I was reading the example given in the Wikipedia article for the Babylonian method of computing the square root, and I wondered why did they set the starting $x_0$ to 600:
To calculate $\sqrt S$, where S = 125348, to six significant figures, use the rough estimation method above to get:
$x_0 = 6 * 10^2$ = 600.000
Did they pick 6 because the number $S$ has 6 digits, and multiplied by $10^2$ to get a number that has half as many digits as $S$?
Update: another reason I'm asking is that I've seen this algorithm that calculates the square root efficiently, but which lacks explanatory comments. I am fairly confident that what it does is compute the square root as if the input was a perfect square of a power of two, but I'm not sure if that's the optimal way in a wider mathematical sense.