Yet another method for calculating square roots by hand... A Microsoft employee sent this to me a long, long time ago. It's essentially the same as the method described by Dr. Math, but it has more explanation and takes into account negative partial dividends. This links to the old web site:
Very detailed instructions and explanation for calculating square roots by hand.

Re: How to Calculate Square Roots without a Calculator How to find the square root of a number without a calculator AND without pen and paper!

For those of you who are over 50 years old, you probably remember the paper method of calculating square roots as described by Dr. Math and Yet another method referenced by Clay.

To find square roots without a scientific calculator, Clay's method works very well. He'll just have to repeat the same step a few times to get a better answer. Like he said, eventually, it will be right. This is the method by iterations, as higher mathematics call it. In fact, there is a method devised by Newton, so we call it Newton's method of calculating square roots.

It is almost identical to Clay's method, except that to get a better answer, we do not simply divide the original number N by the estimate, but we add the residual divided by twice the estimate, E. If it all sounds complicated, an example is in order.
Let's try to calculate the square root of 24.6. As Clay did, the initial estimate is 5. So N=24.6, and
E=5.
The Newton's method says a better estimate, E1 is
E1=E+(N-E*E)/(2E)
=5-(24.6-5*5)/(2*5)
=5-0.4/10
=4.96 (error = 0.0033%)
This can easily be handled using mental arithmetic, and is pretty close to the real value of 4.9598.
Continuing to use mental arithmetic (see Clay's article on multiplication tricks), we can mentally calculate
4.96*4.96=5*5-2*0.04*5+0.0016=24.6016
This time the residue is 0.0016, divided by 2*4.96=9.92 gives the next estimate,
E2=E1-(N-E1*E1)/(2*E1)
=4.96-.0016/9.92¸
Now, we can look at how we can handle the division. The approximation
0.0016/10=0.00016 is a very close. In fact, it gives E2=4.95984, the error is 0.0000261%!

However, if you want to push your mental arithmetic, the division mentally is possible! You notice that dividing by 9.92 instead of 10 gives 0.8% more to the result, which we can simply add to 0.00016+0.00000128=0.00016128
Therefore
E2=E1-0.00016128=4.95983872, compared to the actual value of 4.959838707 (error of 0.0000003%). I think we fare better than calculators displaying up to 8 digits, using just our brains, not even pencil and paper!

By the way, Clay, this may answer one of your reader's question about a colleague being able to do square roots mentally, and amazingly accurately. Now you can too! Feel free to write me for more explanations if I didn't make it clear above.

Re: How to Calculate Square Roots without a Calculator Clay has pointed out an error in the above formulas, although all the results still stand. I apologize for the oversight.
To be consistent with the calculations, the formulas should read
E1=E-(E*E-N)/(2E), and
E2=E1-(E1*E1-N)/(2E1).
Notice that I have switched around the E*E and N to change the sign to a minus.
Hope you enjoy this new Square Root Power!

Check your formulas -- they don't work (still) Sorry - but this is still wrong. The correct formula is:
E1=1/2 (E + N/E)

Using your example

E1 = 1/2 (5 + 24.6/5)

E1 = 1/2 (5 + 4.92)

E1 = 1/2 (9.92)

E1 = 4.96

square root of 24.6 is 4.9598 --- pretty close
The math WORKS in this formula!

Re: How to Calculate Square Roots without a Calculator E1=E+(N-E*E)/(2E)
=5-(24.6-5*5)/(2*5)
=5-0.4/10
=4.96 (error = 0.0033%)


Check your math...

5 + (24.6 - 5*5) / (2*5)

5 + (- 0.4) /10

4.6/10

0.46 (not even close!)

Re: How to Calculate Square Roots without a Calculator Deedee gave an expanded version of this, which is very accurate, but past the first level isn't very feasable to do in your head:

E1=1/2 (E + N/E)

I have a quick and dirty version, it's really just a refinement of Clay's version.

At step 3, after you have 4.92, instead of picking at random a number between 4.92 and 5, split the difference in half and add it to 4.92. 5 - 4.92 = .08, .08 / 2 = .04, 4.92 + .04 = 4.96

Now repeat steps 2 and 3 with your new guess, and it almost nails it. Plus, using splitting the difference as a guide and you can keep boosting your accuracy the most efficient way possible. On paper it's more work, but it's easy as pie when you just do it in your head.

Now I just need to find a trick like the X multiplication trick to make dividing these numbers easier. :P

This site rocks.

Re: How to Calculate Square Roots without a Calculator you're not following the order of operations.

it's (-0.4)/10 first, which is -0.04 then add 5

so:

5 + (24.6 - 5*5)/(2*5) = 5 + (-0.4)/10 = 5 - 0.04 = 4.96, which is very close

Here's another link: http://math.arizona.edu/~kerl/doc/square-root.html