YBC 7289: the square root of two on a school tablet.
A small round clay tablet from the Old Babylonian period shows a square with its diagonals and a number close to the square root of two. Read the numbers as a scribe would.
Start with the object
An educational page from St Lawrence University describes YBC 7289 as a round school tablet of unknown provenance from the Old Babylonian period, held in the Yale Babylonian Collection. It carries a square with both diagonals drawn in. St. Lawrence University educational resource ↗
Read the numbers
On one side of the square is the number 30. Along a diagonal is 1,24,51,10, and below it 42,25,35. These are written in base sixty. Multiplying 30 by 1,24,51,10 gives 42,25,35, so the natural reading is that a square with side 30 has a diagonal of 42,25,35. St. Lawrence University educational resource ↗
How close it is
The ratio of diagonal to side in a square is the square root of two, which cannot be written exactly in base sixty. The number 1,24,51,10 is therefore an approximation. In decimal it is about 1.414213, within a few parts in ten million of the true value. The same coefficient appears in an Old Babylonian coefficient list, so the scribe was using a standard value. St. Lawrence University educational resource ↗
Try the research habit
The research record notes that the exact Yale catalogue entry should replace an educational page as the primary locator. Treat the page as a good guide to the reading, and the museum record as the authority on the object.
A question to carry forward
Why would a school exercise choose a side of 30? The source explains that this choice makes the diagonal equal to the reciprocal of the coefficient. What does that tell you about how scribes valued reciprocals?