Notes on Calculating Roots

On more than one occasion I've had to calculate the square root of a number without a calculator. There's a neat way to approximate square roots in your head, but first I'll show you a way that will give you an answer as accurate as you need.

Note: I did not invent this, and am not claiming I did.

Square Roots in your Head

To approximate the square root of N:

1. Subtract the next lower perfect square.
2. Divide by the square root of the next lower perfect square.
3. Divide by 2.
4. Add to the square root of the next lower perfect square.

For example, let say you need the square root of 19.

• Subtract the next lower perfect square, 16, from 19. You get 3.
• Divide by the square root of 16, or 4. You get ¾.
• Divide by 2. You get 3/8.
• Add the square root of 16, or 4, to get 4.375.

The correct answer is 4.3588989435406735522369819838596... This is just an approximation, of course, but it's pretty close. It's a very good start for the next method.

Square Roots

To get the square root of X:

1. Guess.
2. Divide X by your guess to get R. If R equals your guess (to as many decimal places as you need), you're done.
3. Average your guess and R to get a new guess.
4. Go back to step 2.

You MUST work to one more decimal place than you're looking for. If you don't, the numbers could bounce around the right answer, but never quite get there.

This method converges surprisingly fast, even when you make a really crappy guess.

EXAMPLE: Let's say you need the square root of 19, to three decimal places. Using the above method, I know that the answer is around 4.375, so I'll start there.

• Divide 19 by 4.3750. You get 4.3429.
• Average 4.3750 and 4.3429. You get 4.3589.
• Divide 19 by 4.3589. You get 4.3589. See how you get what you started with? This is your answer.

Nth Roots

This is my own idea - wouldn't it be nice to think that I was the first one to think it up? I can't imagine that I am, but I've never seen this method anywhere else.

It is an extension of the method above, with two changes.

1. Instead of dividing by your guess, you divide by your guess to the power of N-1.
2. Instead of averaging your guess and R, you average your N-1 guesses and R.

This method also converges surprisingly fast. Good thing, because it's rather intensive. Again, you need to calculate to one more decimal place than you need.

To get the Nth root of X:

1. Guess.
2. Divide X by your guess to the power of N-1 to get R. If R equals your guess (to as many decimal places as you need), you're done.
3. Average (N-1) guesses and R to get a new guess.
4. Go back to step 2.

EXAMPLE: Let's say you need the 4th root of 19, to four decimal places. I know off the top of my head that 2⁴ is 16, so we'll start there.

• Divide 19 by 2³. You get 2.37500.
• Average three 2s and 2.37500. (2 + 2 + 2 + 2.375) / 4 = 2.09375.
• Divide 19 by 2.09375³. You get 2.07004.
• Average three 2.09375s and 2.07004. You get 2.08782.
• Divide 19 by 2.08782³. You get 2.08772.
• Average three 2.08782s and 2.08772. You get 08780.
• Devide 19 by 2.08780³. You get 2.08780. Done.

References

• E-mail me at bill@gizmology.net if you find a mistake!

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© 2004 W. E. Johns