Square Root Extraction

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

Here is a fairly normal procedure:



First group is 2. Nearest lower square is 1 of which the square root is 1. Write 1 in the result.

Write 1 to the left. Then write 1 under it. Multiply and write the result 1 under the 2.

To the left, you add 1 and 1 and get 2. Further right, you take 1 away from 2 and get 1.

Next to that 1, you put down the 8 from above, and then ...



2 in 18 is 9. So, one could write 9 in the result after 1.

However, this will lead to 29 * 9 = 267 sorry, 261 which cannot be deduced from 189 once you put down the 9 beside the 18.

Now, erase the 9 from everywhere, I'll skip trying with 8, and go to trying with next lower, 7.

Write 7 in result, 7 after 2 and 7 under that 7. Now multiply 27 * 7, this should land you with 189, which deducted from 189 leaves nothing, meaning 289 is the exact square of 17.

Is 27 * 7 really 189?



Seems so. So, sqrt(289)=17.

Here I show completed process for sqrt(121) = 11:



First group was 1, nearest lower square is 1 of which the square root is 1. Write 1 in result, 1 under the 1 to the left, multiply to get 1 to deduct from 1, add to get 2 to divide the 02, get 1 as result, put 1 in result, put 1 after the 2, put 1 under this 1, then multiply, 21 * 1 = 21, deduct from 021, leaves nothing.

Enjoy! A word of caution, though. I tried this on 2.00|00|00|00|00|00|00|00|00 ... and at about 8 decimals of the result, I got a value which did not square to 1.9999 ... plus some decimals at the end, but to 2.0000 ... plus some decimals at the end. I don't know why, generally, this should not happen, but I haven't detected my mistake, if such.

Hans Georg Lundahl
Nanterre UL
St. Timothy the Deacon
19.XII.2018