h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Why does it work to double the square root(s) in the "quotient" before multiplying it by 10?

It's a bit tricky to explain, but if you look at a number, this method, and the solution, you can do some maths in reverse to show why you do the doubling.

If the solution is a number abc, then the square is (100*a + 10*b + c)^2 = 10,000*a^2 + 2000*a*b + 100*b^2 + 200*a*c + 20*b*c + c^2

We could add more digits, but then things would get a lot more complex. And we could also discuss irrational roots, but lets not.

The square of abc has 5 or 6 digits, the number a has to be the largest one whose square is less than the first digit (if five digits) or the number represented by the first two digits. We can see that this fits with the factor 10,000 which shifts things five digits.

In the next step we take a, double it add b on the end and multiply by b. That's ( 2*a*10 + b ) * b = 20*a*b + b^2 . Multiply this by 100 to shift things to the 3rd and 4th digits of the square and you get 2000*a*b + 100*b^2 which you might recognise from the formula above.

The rest of the explanation is left as an exercise for the reader