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Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of
Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the ...
, from the days of the early
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of ...
ians to the centuries following the fall of
Babylon ''Bābili(m)'' * sux, 𒆍𒀭𒊏𒆠 * arc, 𐡁𐡁𐡋 ''Bāḇel'' * syc, ܒܒܠ ''Bāḇel'' * grc-gre, Βαβυλών ''Babylṓn'' * he, בָּבֶל ''Bāvel'' * peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru'' * elx, 𒀸𒁀𒉿𒇷 ''Babi ...
in 539 BC. Babylonian mathematical texts are plentiful and well edited. 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. In contrast to the scarcity of sources in
Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for count ...
, knowledge of
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c ...
n mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in
Cuneiform script Cuneiform is a logo- syllabic script that was used to write several languages of the Ancient Middle East. The script was in active use from the early Bronze Age until the beginning of the Common Era. It is named for the characteristic wedge-s ...
, 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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet
YBC 7289 YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest ...
gives an approximation to \sqrt 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 The ancient Near East was the home of early civilizations within a region roughly corresponding to the modern Middle East: Mesopotamia (modern Iraq, southeast Turkey, southwest Iran and northeastern Syria), ancient Egypt, ancient Iran ( Elam, ...
, written in
cuneiform script Cuneiform is a logo- syllabic script that was used to write several languages of the Ancient Middle East. The script was in active use from the early Bronze Age until the beginning of the Common Era. It is named for the characteristic wedge-s ...
. Study has historically focused on the
Old Babylonian period The Old Babylonian Empire, or First Babylonian Empire, is dated to BC – BC, and comes after the end of Sumerian power with the destruction of the Third Dynasty of Ur, and the subsequent Isin-Larsa period. The chronology of the first dynast ...
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 Akkadian or Accadian may refer to: * Akkadians, inhabitants of the Akkadian Empire * Akkadian language, an extinct Eastern Semitic language * Akkadian literature, literature in this language * Akkadian cuneiform Cuneiform is a logo-syllabic ...
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.


Babylonian numerals

The Babylonian system of mathematics was a
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
(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. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a
superior highly composite number In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite ...
, 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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
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).


Sumerian mathematics

The ancient
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of ...
ians of
Mesopotamia Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the ...
developed a complex system of
metrology Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fran ...
from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with
geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.


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.


Arithmetic

The Babylonians used pre-calculated tables to assist with
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
. For example, two tablets found at Senkerah on the
Euphrates The Euphrates () is the longest and one of the most historically important rivers of Western Asia. Tigris–Euphrates river system, Together with the Tigris, it is one of the two defining rivers of Mesopotamia ( ''the land between the rivers'') ...
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: :ab = \frac :ab = \frac to simplify multiplication. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that: :\frac = a \times \frac together with a table of reciprocals. Numbers whose only
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s are 2, 3 or 5 (known as 5- smooth or regular numbers) have finite
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s 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: :\frac = \frac = 7 \times \frac \approx 7 \times \frac=7 \times \frac = \frac = \frac + \frac.


Algebra

The
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c ...
n clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
figures, 1;24,51,10,The standard sexagesimal notation using semicolon–commas was introduced by Otto Neugebauer in the 1930s. which is accurate to about six
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
digits, and is the closest possible three-place sexagesimal representation of : :1 + \frac + \frac + \frac = \frac = 1.41421\overline. As well as arithmetical calculations, Babylonian mathematicians also developed
algebraic Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
methods of solving
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
s. Once again, these were based on pre-calculated tables. To solve a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
, the Babylonians essentially used the standard
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
. They considered quadratic equations of the form: :\ 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: :x = - \frac + \sqrt and they found square roots efficiently using division and averaging. 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 ''n''3 + ''n''2 were used to solve certain cubic equations. For example, consider the equation: :\ ax^3 + bx^2 = c. Multiplying the equation by ''a''2 and dividing by ''b''3 gives: :\left ( \frac \right )^3 + \left ( \frac \right )^2 = \frac . Substituting ''y'' = ''ax''/''b'' gives: :y^3 + y^2 = \frac which could now be solved by looking up the ''n''3 + ''n''2 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 function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \f ...
s), and doubling time, the latter in the context of interest on loans. Clay tablets from c. 2000 BC 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.Why the "Miracle of Compound Interest" leads to Financial Crises
, by Michael Hudson


Plimpton 322

The Plimpton 322 tablet contains a list of " Pythagorean triples", i.e., integers (a,b,c) such that 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 '' '' is estimated as 3. They were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near
Susa Susa ( ; Middle elx, 𒀸𒋗𒊺𒂗, translit=Šušen; Middle and Neo- elx, 𒋢𒋢𒌦, translit=Šušun; Neo- Elamite and Achaemenid elx, 𒀸𒋗𒐼𒀭, translit=Šušán; Achaemenid elx, 𒀸𒋗𒐼, translit=Šušá; fa, شوش ...
in 1936 (dated to between the 19th and 17th centuries BC) gives a better approximation of as 25/8 = 3.125, about 0.5 percent below the exact value. 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 rule was also known to the Babylonians. 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 formulas 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. The
Babylonian astronomers Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations m ...
kept detailed records of the rising and setting of
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s, the motion of the
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s, and the solar and lunar
eclipse An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three c ...
s, all of which required familiarity with angular distances measured on the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...
. They also used a form of Fourier analysis to compute an
ephemeris In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...
(table of astronomical positions), which was discovered in the 1950s by Otto Neugebauer. 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 The British Museum is a public museum dedicated to human history, art and culture located in the Bloomsbury area of London. Its permanent collection of eight million works is among the largest and most comprehensive in existence. It docum ...
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 A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
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 Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...
had traveled in a certain amount of time.


See also

*
Babylonia Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c ...
*
Babylonian astronomy Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellation ...
* History of mathematics *
Islamic mathematics Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics ( Aryabhata, Brahmagupta). Important progress was made, such as ...
for mathematics in Islamic Iraq/Mesopotamia


Notes


References

* * (1991 pbk ed. ). * * * * * * * * * {{DEFAULTSORT:Babylonian Mathematics