Ancient Babylonian Mathematics

YBC 7289 is an Old Babylonian clay tablet. Begin with what is visible: a square, two diagonals and grouped wedge marks. Do not name the numbers yet.

An artifact can support an observation before it supports an interpretation. The photograph records the surviving tablet; our normalized sign diagrams are teaching reconstructions.

A normalized teaching diagram uses one sign for 1 and another for 10. Repeated signs build the values within one place. The ancient impressions varied; the diagram is not a claim that every wedge had one identical shape.

Successive places have values …, 60², 60¹, 60⁰, 60⁻¹, 60⁻², …. In modern transcription, 1,3 means one group of sixty and three units: 1 × 60 + 3 = 63. The comma is modern punctuation.

Try reading 1,12 and 2,5, then encode 125. Since 125 = 2 × 60 + 5, its sexagesimal transcription is 2,5.

Places can continue to the right of the units place. We write 0;30 = 30/60 = 1/2. The semicolon is a modern transcription convention, not a symbol pressed into the tablet.

Multiplying by 0;30 divides by 2 because 2 × 0;30 = 1. Reciprocal tables made many divisions into multiplications, a natural fit for a positional base-60 system.

Use the reciprocal 1/6 = 0;10. Six groups of ten sixtieths make one whole.

Then 37 ÷ 6 = 37 × 0;10 = 6;10. In modern fraction notation, 6;10 = 6 + 10/60 = 6 + 1/6.

Work silently for three minutes before requesting a hint. Read A as four consecutive sexagesimal places and write an exact place-value sum. Then conjecture what A might describe.

Treat A and B as two undecoded inscriptions at first. The staged reveal preserves the distinction between decoding marks, finding a numerical relationship and interpreting what the numbers mean.

For a modern unit square, the Pythagorean theorem gives d² = 1² + 1² = 2, so the positive diagonal length is d = √2.

This derivation explains why the decoded value is relevant to a square. It is our modern explanatory bridge. It does not claim that YBC 7289 contains Greek notation, this proof, or a recorded algorithm for producing the approximation.

A problem associated with YBC 6967 asks for a smaller and a larger number. Their difference is 7, and at the scale used here their product is 1,0 — that is, 60 in decimal notation. A rectangular area story can help us picture the problem, but that setting is a modern classroom adaptation.

Try factor pairs first. Then compare your method with a translated procedure: halve the difference, square it, add the product, take the square root, then add and subtract the half-difference.

Artifact evidence includes the surviving layout and marks. Reading those marks as sexagesimal numbers is a scholarly interpretation expressed through modern transcription. The unit-square equation is a derivation. A proposed square-root algorithm is a reconstruction unless the procedure itself is documented.

Plimpton 322 can prompt an open question about the purpose of a mathematical table, but contested interpretations should remain questions rather than settled labels.

Base 60 changes the size of each place and makes many useful fractions terminate. A reciprocal table turns division into multiplication. These are connected features of a calculation system, not isolated tricks.

Exit recall: state the place values in 1;24, explain one step in 37 ÷ 6, and label one statement from today as artifact evidence, derivation, numerical experiment, reconstruction or open question.