Mesopotamian Mathematics

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math exam question on calculating a square's area

The Mesopotamians are credited with inventing mathematics. The people of Mesopotamia developed mathematics about 5,000 years ago. Early mathematics was essentially a form of counting, and was used to count things like sheep, crops and exchanged goods. Later it was used to solve more sophisticated problems related to irrigation and perhaps architecture. By the Late Babylonian period was used to solve complicated astrological and geometrical problems.

The considerable mathematical knowledge of the Babylonians was uncovered by the Austrian mathematician Otto E. Neugebauer, who died in 1990. Scholars since then have turned to the task of understanding how the knowledge was used. The archaeological collections at Columbia, Yale and the University of Pennsylvania have offered insights into this issue.

Kenneth Chang wrote in the New York Times: “Early Babylonian mathematicians who lived between 1800 B.C. and 1600 B.C. had figured out, for example, how to calculate the area of a trapezoid, and even how to divide a trapezoid into two smaller trapezoids of equal area. For the most part, Babylonians used their mathematical skills for mundane calculations, like figuring out the size of a plot of land. But on some tablets from the later Babylonian period, there appear to be some trapezoid calculations related to astronomical observations.In the 1950s, an Austrian-American mathematician and science historian, Otto E. Neugebauer, described two of them.” [Source: Kenneth Chang , New York Times, January 28, 2016]

The so-called Pythagorean theorem (“the sum of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides”) was known to the Sumerians as early as 2000 B.C. A cuneiform tablet from Tell Hamal, dated to 1800 B.C., shows an algebraic-geometrical table with triangles described by perpendicular lines drawn from the right angle to the hypotenuse. Another shows an algebraic-geometrical problem involving a rectangle whose diagonal area is given and the length and width need to be determined. The are also tablets with quadratic equations.

Websites on Mesopotamia: Internet Ancient History Sourcebook: Mesopotamia sourcebooks.fordham.edu ; International Association for Assyriology iaassyriology.com ; Institute for the Study of Ancient Cultures, University of Chicago isac.uchicago.edu ; University of Chicago Near Eastern Languages and Civilizations nelc.uchicago.edu ; University of Pennsylvania Near Eastern Languages & Civilizations (NELC) nelc.sas.upenn.edu; Penn Museum Near East Section penn.museum; Ancient History Encyclopedia ancient.eu.com/Mesopotamia ; British Museum britishmuseum.org ; Louvre louvre.fr/en/explore ; Metropolitan Museum of Art metmuseum.org/toah ; Ancient Near Eastern Art Metropolitan Museum of Art metmuseum.org; Iraq Museum theiraqmuseum ABZU etana.org/abzubib; Archaeology Websites Archaeology News Report archaeologynewsreport.blogspot.com ; Anthropology.net anthropology.net : archaeologica.org archaeologica.org ; Archaeology in Europe archeurope.com ; Archaeology magazine archaeology.org ; HeritageDaily heritagedaily.com; Live Science livescience.com/

Base 60 Numerical System and the 360-Degree Circle

The Mesopotamians numerical system was based on multiples of 6 and 10. The first round of numbers were based on ten like ours, but the next round were based on multiples of six to get 60 and 600. Why it was based on multiples of six no one knows. Perhaps it is because the number 60 can be divided by many numbers: 2, 3, 4, 5, 6, 12, 15 , 20 and 30.

The Sumerians developed a numerical system based on 60. The base 6 numerical system is the reason why Babylonians chose 12 months instead of 10 for their calendar and why hours and minutes are divided into 60 units and why we have dozens and a circle has 360 degrees. Babylonian astronomers knew the true number of days in a year, but kept it at 360 because that number was believed to be possessed with magical properties.

Babylonians devised the system of dividing a circle into 360 degrees (some say it was the Assyrians who first divided the circle). The tiny circle as a sign for a degree was probably originally a hieroglyph for the sun from ancient Egypt. A circle was used by the ancient Babylonian and Egyptian astronomers to the circle the zodiac. The degree was a way of dividing a circle and designating the distance traveled by the sun each day. It is no coincidence then that the number of degrees in a circle (360) corresponds with the days of the year on the Babylonian calendar.

Early Mesopotamia Mathematics and Irrigation

YBC 7289 shows the square root of two

J J O'Connor and E F Robertson wrote: Under the Sumerians, “writing developed and counting was based on a sexagesimal system, that is to say base 60. Around 2300 B.C. the Akkadians invaded the area and for some time the more backward culture of the Akkadians mixed with the more advanced culture of the Sumerians. The Akkadians invented the abacus as a tool for counting and they developed somewhat clumsy methods of arithmetic with addition, subtraction, multiplication and division all playing a part.[Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

On the use of mathematics in the irrigation systems of the early civilisations in Mesopotamia, Kazuo Muroi wrote: “It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers. ==

“There are several Old Babylonian mathematical texts in which various quantities concerning the digging of a canal are asked for. They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian ..., and YBC 9874 and BM 85196, No. 15, which are written in Akkadian .... From the mathematical point of view these problems are comparatively simple”. ==

Sumerian Mathematics

Nicholas Wade wrote in the New York Times: “Sumerian math was a sexagesimal system, meaning it was based on the number 60. The system “is striking for its originality and simplicity,” the mathematician Duncan J. Melville of St. Lawrence University, in Canton, N.Y. A 59 x 59 multiplication table might not seem simple, and indeed is far too large to memorize, so tablets were needed to provide essential look-up tables. But cuneiform numbers are simple to write because each is a combination of only two symbols, those for 1 and 10. [Source: Nicholas Wade, New York Times, November 22, 2010 ^=^]

“Why the Sumerians picked 60 as the base of their numbering system is not known for sure. The idea seems to have developed from an earlier, more complex system known from 3200 B.C. in which the positions in a number alternated between 6 and 10 as bases. For a system that might seem even more deranged, if it weren’t so familiar, consider this way of measuring length with four entirely different bases: 12 little units, called inches, make a foot, 3 feet make a yard, and 1,760 yards make a mile. ^=^

“Over a thousand years, the Sumerian alternating-base method was simplified into the sexagesimal system, with the same symbol standing for 1 or 60 or 3,600, depending on its place in the number, Dr. Melville said, just as 1 in the decimal system denotes 1, 10 or 100, depending on its place.The system was later adopted by Babylonian astronomers and through them is embedded in today’s measurement of time: the “1:12:33” on a computer clock means 1 (x 60-squared) second + 12 (x 60) seconds + 33 seconds.” ^=^

Lessons in Sumerian Math

another math exam similar to the one above

As well as providing a medium for the first writing, cuneiform clay tablet were the first recording medium to be used in education. Nicholas Wade wrote in the New York Times: Many of the 13 tablets at a 2010 exhibition at the Institute for the Study of the Ancient World, part of New York University, were “exercises of students learning to be scribes. Their plight was not to be envied. They were mastering mathematics based on texts in Sumerian, a language that even at the time was long since dead. The students spoke Akkadian, a Semitic language unrelated to Sumerian. But both languages were written in cuneiform, meaning wedge-shaped, after the shape of the marks made by punching a reed into clay. [Source: Nicholas Wade, New York Times, November 22, 2010 ^=^]

“They include two celebrated tablets, known as YBC 7289 and Plimpton 322, that have played central roles in the reconstruction of Babylonian math. YBC 7289 is a small clay disc containing a rough sketch of a square and its diagonals. Across one of the diagonals is scrawled 1,24,51,10 — a sexagesimal number that corresponds to the decimal number 1.41421296. Yes, you recognized it at once — the square root of 2. In fact it’s an approximation, a very good one, to the true value, 1.41421356.^=^

“Below is its reciprocal, the answer to the problem, that of calculating the diagonal of a square whose sides are 0.5 units. This bears on the issue of whether the Babylonians had discovered Pythagoras’s theorem some 1,300 years before Pythagoras did. No tablet bears the well-known algebraic equation, that the squares of the two smaller sides of a right-angled triangle equal the square of the hypotenuse. But Plimpton 322 contains columns of numbers that seem to have been used in calculating Pythagorean triples, sets of numbers that correspond to the sides and hypotenuse of a right triangle, like 3, 4 and 5. ^=^

“Plimpton 322 is thought to have been written in Larsa, just north of Ur, some 60 years before the city was captured by Hammurabi the lawgiver in 1762 B.C. Other tablets bear lists of practical problems, like calculating the width of a canal, given information about its other dimensions, the cost of digging it and a worker’s daily wage. With some tablets the answers are stated without any explanation, giving the impression that they were for show, a possession designed to make the owner seem an academic.” ^=^

Babylonian Mathematics

J J O'Connor and E F Robertson wrote: “The Babylonians had an advanced number system, in some ways more advanced than our present systems. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number... As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation. ==

“Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 B.C.. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 which stands for 82 = 1, 4 = 1 × 60 + 4 = 64 and so on up to 592 = 58, 1 (= 58 × 60 +1 = 3481). ==

“The Babylonians used the formula ab = [(a + b)2 - a2 - b2]/2 to make multiplication easier. Even better is their formula ab = [(a + b)2 - (a - b)2]/4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer. ==

“Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a/b = a × (1/b) so all that was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion. Of course these tables are written in their numerals, but using the sexagesimal notation we introduced above, ==

“Now the table had gaps in it since 1/7, 1/11, 1/13, etc. are not finite base 60 fractions. This did not mean that the Babylonians could not compute 1/13, say. They would write 1/13 = 7/91 = 7 × (1/91) = (approx) 7 × (1/90) and these values, for example 1/90, were given in their tables. In fact there are fascinating glimpses of the Babylonians coming to terms with the fact that division by 7 would lead to an infinite sexagesimal fraction. A scribe would give a number close to 1/7 and then write statements such as (see for example [5]):-

Babylonian numerals

Image Sources: Wikimedia Commons

Text Sources: Internet Ancient History Sourcebook: Mesopotamia sourcebooks.fordham.edu , National Geographic, Smithsonian magazine, especially Merle Severy, National Geographic, May 1991 and Marion Steinmann, Smithsonian, December 1988, New York Times, Washington Post, Los Angeles Times, Discover magazine, Times of London, Natural History magazine, Archaeology magazine, The New Yorker, BBC, Encyclopædia Britannica, Metropolitan Museum of Art, Time, Newsweek, Wikipedia, Reuters, Associated Press, The Guardian, AFP, Lonely Planet Guides, “World Religions” edited by Geoffrey Parrinder (Facts on File Publications, New York); “History of Warfare” by John Keegan (Vintage Books); “History of Art” by H.W. Janson Prentice Hall, Englewood Cliffs, N.J.), Compton’s Encyclopedia and various books and other publications.

Last updated July 2024