Example

**Input**: 420

**Output**: 20.493901531919196

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The square root of a number S is denoted $\sqrt{S}$ with the property that $\sqrt{S} \times \sqrt{S} = S$. It's easy to calculate for square numbers, like $3 \times 3 = 9$ so $\sqrt{9} = 3$. For other numbers it's not as easy but the ancient Babylonians seemed to know how to calculate them pretty accurately (see the figure and caption).

We have no idea how they calculated square roots so accurately, but a method they might have used has been called the Babylonian method. The basic idea is to start with an initial guess $x_0$ for $\sqrt{S}$. Then we can keep improve on this guess: if our current guess, $x_n$, is an overestimate to $\sqrt{S}$ then $S/x_n$ will be an underestimate and vice versa, so averaging the two should produce a better guess: $$ x_{n+1} = \frac{1}{2} \left( x_n + \frac{S}{x_n} \right) $$ We can use this formula to calculate better and better guesses. Given a number $S$, calculate and return its square root using the Babylonian method.

**Input**:
A number $S \ge 0$.

**Output**:
The square root of $S$ approximated using the babylonian method. Your answer should be accurate to 10 significant
figures. That is, it should be correct to within 1 part in 10 billion, or the relative error should be lower than
$10^{-10}$). You can calculate the relative error of your guess $x_n$ using $|x_n^2 - S| / S$ where $|\cdotp|$ is
the absolute value (or `abs`) function.

Difficulty | Timesink | ||
---|---|---|---|

Function | babylonian_sqrt(n) |

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- The method we used here is also called Heron's method after the Hero of Alexandria who explicitly described it in 60 AD.
- The Babylonians did lots of other cool stuff that was way ahead of their time like calculate Jupiter’s position from the area under a time-velocity graph .

- Babylon and the square root of 2, The Azimuth Project
- Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context , David Fowler & Eleanor Robson, Historia Mathematica, Volume 25, Issue 4, pp. 366-378 (1998).

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