My attention was wandering at some meeting or other so I started taking the square root of 6 by hand. Nothing like an arithmetic algorithm to occupy empty time and keep me calm. I chose six because it gave me so much trouble using mental math. I like algorithm so much that I decided to show that it was somehow more efficient that Newton’s method. Fat chance.
Here is the work by hand.
Notice how the correct digits appear one by one as the work of multiplying and subtraction also increases by one digit of a time.
Here is Newton’s method.
For each iteration of the formula, the accurate digits of the square root double.
In both methods the algorithm ends with a divide-like operations but the work increases on each sub-step by hand and it stays the same (a property of the division algorithm) using Newton’s method and besides the accuracy doubled. Oh well.