE) gives a better approximation of π as 25/8=3.125,

about 0.5 percent below the exact value.

The "Babylonian mile" was a measure of distance equal to about 11.3 km (or about seven modern miles).

This measurement for distances eventually was converted to a "time-mile" used for measuring the travel

of the Sun, therefore, representing time.

The ancient Babylonians had known of theorems on the ratios of the sides of similar triangles for many

centuries, but they lacked the concept of an angle measure and consequently, studied the sides of triangles

instead.

The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of

the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances

measured on the celestial sphere. They also used a form of Fourier analysis to compute ephemeris (tables

of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.

Babylonian Mathematician

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Nabu - rimanni and Kidinu are two of the only known mathematicians from Babylonia.

However, not much is known about them. Historians believe Nabu - rimanni lived around 490

BC and Kidinu lived around 480 BC.

Conclusion

That Babylonian mathematics may seem to be further advanced than that of Egypt may be due to the

evidence available. So, even though scope, there remain many similarities. For example, problems

contain only specific cases. There seem to be no general formulations. The lack of notation is clearly

detrimental in the handling of algebraic problems.

Geometric considerations play a very secondary role in Babylonian algebra, even though geometric

terminology may be used. Areas and lengths are freely added, something that would not be possible in

Greek mathematics.

Overall, the role of geometry is diminished in comparison with algebraic and numerical methods.

Questions about solvability or insolvability are absent. The concept of proof is unclear and uncertain.

Overall, there is no sense of abstraction. In sum, Babylonian mathematics, like that of the Egyptians, is

mostly utilitarian but apparently more advanced.

Reference

Boyer, Merzbach (1989). A History of Mathematics Second Edition. John Wiley & Sons,.

Bunt, Jones, and Bedient (1988). The Historical Roots of Elementary Mathematics. Dover

Publications.

Burton, D. M. (2003). Burton’s The History of Mathematics: An Introduction, 5th Edition. New

York: McGraw Hill.

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……………. History of Babylonian Mathematics

http://www.math.wichita.edu/history/topics/num-sys.html#babylonian retrieved on 24 September

2015.

……………. Egytption Mathematics http:// www.slideshare.net/Mabdulhady/egyptian-

mathematics?from_search=1 retrieved on 24 September 2015.

……………. Egytption Mathematics http:// en.wikipedia.org/wiki/Egyptian_mathematics

retrieved on 24 September 2015.

……………. Egytption Mathematics

http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_arith.html retrieved on 24

September 2015.

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