Sqrts

Assorted retorts from yahoo boards and elsewhere : Sqrts · Recipes from Home and Abroad : Square Root Extraction

How to find the square root of a decimal number without a calculator
Tibees | 1.XII.2018
https://www.youtube.com/watch?v=nAZvUnWbS8c

I

0:44 In math and science books back then, there was given an algorithm for finding square roots. Works same way before and after decimal comma, but depends on dividing the number into groups of two (decimal comma being at one division, always). (This means, first group of integers or last numeral after a comma may be a sole numeral in its group)

If I want to make this for 144, I first divide this into two groups : 1 | 44.

Then I figure out square root of first group (or of largest square number below that group), in this case 1. I put a 1 after = sign, and a 1 to the left.

THEN I do some interesting stuff.

I put a 1 below the 1 to the left. I Then multply them and put product under first group to deduct from it : 1 - 1 = 0. The 0 is put under a line.

I then put a line under the two vertical 1, I make an addition. Same height as the 0, ideally.

1+1=2. Now, I put down first numeral in second group next to the zero : 04.

I divide 04 by 2 in my head and get 2. Now I need to check it is not excessive. I put a 2 after the 1 after =, I put a 2 after the 2 under the line to the left, I put a 2 under that 2, and multiply: 2 * 22 and as I get 44, I put that under 044 (I now have added the other numeral of second group). 044 - 44 = 0.

12 is the very exact sqrt of 144 ... which we already knew, but I was demonstrating the method.

There is a similar one for Cube roots, but it's more complex.

BONUS : if the antiquated book you get this from is in Swedish, you are also likely to get an older spelling, previous to 1906!

II

1:36 No, I know that method too, but it's not the fastest one.

2:42 Wait, no, it starts looking like the theory behind the algorithm I just gave.

4:02 Yes, starts looking like what I do when offline ...

8:39 "several attempts"

I confirm. If you are putting down a first digit for a next group after the remainder from first and the division from the sum to the far left as divisor gives you a 9, you may have to lower it to 8 (I think I may even have seen 7).