Estimating Square Roots

Published on Sunday, July 08, 2012 in , ,

purzen's (openclipart.org) thought bubble surrounding David Vignoni's square root icon over Josh Green's (subtlepatterns.com) Old Mathematics patternI've shown how to find square roots of perfect squares in past tutorials, but how do you handle numbers that aren't perfect squares?

In this tutorial, you'll learn how to quickly determine an approximate square root for any number from 1 to 1,000. The method is a little challenging, but the results are impressive and worth the work.

In the feat itself, you're going to have someone have someone enter the square root into their calculator in a way that's easy for the calculator to understand, and then have them square it. If they give you the number 269, you instantly tell them to divide 13 by 33, and then add 16, explaining that you've determined 16 and 13/33rds to be the approximate square root of 269. When they square that number, they'll see that the answer is quite close (roughly 268.761).

Before you learn this feat, there are a couple of other feats you should learn. You should be comfortable with squaring 2-digit numbers, and being able to find the square roots of perfect squares. You'll also need to know the squares of the numbers from 1 to 31 off the top of your head, in order to handle the numbers from 1 to 1,000.

During this feat, you'll be subtracting 3 digits numbers. You can brush up on your mental 3-digit subtraction with help from this video.

That's enough for the preparation, how do you actually do the feat?

Start by asking someone to take out their calculator, make sure it's cleared, and then ask them for any number from 1 to 1,000. As an example, we'll use 149, which we'll refer to as the given number.

Step 1: Find the closest perfect square that is less than or equal to the given number. We'll refer to as the reference square or ref. square, and the root of the reference square will be called the reference root or ref. root. If they happen to give you a perfect square, you can state the square root instantly (and impressively).

With 149, you should instantly recognize that the closest perfect square below it is 144 (122). So our reference square is 144 and the reference root is 12.

Step 2: From the given number, subtract the reference square, and remember this result. This result will be known as the numerator.

Starting with 149, we subtract 144 (the reference square) to get 5.

Step 3: Ask the person who suggested the given number to enter the numerator into their calculator, and then press the division key (÷).

Continuing with our example, they would enter 5, and then press the ÷ key.

Step 4: While they're entering the information from step 3 into the calculator, double your reference root and then add 1. This total will be referred to as the denominator.

The reference root is 12, so we double that to get 24, then add 1, giving a total of 25. 25 will be our denominator.

Note: You might be curious as to why you're doubling the reference root and adding 1. This is a short cut for finding the differences between your reference square, and the next perfect square.

If you think of the reference root as x, then the next number must be x + 1. For any perfect square x2, the next perfect square is (x + 1)2. The long way to determine the difference between them would be to work through the equation (x + 1)2 - x2.

However, it turns out that (x + 1)2 - x2 simplifies to 2x + 1! Doubling our reference root and adding 1 is much quicker than working through exponents!

Step 5: Have them enter the denominator into the calculator, and press the equals (=) button. The answer displayed will now be a decimal equal to the numerator divided by the denominator.

In our example, they've divided 5 by 25, which is .2.

Note: What the calculator is displaying at this point has a very useful double meaning. Our reference square in this example is 144, which is 122 (as we've already determined). The next perfect square is 132, or 169.

Picture the range of 144 to 169 as a line, and 149 as a single point along that line, as in this Wolfram|Alpha diagram. The first meaning of the 5/25 is that our given number 149 is 5/25 of the way between two perfect squares.

Since 149 is 5/25 of the way between 144 and 169, then it's reasonable to assume that 149's square root would be about 5/25 of the way between 12 and 13. This is the second meaning: It's the fraction we need to add to the reference root.

What we've been doing up to this point, then, is finding out how far between two perfect squares we have to travel, and expressing that as a fraction. Because of the way squaring works, this won't be an exact square root, but will come very close.

Step 6: Have them enter the addition (+) key, then enter the reference root, and then the equals (=) key.

Continuing with the example, we'd have them enter + 12 (the ref. root) =, so the calculator should now display 12.2.

Step 7: To prove how good your mental estimate is, have them press the x2 button on their calculator. If they don't have one, the same result can be achieved by pressing the × button, followed immediately by the = button.

With 12.2 displayed, they now press x2, and see a number approximately equal to 148.8399, which is very close to the given number 149!

Just to lock it in, let's try with another example. Let's say you're given a much higher number, such as 806.

Working through the process as above, we find the reference square, the reference root, and work through the process from there:

Once again, the squared result of 805.763 is very close to the given number 806!

As mentioned above, this works because we're working out the distance between two squares, and then seeing how far along that distance is the given number. Working that out as a fraction allows us to scale this answer down to be used as part of the given number's root. You can use this online web app I've developed to understand this concept more completely.

You might be wondering how close your estimates, when squared, will be to the original given number. The range of numbers with the biggest divisors, of course, will be the numbers from 961 up to 1,000. Using the process I teach above, here's a list of the results you'll get for each of those numbers.

Notice that the results are all just under the given number. For example, when you're given 962, your estimated square root, when squared, will return an approximate result of 961.984. If we look at just the margins of error for each number from 961 to 1,000, you'll note that it never gets farther away than .25 (or ¼)!

If you're presenting this as a bet, you can include the proposition that you have to be within plus or minus ½ in your estimate. This is only a smoke screen, as you know the resulting square will always be less than the given number, and it will never be off by more than ¼.

Instead of verbally instructing someone to enter the numbers in the calculator, you could write the answer down first. In this case, you would work through the process almost exactly backwards. Let's use 638 as an example.

The reference square, in this case, would be 625, and the reference root would be 25. Write down the reference root on the paper first.

Next, work out the denominator by doubline the reference root, then adding 1 to it. Write this as the denominator of the fraction on the paper.

The ref. root in this example is 25. We double that to get 50, and add 1 for a denominator of 51.

Finally, subtract the reference square from the given number to get the numerator.

638 - 625 is 13, so 13 is the numerator.

Sure enough, 25 and 13/51sts, when squared, gives approximately 637.81! Note that this is within our established -¼ margin of error.

You can practice using the random number generators at Wolfram|Alpha or Random.org and any handy calculator.

Naturally, the more squares you memorize, the higher you can go. If you memorize the squares of numbers up to 100, then you'll be able to estimate square roots of any number from 1 to 10,000! And yes, even at that scale, the resulting square will still never vary from the given number by more than ¼.

You can find out more about presenting this feat by reading the next post, Estimating Square Roots: Tips & Tricks.

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3 Response to Estimating Square Roots

7:12 PM

Use the original square root method from fourth grade math.

Or memorize the one page thee place log table.

9:18 PM

This is a good method for a quick estimate. If you are proficient with mental calculation then this article has great algorithms and solutions for square roots to almost arbitrary precision and many other types of calculations, including logarithms


Dead Reckoning the book is, in my opinion, the next step after Secrets of Mental math. I've had it for years and am still finding great stuff out of it. It is very hard and will take a lifetime to master everything, but it is sooo much fun to practice.

11:51 PM

this is a great method that reduces the computational intensity to almost nothing. Memorizing the squares up to 31 is pretty easy as most young people should have the squares up to 12 memorized (in America- I believe it's 15 or 30 in India?), the squares in the teens are relatively easy, the squares of any even number can be 4 times 1/2 of the square of the number you're trying to square, the square of numbers ending in 5 is simple, and the rest can be found and committed to memory easily enough.

Simply finding the squares of numbers gives one the most important tool for learning this method, and once learned, the answers almost fall out from one's mind.