Square Root Process Similar to Long Division

[Thomas Okken]

I learned the long-division-like square root algorithm from a book called Mathematics for Statistics, by W. L. Bashaw. The book was in English, a language I had only just started learning at the time, but the algorithm was explained with a worked-out example that made it easy to follow. I was a bit surprised that no one else seemed to know this algorithm and it wasn't taught in school, even though calculators were not standard equipment yet at the time.

Regarding the numerical approach, I knew the r -> (r + x/r)/2 iteration from applying Newton's method to r^2-x=0. It does require one division per step, but since it converges pretty quickly, that didn't seem like a big issue to me. I always assumed that the four-banger calculators back then used this algorithm, which would make sense because it's easy to implement, and it would be consistent with √ typically being slower than the arithmetic functions, but not very slow.