# What was babylonians estimation for square root 3?

We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

But how close could they approximate the value of SQRT3? I have a document that talks about the subject, but I cannot trace the answer directly from it: http://www.helsinki.fi/~whiting/roots.pdf

Maybe someone has knowledge of the method they used and reference to the old Babylonian clay tablets, where it can be verified and seen in use.

• I wrote a blog article about this a few years back. It is not difficult to calculate good approximations to $\sqrt n$, and methods for doing so are straightforward. – MJD Sep 2 '14 at 15:22
• @MJD It's interesting that essentially the same method can be used to estimate logarithms (of whole numbers) to (whole number). bases. E.g., to approximate $\log_2 5$, list powers of $2$ and powers of $5$ until you find a close match. Noticing that $2^7 = 128 \approx 125 = 5^3$ we find that $2^{7/3} \approx 5$ and therefore $\log_2 5 \approx 7/3$. – mweiss Sep 2 '14 at 17:01
• Yes, although it doesn't work as well, because perfect powers are spaced much farther apart than squares, so it's harder to find coincidences. – MJD Sep 2 '14 at 17:11

## 1 Answer

The tablet MS 3051 (Friberg, A Remarkable Collection Babylonian Mathematical Texts) deals with the calculation of the area of an equilateral triangle; the answer given there leads to the simple approximation $\sqrt{3} \approx 7/4$. This does not mean that they couldn't have done better.