There are many well known ancient civilization like the Indus Valley, Egyptian, Mexican, and Mesopotamian Civilization. Modern humans is always curious to know about these ancient cultures.
We have been lucky enough because our ancestors left so many clues for us to know them. In the case of Egyptian and Mesopotamian culture, we have discovered so many artifacts by which we are able to understand almost every information about those ancient civilizations.
Here, you will find 5 ancient Mesopotamian clay tablets which will give you an Idea about Mesopotamian mathematics.
Pythagoras Clay Tablet
We already know that each and every information in ancient Mesopotamia was written on the clay tablets in cuneiform script. A Mesopotamian version of the Pythagoras theorem is available on one of the clay tablets which is believed to have been written about 1800 BC.
That tiny clay tablet has a table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists two of the three numbers in what is now called Pythagorean triples, i.e.
integers a, b, and c satisfying a2 + b2 = c2
From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem.
The main content of the tablet is a table of numbers, with four columns and fifteen rows, in Babylonian sexagesimal notation or base 60. A fourth column is just a row number, in order from 1 to 15. The second and third columns are completely visible in the surviving tablet.
Let’s take one example from above table. Equation
That tablet shows that ancient Mesopotamian people had a knowledge of the Pythagorean theorem in a more general framework. It might not exactly match with Pythagoras theorem but Mesopotamian people had their own way to use it and find out desired results from their equation.
Square Root Tablet
The tablet known as 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 known computational accuracy in the ancient world”.
The tablet is believed to be the work of a student in southern Mesopotamia from some time in the range from 1800–1600 BC.
The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers.
The first of these two, 1;24,51,10 represents the number
1 + 24/60 + 51/60*60 + 10/60*60*60 = 305470/216000 ≈ 1.414213
A numerical approximation of the square root of two that is off by less than one part in two million.
The second of the two numbers is
42;25,35 = 42 + 25/60 + 35/60*60 = 152735/3600 ≈ 42.426.
This number is the result of multiplying 30 by the given approximation to the square root of two and approximates the length of the diagonal of a square of side length 30.
Well explained: Diagonal of square of length a will √2*a
The small round shape of the tablet, and the large writing on it, suggests that it was a “hand tablet” of a type typically used for rough work by a student who would hold it in the palm of his hand.
YBC 4652 is a small fragment of an early Old Babylonian mathematical theme text from some southern site, possibly Ur. YBC 4652 is a fairly well-organized theme text written entirely in Sumerian. It originally contained 22 exercises, all on the theme,
na 4 i. pa ki.lá nu.na.tag ‘I found a stone, the weight unmarked(?)’.
Of the 22 exercises, only 7, 7-9, and 19-22 are so well preserved that they allow a detailed interpretation.
The stated problems can be reduced to “chains” of linear equations. The relatively simple solution procedures are never indicated. Here are a couple of examples:
The solution can be found in three easy steps, by repeated use of the rule of false value
- Set w3 = 13. Then w3 – 1/13-w3 = 13 – 1 = 12, igi 12. 1(00) sh. = 5 sh., w3 = 13.5 sh. = 105 sh.
- Set w2 = 11. Then w2+1/11 w2 = 11 + 1 = 12, igi 12. 105 sh. = 5:25 sh., W2 = 11. 5:25 sh. = 59:35 sh.
- Set w1 = 7. Then w1 – 1/7-w1 = 7 – 1 = 6, igi 6-59,35 sh. = 9:55 50 sh., w = 7 – 9:55 50 sh. = 1 09:30 50 sh.
Thus, the initial weight was 1 09:30 50 sh. = 1 mina 9 1/2 shekels 2 1/2 barley-corns.
This Clay tablet I am referring from the book “A Remarkable Collection of Babylonian Mathematical Texts”.
Division of Sexagesimal Number
MS 5112 clay tablet is an Old Babylonian cay tablet (ca. 1900-1600 BC) which explains about metric algebra problems for one or several squares and the division of numbers.
The problem texts were the higher mathematics of the time and were available for better students only. It was a collection of 16 mathematical problem texts. The upper half of a tablet is 8,9×9,8×2,7 cm, 2+2 columns, 125 lines in a clear minute cuneiform script.
In case of division, in most cases, divisions in Old Babylonian mathematical texts were transformed into multiplications with the reciprocal value of divider. Here are some examples of the application of this method in various exercises in MS 5112.
Line 1 from above examples
|30/16 = 30*(1/16) = 1 52 30 (Sexa) = 1.875 (Decimal)|
|30/16 = 1.875 (Modern Division)|
Line 2 from above examples
|20/1 20 = 20/1.3333333 (Deimal) = 15 (Sexa) = 15 (Decimal)|
|20/1.3333333 = 15 (Modern Division)|
The oldest known clay tablet containing mathematical computations (MS 3047) is a multiplication table from the 27th century BCE (ca. 2600-2500 BC) giving area as a product of recorded lengths. The tablet measures less than 3 inches in height and width and is less than an inch thick.
The sides of the 6 rectangles in the ratio 1:60 demonstrates that the mathematics taught in the scribe schools in Sumer before the middle of the third millennium BC was unexpectedly sophisticated.
It is a curious table of areas of a series of six rectangles, all (unrealistically) with the length 60 times as long as the width. The text ends by mentioning the sum of the six computed areas.
The first column is width, Second column is length which is 60 times of width and third column is area of rectangle by performing simple multiplication. Another interesting thing to note here is, they had a different unit to show a different range of numbers as we have in modern math (hundred, thousand, millions, etc)
As we can see that Sumerian culture were so rich in mathematics as we have discovered clay tablets for division, multiplication, linear equation and many places we see sophisticated calculation and learning examples as well. There are strong chances that modern mathematics has evolve from these ancient mathematics system.
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