How do you work out "square roots" on paper ?

Question

Always wanted to know this, and have asked a few "maths teachers" but never got an answer !

Answer 1

There is an process which looks sort of like long division,

which is sometimes taught in school, but I don't remember it.

The following method of repeated estimation works quite well though.

Let's say you want the square root of N.

First take a guess (it doesn't really matter how good or bad it is).

Let's call it G.

You then average G and N/G and make that your next guess.

You keep doing that until two successive guesses are close enough.

Example:

Square root of 8733912.

We'll make a really bad guess to start with: 2.

1/2 (2 + 8733912/2) = 2183479

1/2 (2183479 + 8733912/2183479) = 1091741

1/2 (1091741 + 8733912/1091741) = 545874

1/2 (545874 + 8733912/545874) = 272944

1/2 (272944 + 8733912/272944) = 136487

1/2 (136487 + 8733912/136487) = 68275

1/2 (68275 + 8733912/68275) = 34201

1/2 (34201 + 8733912/34201) = 17228

1/2 (17228 + 8733912/17228) = 8867

1/2 (8867 + 8733912/8867) = 4925

1/2 (4925 + 8733912/4925) = 3349

1/2 (3349 + 8733912/3349) = 2978

1/2 (2978 + 8733912/2978) = 2955

2955*2955 = 8732025

8733912

2956*2956 = 8737936

That took 13 steps.

If you needed a more precise answer, you could keep going.

A slightly better guess, say 500, which still isn't very good,

takes 5 steps.

1/2 (500 + 8733912/500) = 8983

1/2 (8983 + 8733912/8983) = 4977

1/2 (4977 + 8733912/4977) = 3365

1/2 (3365 + 8733912/3365) = 2980

1/2 (2980 + 8733912/2980) = 2955

If we make a better guess, say 6000:

it only takes 4 steps.

1/2 (6000 + 8733912/6000) = 3727

1/2 (3727 + 8733912/3727) = 3035

1/2 (3035 + 8733912/3035) = 2956

1/2 (2956 + 8733912/2956) = 2955

Making a good initial guess shortens the process,

but even a really bad guess doesn't take too much longer.

.