Babylonian Mathematics develops from the times of the early Sumerians to the fall of Babylon in 539 BC in Mesopotamia, and is especially known for the development of the Babylonian Numeral System
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Whatwas math used for?
Sumerian mathematicsdeveloped probably as early as the6th Millenium BC, as a response tobureaucratic needsfor land measurement, taxation of individuals,etc. Furthermore, the Sumerians and Babylonians needed to describe rather large tochart the course of the night skyanddevelopthelunar calendar.
To make it easier to describe large numbers, the Sumerians were among the first to assign symbols to groups of objects. Starting fromthe 4th Millennium BCE, they began to useclay shapes of various sizes to represent different numerical values.To ensure that the numbers could be written with the same stylus used for the words of the text, these objects were replaced bycuneiform equivalents. Probablyin Sumeria as early as 2700 – 2300 BCE,a rudimentary model of the abacus was in use.
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The numeric system
Sumerian and Babylonian mathematicswas based on asexagesimal number system, or base 60.The Babylonian numbers, unlike those of the Egyptians, Greeks, and Romans,in the left column the figures represented the largest values,as in the modern decimal system, but withbase 60 and not 10. Furthermore, two distinct symbols were used to representthe numbers 1 – 59,a unit symbol (1) and a ten symbol (10)which were combined in a similar way to the familiar system ofRoman numerals(e.g. 23 would be shown as 23).The number 60 was represented by the same symbol as the number 1 and,due to the lack of a comma, the actual positional value of a symbol often had to be inferred from the context.
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It has been speculated that Babylonian advances in mathematics were facilitated by the fact that60 has many dividersand thecontinuous modern use of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are evidence of the ancient Babylonian system.It is for similar reasons that12has historically been a popular multiple(e.g. 12 inches, 12 months, 2 x 12 hours, 12 pence, etc). The Babylonians also developeda circular character for zero.
The use of a sexagesimal system was due to the Babylonians’ desire todevelop accurate calendars both to track the change of seasons and to predict the best times for sowing.Initially, the Babylonians believed that therewere 360 days in a year, and this formed the basis of their numerical system. Our entire system of geometry, astronomy, and dividing the day into seconds, minutes, and hours comes from this historical period.
Fractions
In thesexagesimal system, anyfraction in which the denominator is a regular number(having only 2, 3, and 5 in its prime factorization)may be expressed exactly.The table below shows the sexagesimal representation of all fractions of this typein which the denominator is less than 60. The sexagesimal values in the table belowmay be interpreted as giving the number of minutes and seconds in a given fraction of an hour; for instance, 1/9 of an hour is 6 minutes and 40 seconds.
| Fraction: | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | 1/8 | 1/9 | 1/10 |
|---|---|---|---|---|---|---|---|---|
| Sexagesimal: | 30 | 20 | 15 | 12 | 10 | 7,30 | 6,40 | 6 |
| Fraction: | 1/12 | 1/15 | 1/16 | 1/18 | 1/20 | 1/24 | 1/25 | 1/27 |
| Sexagesimal: | 5 | 4 | 3,45 | 3,20 | 3 | 2,30 | 2,24 | 2,13,20 |
| Fraction: | 1/30 | 1/32 | 1/36 | 1/40 | 1/45 | 1/48 | 1/50 | 1/54 |
| Sexagesimal: | 2 | 1,52,30 | 1,40 | 1,30 | 1,20 | 1,15 | 1,12 | 1,6,40 |
Howevernumbers that are not regular form more complicated repeating fractions.For example:
1/7 = 0;8,34,17,8,34,17 … (with the sequence of sexagesimal digits 8,34,17 repeating infinitely many times) = 0;8,34,17
1/11 = 0;5,27,16,21,49
1/13 = 0;4,36,55,23
1/14 = 0;4,17,8,34
1/17 = 0;3,31,45,52,56,28,14,7
1/19 = 0;3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41
In arithmetic the two numbers 59 and 61, are bothprime numbers implies that simple repeating fractions that repeat with a period of one or two sexagesimal digits can only have 59 or 61 as their denominators(1/59 = 0;1; 1/61 = 0;0,59), and that其他非正式质数repea分数t with a longer period.
From numbers to geometry
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The idea ofsquare numbers and quadratic equations, where the unknown quantity is multiplied by itself, was born forthe measurement of land, and theBabylonian mathematical tablets are the first proof of the solution of these equations. The Babylonian approach to solving them revolved around ageometric gamein which shapes were sliced and rearranged. At leastsome of the examples foundseem toindicate problem-solving for an end in itself rather than solving a concrete practical problem.
The Babyloniansin their buildings, design and in gaming dice as well as backgammon, usedgeometric shapes.Their geometry included the calculation of the volumes of simple shapes such ascylinders and bricks, such as theareas of triangles, rectangles, and trapezoids.
ThePlimpton 322 clay tablet(around 1800 BCE), indicates that the Babylonians may have recognized, that the square of the hypotenuse equals the sum of the square of the other two sides, centuries beforePythagoras.15 perfect Pythagorean triangles with full sides are listed on the tablet, although some claim that they were simply exercises.
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Whoinherited and developed this knowledge?
The Babylonians, Sumerians, and other inhabitants of the Euphrates valley developedthe basis of numerical notation, arithmetic, and using fractions.Their work was adopted by the Greeks, who learned mathematical techniques.Alexander the Greatafter conquering the area alongthe Silk Route from Anatolia (Turkey) to China,sent astronomical documents from Babylonia to Aristotle.The Greeks began to explore the relationships between numbers.
Info sources:
https://en.wikipedia.org/wiki/Sexagesimal
