Please excuse me if this is a stupid question, I have an American High School education and we didn’t get into square roots too much.

Is there a method for working out square roots on paper? The same way one might do long division or muiltiplication?

I know that the square root of 25 is 5, but I only know that from working backwards through multiplication. If I need to know the square root of 92, I have to use a calculator with a square root function. This annoys me in some aesthetic way. Is there a way to work out square roots on paper?

The iteration method gets you closer and closer with each iteration. There is also a direct method similar to long division but I do not remember how to do it. Maybe a Google search would find it.

Here and here are a couple of the many sites describing the ‘long division’ method (sometimes called the ‘exact’ method). A Google search for [“square roots” “long division”] finds more. But compared to this, Newton’s method has the strong advantage that it’s fail-safe; if you make a mistake, it’ll still keep converging.

Newton’s method for finding square roots, which Melraidin shows, is equivalent to this:

Guess the square root.

Divide that into the original number (if your guess is close, this answer should also be close to the square root right?)

Take the average of that answer and the guess

Use that as your new guess.

The “long division” algorithm that raygirvan linked to, uses the idea that (a + b)[sup]2[/sup] is a[sup]2[/sup]+2ab+b[sup]2[/sup]. Once you have a guess a, subtract off a[sup]2[/sup] to leave (2ab+b[sup]2[/sup]), which you can divide by 2a to estimate b, since b[sup]2[/sup] is so small (if you’ve done it right). a+b is your new “guess.”

I can’t remember my calculus that well, but isn’t there a relatively simple technique for finding square roots involving the use of complex numbers and rotations?