Extracting square roots: the main algorithms

There are countless methods for extracting square roots without a calculator or computer. They developed over the centuries, sometimes using subtle techniques, often devised by famous mathematicians who, of course, performed their calculations "by hand."

Geometric inspiration -------------------------------

The Babylonians already knew how to calculate approximate values of certain square roots.

For example, on the YBC 7289 tablet, immediately above the horizontal diagonal, we find a sexagesimal approximation of $\sqrt{2}\backslash sqrt\{2\}$ equal to 1 24 51 10, namely $1+{\displaystyle \frac{24}{60}}+{\displaystyle \frac{51}{{60}^2}}+{\displaystyle \frac{10}{{60}^3}},1+\; \backslash dfrac\{24\}\{60\}\; +\; \backslash dfrac\{51\}\{60^2\}\; +\; \backslash dfrac\{10\}\{60^3\},$ which gives approximately 1.41421296—already a remarkable result.

The other value, 1 42 25 28, or $1+{\displaystyle \frac{42}{60}}+{\displaystyle \frac{25}{{60}^2}}+{\displaystyle \frac{28}{{60}^3}},1+\; \backslash dfrac\{42\}\{60\}\; +\; \backslash dfrac\{25\}\{60^2\}\; +\; \backslash dfrac\{28\}\{60^3\},$ shown a little lower down, is the diagonal of the square in the diagram, whose side length is 30.