Plimpton 322

Plimpton 322 is an ancient Babylonian clay tablet from around 1800 BC. It has a mathematical table with numbers written in cuneiform script. The table is special because it shows groups of three numbers that follow the Pythagorean theorem. This theorem helps us understand right triangles. It shows that people in Babylon knew this rule long before Greek mathematicians like Pythagoras were born.

When experts looked at the tablet in the 1940s, they saw that the Babylonians had a smart way to make these number groups, called Pythagorean triples. Some numbers are very large, which suggests they had a special method. One row shows the numbers 12709, 13500, and 18541, which all fit the Pythagorean rule.

The tablet only shows part of each number group, focusing on numbers that were easy to use in the Babylonian base-60 (sexagesimal) system. This makes scholars think the tablet might have been used for teaching math or for 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 knew a lot about numbers and shapes.

Provenance and dating

Plimpton 322 is a broken clay tablet. It is 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. It was later given to Columbia University. The tablet was probably written around 1800 BC. This is based on the style of its cuneiform script. It likely came from the ancient city of Larsa in southern Iraq.

Content

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

The table shows groups of three numbers that follow 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 show how the short side, long side, and diagonal (or hypotenuse) of these triangles are related. This tablet proves 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 learn more about what the writers meant. Researchers have found a few small copying mistakes, but overall, the tablet gives us a look at how ancient people solved math problems.

Construction of the table

Scholars have different ideas about how the numbers in the table were made. 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.

Purpose and authorship

People wonder why the ancient Babylonians made Plimpton 322. Some think it was a list of special number groups that form right triangles. Others say it might have been used to teach math problems.

One researcher, Eleanor Robson, thinks the tablet was probably made by a scribe, maybe a teacher. She believes it was a set of practice exercises. She says the Babylonians used simple math methods, like those used in their schools, instead of more complex math. The tablet’s style matches what scribes used for keeping records and teaching.