Estimating Square Roots of Non-Perfect Squares

Mental Calculation, Mathematics, and Puzzles
1

On my blog, I just posted a method for quickly and accurately working out square root estimates for any whole number from 1 to 1,000:
http://headinside.blogspot.com/2012/07/estimating-square-roots.html

Best of all, there’s no iterations - it’s simple and direct.

You need to know the squares of all the numbers 1 through 31 (The more you know, the higher you can go with this feat). That knowledge will, of course, give you the whole number part of the square root.

You’re basically working out how far between perfect squares a given number is, expressing that as a fraction, and then using that fraction as part of the square root.

Let’s say you’re given the number 237. First, you ask yourself what the nearest perfect square is that is equal to or less than the given number. With 237, that would be 225. 225 will be our “reference square”. Its square root, 15, will be our “reference root”.

Write down or say the reference root (15 in our example).

Next, take the given number and subtract the reference square. In our example, 237 - 225 = 12. At this point, you say or write this number as the numerator of a fraction. So, you’d continue, “and 12…”.

Finally, to work out the denominator, simply double the reference root and add 1. Our example reference root is 15, doubled is 30, plus 1 is 31. This will be your denominator, so it should be said as such, as in: “31sts”.

This doubling and adding 1 process is a shortcut for finding the difference between two consecutive perfect squares. If you have a number x, it’s square is x^2, and the next perfect square after x^2 is (x + 1)^2. To find the difference just subtract and work out the algebra:
(x + 1)^2 - x^2
(x + 1)(x + 1) - x^2
x^2 + 2x + 1 - x^2
x^2 - x^2 + 2x + 1
2x + 1

What you’ve said or written at this point is something like “15 and 12 31sts”, which is, indeed, a VERY close estimate for the square root of 237. Check on your calculator, and see for yourself.

The larger the given number, and the closer it is to a perfect square, the more accurate will be your estimate, but at worst, your estimate, when squared, will never be farther from the given number than .25!

I’ve posted a web app that will help you visualize the process:
https://dl.dropbox.com/u/23420016/geogebra/EstSquareRoots.html

There’s more details at the blog post.

Guessing square roots quickly
2

Another good way is the duplex method that i use, because i already know my squares through 100 so if i were to find the square root of 237 i already know its 15 point something. Then I find the next decimal from guess and check a couple of times. And after that I use the duplex method to get several more decimal places after that. Please let me know if you want me to explain to you how the duplex method works.

3

My main goal with the method I posted was to have a mental approach that would deliver an accurate estimate quickly, and without iteration.

In the method I posted, the accuracy increases as the numbers get bigger (and also with how close it is to a perfect square). The error between the actual square root and my method is graphed here:

As mentioned in the post above, the error of the squared result isn’t ever off by more than .25:

The duplex method looks interesting, from what I read about at the following link, but wasn’t what I was looking for when I was investigating square root methods:
http://blogannath.blogspot.com/search/label/Square%20Root

4

Ok I’m pasting this from wikipedia.

Consider the perfect square 2809 = 53^2. Use the duplex method to find the square root of 2,809.
Set down the number in groups of two digits.
Define a divisor, a dividend and a quotient to find the root.
Given 2809. Consider the first group, 28.
Find the nearest perfect square below that group.
The root of that perfect square is the first digit of our root.
Since 28 > 25 and 25 = 5^2, take 5 as the first digit in the square root.
For the divisor take double this first digit (2 · 5), which is 10.
Next, set up a division framework with a colon.
28: 0 9 is the dividend and 5: is the quotient.
Put a colon to the right of 28 and 5 and keep the colons lined up vertically. The duplex is calculated only on quotient digits to the right of the colon.
Calculate the remainder. 28: minus 25: is 3:.
Append the remainder on the left of the next digit to get the new dividend.
Here, append 3 to the next dividend digit 0, which makes the new dividend 30. The divisor 10 goes into 30 just 3 times. (No reserve needed here for subsequent deductions.)
Repeat the operation.
The zero remainder appended to 9. Nine is the next dividend.
This provides a digit to the right of the colon so deduct the duplex, 3^2 = 9.
Subtracting this duplex from the dividend 9, a zero remainder results.
Ten into zero is zero. The next root digit is zero. The next duplex is 2(3·0) = 0.
The dividend is zero. This is an exact square root, 53.

5

@greymatters, could you reupload the webapp in the dropbox link. It seems to have been removed.

6

It’s still on there, but DropBox doesn’t allow HTML to be run anymore.

Here’s a link where you can download the whole thing, and run it from your computer on your browser: Dropbox