Also called the Babylonian method, Heron's method is an algorithm devised by Heron of Alexandria in order to approximate the square root of a number $x$ to an arbitrary precision level. Starting with a random guess $g$, you improve the guess $g \gt 0$ by averaging it with $\frac{x}{g}$. You can continuously improve that guess by taking the result of $ \frac{g + (x / g)}{2} $ and replacing $g$ with it. This process can be expressed as the following sequence $ v_{n+1} = \frac{v_{n} + (x / v_{n})}{2} $ You can stop improving the guess $g$ once the results becomes precise enough. Precise enough means the difference between two successive guesses is less than the predetermined level of precision, or is $0$. This algorithm is an example of a function leading naturally towards its fixed point when iterated. We prove that $\sqrt{ x }$ is the fixed point of Heron's method by plugging $\sqrt{ x }$ into it. $ \begin{align} \frac{g + (x / g)}{2} & = \frac{\sqrt{ x } + \frac{x}{\sqrt{ x }}}{2} \\ & = \frac{\frac{x}{\sqrt{ x }}+ \frac{x}{\sqrt{ x }}}{2} \\ & = \frac{\frac{2x}{\sqrt{ x }}}{2} \\ & = \frac{x}{\sqrt{ x }} \\ & = \frac{x \times \sqrt{ x }}{\sqrt{ x } \times \sqrt{ x }} \\ & = \frac{x \times \sqrt{ x }}{x} \\ & = \sqrt{ x } \end{align} $ Plugging $\sqrt{ x }$ into the formula yields back $\sqrt{ x }$ indeed. The intuition for Heron's method is that $g$ is either an over or underestimate of $\sqrt{ x }$, and that subsequently $\frac{x}{g}$ is respectively either an under or overestimate of $\sqrt{ x }$. It follows the average of both is closer to $\sqrt{ x }$ than any of them. For a proof using Taylor series, see [Estimating Square Roots in Your Head](https://gregorygundersen.com/blog/2023/02/01/estimating-square-roots/).