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Babylonian mathematics
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Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (Ancient 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 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 which include fractions, algebra, quadratic and cubic equations, the Pythagorean theorem, and the calculation of Pythagorean triples and possibly trigonometric functions (see Plimpton 322). The Babylonian tablet YBC 7289 gives an approximation to accurate to nearly six decimal places.
Babylonian numeralsThe Babylonian system of mathematics was 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 (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 composite number, having divisors 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, facilitating calculations with fractions. Additionally, unlike the Egyptians and 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).
Sumerian mathematics (3000-2300 BC)The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They 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.
Old Babylonian mathematics (2000-1600 BC)The Old Babylonian period is the period to which most of the clay tablets on Babylonian mathematics belong, 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.
ArithmeticThe 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.
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 (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
AlgebraAs 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
and 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.
GeometryThe Babylonians may have known the general rules for measuring areas and volumes. 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 p 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. Also, there was a recent discovery in which a tablet used p as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time.
TrigonometryThere is also evidence that the Babylonians first used trigonometric functions, based on a table of numbers written on the Babylonian cuneiform tablet, Plimpton 322 (circa 1900 BC), which can be interpreted as a table of secants.
Babylonian mathematics in AlexandriaIslamic mathematics in MesopotamiaSee also
External links- , with particular emphasis on Pythagorean triples.
- , taken by Bill Casselman at the Yale Babylonian Collection
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