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18th-century BC inscriptions1922 archaeological discoveries1922 in IraqBabylonian mathematics

Plimpton 322

Adapted from Wikipedia · Discoverer experience

An ancient Babylonian clay tablet showing early mathematical calculations with Pythagorean triples.

Plimpton 322 is an ancient Babylonian clay tablet that was written around 1800 BC. It contains a mathematical table with numbers written in cuneiform script. The table is very special because it shows sets of three numbers that fit the Pythagorean theorem, which helps us understand right triangles. This means the people of Babylon knew about this important rule long before Greek mathematicians like Pythagoras were born.

When experts first studied the tablet in the 1940s, they realized it showed the Babylonians had a smart way to create these number sets, called Pythagorean triples. Some of the numbers are very large, which suggests they had a special method instead of just guessing. One row, for example, shows the numbers 12709, 13500, and 18541, which all fit the Pythagorean rule.

The tablet only shows part of each number set, focusing on numbers that are easy to work with in the Babylonian base-60 (sexagesimal) system. This makes scholars think the tablet might have been used for teaching math problems or for practical tasks like building or measuring land. Even though we don’t know exactly why they made it, Plimpton 322 shows that ancient Babylonian mathematicians were very clever and had deep knowledge of numbers and shapes.

Provenance and dating

Plimpton 322 is a partly broken clay tablet, measuring about 13 cm wide, 9 cm tall, and 2 cm thick. It was bought by New York publisher George Arthur Plimpton from an archaeological dealer, Edgar J. Banks, around 1922 and later given to Columbia University. The tablet is thought to have been written around 1800 BC, based on the style of its cuneiform script, and likely comes from the ancient city of Larsa in southern Iraq.

Content

Plimpton 322 is an ancient Babylonian clay tablet with a mathematical table written around 1800 BC. The tablet has four columns and fifteen rows of numbers, written in a special ancient system called sexagesimal. Most of the numbers are still clear, but some parts are broken off.

The table shows sets of three numbers that fit a rule called the Pythagorean theorem. This theorem helps us understand right triangles — shapes with one corner that is a square of 90 degrees. The numbers on the tablet help show how the shorter side, longer side, and diagonal (or hypotenuse) of these triangles relate to each other. This tablet shows that people in Babylon were doing advanced math thousands of years ago, long before famous Greek mathematicians were born.

The columns have special names in an ancient language, and some parts of the tablet are still being studied to understand exactly what the writers meant. Researchers have also found a few small mistakes in copying the numbers, but overall, the tablet gives us a glimpse into how ancient people solved math problems.

takiltum of the diagonal
from which 1 is torn
out so that the
width comes up		ÍB.SI8 of the
width		ÍB.SI8 of the
diagonal		its
line
(1) 59 00 151 592 491st
(1) 56 56 58 14 56 15
(1) 56 56 58 14 [50 06] 15		56 073 12 01
[1 20 25]		2nd
(1) 55 07 41 15 33 451 16 411 50 493rd
(1) 53 10 29 32 52 163 31 495 09 014th
(1) 48 54 01 401 051 375th
(1) 47 06 41 405 198 016th
(1) 43 11 56 28 26 4038 1159 017th
(1) 41 33 59 03 45
(1) 41 33 [45 14] 03 45		13 1920 498th
(1) 38 33 36 369 01
01		12 499th
(1) 35 10 02 28 27 24 26 401 22 412 16 0110th
(1) 33 45451 1511th
(1) 29 21 54 02 1527 5948 4912th
(1) 27 00 03 457 12 01
[2 41]		4 4913th
(1) 25 48 51 35 06 4029 3153 4914th
(1) 23 13 46 4056
56
(alt.)		53
[1 46]
53 (alt.)		15th
d 2 / l 2 {\displaystyle d^{2}/l^{2}} or s 2 / l 2 {\displaystyle s^{2}/l^{2}} Short Side s {\displaystyle s} Diagonal d {\displaystyle d} Row #
(1).98340281191691
(1).94915863,3674,8252
(1).91880214,6016,6493
(1).886247912,70918,5414
(1).815007765975
(1).78519293194816
(1).71998372,2913,5417
(1).69270947991,2498
(1).64266944817699
(1).58612264,9618,16110
(1).562545*75*11
(1).48941681,6792,92912
(1).450017416128913
(1).43023881,7713,22914
(1).387160556*106*15

Construction of the table

Scholars have different ideas about how the numbers in the table were created. One idea is that the numbers come from special pairs of values. If we have two numbers, p and q, where p is bigger than q, we can make a triangle with sides (p² − q², 2pq, p² + q²). This creates a right triangle, where the squares of the two shorter sides add up to the square of the longest side.

Another idea is that the numbers come from pairs of "reciprocal" values. If we start with a fraction x and its reciprocal 1/x, we can use these to make a triangle. This method was used to solve other math problems at the same time, suggesting that the people who made the tablet might have used similar ideas.

Problemx1/xwidthlengthdiagonal
MS 3052 § 221/23/415/4
MS 3971 § 3a16/15(?)15/16(?)31/480(?)1481/480(?)
MS 3971 § 3b5/33/58/15117/15
MS 3971 § 3c3/22/35/12113/12
MS 3971 § 3d4/33/47/24125/24
MS 3971 § 3e6/55/611/60161/60

Purpose and authorship

People have tried to figure out why the ancient Babylonians created Plimpton 322. Some think it was a list of special number triples that work for right triangles. Others believe it might have been used to teach math problems.

One researcher, Eleanor Robson, suggests the tablet was likely made by a scribe, maybe even a teacher, as a set of practice exercises. She thinks the Babylonians used simple math methods typical for their schools, rather than advanced trigonometry. The tablet’s format fits what scribes used for record-keeping and teaching.

This article is a child-friendly adaptation of the Wikipedia article on Plimpton 322, available under CC BY-SA 4.0.

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