Calculating Square Roots

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?

A reasonable explanation of the Newtonian method is here:

Basically, guess.


  1. subtract the input from the square of your guess
  2. divide this by double your guess
  3. subtract this from your guess

This will give you a better approximation.

Replace your guess with the new approximation and repeat these steps to arrive at a new approximation.

The more repetitions of this algorithm you do the closer you’ll get to an accurate answer.

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’s the algorithm I learned in high school, in the United States, no less: A geometric view of the square root algorithm

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.

  1. Divide that into the original number (if your guess is close, this answer should also be close to the square root right?)
  2. Take the average of that answer and the guess
  3. 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.”

Woohoo! I was finally able to contribute to these forums with a reasonable answer!

Sorry, I’ve read this forums for quite awhile, and have simply never been able to add to the answers provided by the resident geniuses before. :slight_smile:

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?