YBC 7289: the square root of two on a school tablet
A small Babylonian tablet shows a square and an approximation of the square root of two. Try reading its number the way the scribe wrote it.
The square on YBC 7289
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 ↗
A teaching page or an object record?
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.
Why choose a side of 30?
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?