# Babylonian mathematics

Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits.
1 + 24/60 + 51/602 + 10/603 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...

Babylonian mathematics (also known as Assyro-Babylonian mathematics[1][2][3][4][5][6]) are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited.[7] With respect to 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. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia.[7]

In contrast to the scarcity of sources in Egyptian mathematics, 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 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to ${\displaystyle {\sqrt {2}}}$ accurate to three significant sexagesimal digits (about six significant decimal digits).

## Origins of Babylonian mathematics

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.[8] 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.[9]

## Babylonian numerals

The Babylonian system of mathematics was a sexagesimal (base 60) numeral system. From this we derive the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle.[10] The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (including those that are themselves composite), facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians 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).[11]

## Sumerian mathematics

The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[12]

## Old Babylonian mathematics (2000–1600 BC)

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.

Clay tablet, mathematical, geometric-algebraic, similar to the Pythagorean theorem. From Tell al-Dhabba'i, Iraq. 2003-1595 BCE. Iraq Museum
Clay tablet, mathematical, geometric-algebraic, similar to the Euclidean geometry. From Tell Harmal, Iraq. 2003-1595 BCE. Iraq Museum

### 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:

${\displaystyle ab={\frac {(a+b)^{2}-a^{2}-b^{2}}{2}}}$
${\displaystyle ab={\frac {(a+b)^{2}-(a-b)^{2}}{4}}}$

to simplify multiplication.

The Babylonians did not have an algorithm for long division.[13] Instead they based their method on the fact that:

${\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}$

together with a table of reciprocals. Numbers whose only prime factors are 2, 3 or 5 (known as 5-smooth or regular numbers) 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 approximation such as:

${\displaystyle {\frac {1}{13}}={\frac {7}{91}}=7\times {\frac {1}{91}}\approx 7\times {\frac {1}{90}}=7\times {\frac {40}{3600}}={\frac {280}{3600}}={\frac {4}{60}}+{\frac {40}{3600}}.}$

### Algebra

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of 2 in four sexagesimal figures, 1;24,51,10,[14] which is accurate to about six decimal digits,[15] and is the closest possible three-place sexagesimal representation of 2:

${\displaystyle 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {30547}{21600}}=1.41421{\overline {296}}.}$

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:

${\displaystyle \ x^{2}+bx=c}$

where b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is:[citation needed]

${\displaystyle x=-{\frac {b}{2}}+{\sqrt {\left({\frac {b}{2}}\right)^{2}+c}}}$

and they found square roots efficiently using division and averaging.[16] They always used the positive root because this made sense when solving "real" problems[citation needed]. 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:

${\displaystyle \ ax^{3}+bx^{2}=c.}$

Multiplying the equation by a2 and dividing by b3 gives:

${\displaystyle \left({\frac {ax}{b}}\right)^{3}+\left({\frac {ax}{b}}\right)^{2}={\frac {ca^{2}}{b^{3}}}.}$

Substituting y = ax/b gives:

${\displaystyle y^{3}+y^{2}={\frac {ca^{2}}{b^{3}}}}$

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.

### 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.[17][18]

### Plimpton 322

The Plimpton 322 tablet contains a list of "Pythagorean triples", i.e., integers ${\displaystyle (a,b,c)}$ such that ${\displaystyle a^{2}+b^{2}=c^{2}}$. The triples are too many and too large to have been obtained by brute force.

Much has been written on the subject, including some speculation (perhaps anachronistic) as to whether the tablet could have served as an early trigonometrical table. Care must be exercised to see the tablet in terms of methods familiar or accessible to scribes at the time.

[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems.

(E. Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 202).

### 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. They 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.[19] 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.[20][21][22]

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.[23]

The ancient Babylonians had known of theorems concerning 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.[24]

The Babylonian astronomers kept detailed records of 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.[25]

They also used a form of Fourier analysis to compute ephemeris (tables of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.[26][27][28][29] To make calculations of the movements of celestial bodies, the Babylonians used basic arithmetic and a coordinate system based on the ecliptic, the part of the heavens that the sun and planets travel through.

Tablets kept in the British Museum provide evidence that the Babylonians even went so far as to have a concept of objects in an abstract mathematical space. The tablets date from between 350 and 50 B.C.E., revealing that the Babylonians understood and used geometry even earlier than previously thought. The Babylonians used a method for estimating the area under a curve by drawing a trapezoid underneath, a technique previously believed to have originated in 14th century Europe. This method of estimation allowed them to, for example, find the distance Jupiter had traveled in a certain amount of time.[30]

## Notes

1. ^ Lewy, H. (1949). "Studies in Assyro-Babylonian mathematics and metrology". Orientalia. NS. 18: 40–67, 137–170.
2. ^ Lewy, H. (1951). "Studies in Assyro-Babylonian mathematics and metrology". Orientalia. NS. 20: 1–12.
3. ^ Bruins, E. M. (1953). "La classification des nombres dans les mathématiques babyloniennes". Revue d'Assyriologie. 47 (4): 185–188. JSTOR 23295221.
4. ^ Cazalas (1932). "Le calcul de la table mathématique AO 6456". Revue d'Assyriologie. 29 (4): 183–188. JSTOR 23284034.
5. ^ Langdon, S. (1918). "Assyriological notes: Mathematical observations on the Scheil-Esagila tablet". Revue d'Assyriologie. 15 (3): 110–112. JSTOR 23284735.
6. ^ Robson, E. (2002). "Guaranteed genuine originals: The Plimpton Collection and the early history of mathematical Assyriology". In Wunsch, C. (ed.). Mining the Archives: Festschrift for Christopher Walker on the occasion of his 60th birthday. Dresden: ISLET. pp. 245–292. ISBN 3-9808466-0-1.
7. ^ a b Aaboe, Asger (1991). "The culture of Babylonia: Babylonian mathematics, astrology, and astronomy". In Boardman, John; Edwards, I. E. S.; Hammond, N. G. L.; Sollberger, E.; Walker, C. B. F. (eds.). The Assyrian and Babylonian Empires and other States of the Near East, from the Eighth to the Sixth Centuries B.C. Cambridge University Press. ISBN 0-521-22717-8.
8. ^ Henryk Drawnel (2004). An Aramaic Wisdom Text From Qumran: A New Interpretation Of The Levi Document. Supplements to the Journal for the Study of Judaism. Vol. 86 (illustrated ed.). BRILL. ISBN 9789004137530.
9. ^ Jane McIntosh (2005). Ancient Mesopotamia: New Perspectives. Understanding ancient civilizations (illustrated ed.). ABC-CLIO. p. 265. ISBN 9781576079652.
10. ^ Michael A. Lombardi, "Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?", "Scientific American" March 5, 2007
11. ^ Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient (2001). The Historical Roots of Elementary Mathematics (reprint ed.). Courier Corporation. p. 44. ISBN 9780486139685.{{cite book}}: CS1 maint: multiple names: authors list (link)
12. ^ Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University.
13. ^ "Babylonian mathematics". Maths History.
14. ^ The standard sexagesimal notation using semicolon–commas was introduced by Otto Neugebauer in the 1930s. Neugebauer, Otto; Sachs, Abraham Joseph; Götze, Albrecht (1945), Mathematical Cuneiform Texts, American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2
15. ^
16. ^ Allen, Arnold (January 1999). "Reviews: Mathematics: From the Birth of Numbers. By Jan Gullberg". The American Mathematical Monthly. 106 (1): 77–85. doi:10.2307/2589607. JSTOR 2589607.
17. ^ Why the "Miracle of Compound Interest" leads to Financial Crises, by Michael Hudson
18. ^ Have we caught your interest? by John H. Webb
19. ^ David Gilman Romano, Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion, American Philosophical Society, 1993, p. 78. "A group of mathematical clay tablets from the Old Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of π was 3⅛ or 3.125." E. M. Bruins, Quelques textes mathématiques de la Mission de Suse, 1950. E. M. Bruins and M. Rutten, Textes mathématiques de Suse, Mémoires de la Mission archéologique en Iran vol. XXXIV (1961). See also Beckmann, Petr (1971), A History of Pi, New York: St. Martin's Press, pp. 12, 21–22 "in 1936, a tablet was excavated some 200 miles from Babylon. [...] The mentioned tablet, whose translation was partially published only in 1950, [...] states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60)2 [i.e. π = 3/0.96 = 25/8]". Jason Dyer, On the Ancient Babylonian Value for Pi, 3 December 2008.
20. ^ Neugebauer 1969, p. 36. "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse."
21. ^ Høyrup, p. 406. "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyors’ environment, possibly as a spin-off from the problem treated in Db2-146, somewhere between 2300 and 1825 BC." (Db2-146 is an Old Babylonian clay tablet from Eshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)
22. ^ Robson 2008, p. 109. "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste ‘algebra’ on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."
23. ^ Eves, Chapter 2.
24. ^ Boyer (1991). "Greek Trigonometry and Mensuration". A History of Mathematics. John Wiley & Sons. pp. 158–159. ISBN 9780471543978.
25. ^ Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 0-691-09541-8.
26. ^ Prestini, Elena (2004). The evolution of applied harmonic analysis: models of the real world. Birkhäuser. ISBN 978-0-8176-4125-2., p. 62
27. ^ Rota, Gian-Carlo; Palombi, Fabrizio (1997). Indiscrete thoughts. Birkhäuser. ISBN 978-0-8176-3866-5., p. 11
28. ^
29. ^ Brack-Bernsen, Lis; Brack, Matthias (2004). "Analyzing shell structure from Babylonian and modern times". International Journal of Modern Physics E. 13 (1): 247–260. arXiv:physics/0310126. Bibcode:2004IJMPE..13..247B. doi:10.1142/S0218301304002028. S2CID 15704235.
30. ^ Emspak, Jesse. "Babylonians Were Using Geometry Centuries Earlier Than Thought". Smithsonian. Retrieved 2016-02-01.