As I was adding up some expenses on a calculator yesterday I noticed the square root button, and the memory of calculating a square root by hand just popped into my head.
I believe it was in 8th grade Math. It wouldn’t have been covered in Algebra, Geometry, Trig or PreCalc. At least I don’t think so. We were taught the method but not the theory – probably because the theory was too complex for us to absorb at our current level of math education.
Anyway, having the kind of mind that is easily distracted, I abandoned my calculations and decided to prove to myself that I could still do it, even though I probably haven’t used the knowledge in almost sixty years. I got a pencil and paper and set to work, selecting a random four digit number (3,346).
I was amazed that I was able to immediately do the calculation. I just wrote the number down, enclosed it in the square root sign and began as if I had been doing it routinely all my life. Here is my calculation. The original work was a wee bit sloppy, so I reproduced it more neatly.
The process is almost identical to long division.
(1) You start out with the number, and from the decimal point, in each direction, mark off every two digits with an apostrophe. Obviously with a whole number there will be nothing but zeroes to the right of the decimal point.
(2) Starting from the extreme left determine what whole number squared is equal to or less than that one or two digit number. In this case it is 5, because 6 squared is greater than 33. Place that number in the solution row, above 33.
(3) Square the number, in this case: 25, and subtract it from 33 and bring down “46” just as you would in long division, except that you bring down two digits. This gives you 846.
(4) Now multiply the partial answer you have achieved so far (5) by 20, resulting in 100. Determine what number when multiplied by 100 plus that number comes closest to 846 without exceeding it – and do that calculation. The choices are 1 x 101, 2 x 102, 3 x 103 etc. In this case the correct number is 7. Subtract the answer (749) from 846 and bring down “00” next to the result – just like in long division.
(5) Multiply the new partial answer (57) by 20 and proceed as in the above step – resulting in 1148 x 8 and on and on. Continue the process through as many decimals you want in your answer. In the case of 3,346 the decimal string extends to infinity – so I stopped at six.
This knowledge is totally useless nowadays. We have only to load the number and click on or press the square root key in our phone’s calculator or a hand held calculator or our PC, and voilá – an instant answer. But of course, back in the ancient times of my schooldays our only alternatives were to either find an approximation on a slide rule or actually make the calculation as above.