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Learn how square roots are used in the real world, how you can easily estimate the value of a square root in your head, and how you can use an ancient algorithm to calculate the value of a square root by hand to as high a precision as needed.

By
Jason Marshall, PhD,
May 4, 2012
Episode #104

If you’ve read last week’s article, figuring out what the square root of 16, 81, and every other number that’s a perfect square should be easy since you know that you just have to find the number that was squared to create the perfect square. Figuring out what the square root of a negative number like –1 or –42 should be easy as well since we saw that negative numbers don’t actually have square roots (at least none that are any numbers we know about yet). And figuring out what the square root of a non-perfect square like π or 55 should also be a piece of cake since you can always use a calculator.

But what if you don’t have a calculator handy when you really need to find that square root? What can you do? Keep on reading because today we’ll learn how to easily calculate square roots in your head.

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## Recap: What Are Square Roots?

Before we get into the details of how to estimate square roots in your head and how to accurately calculate their values without a calculator, let’s quickly recap what square roots are. The idea is actually very simple: Whenever you multiply a number by itself—and keep in mind this goes for any number whether it’s an integer, rational, or irrational—that’s called squaring the number.

As an example with integers 15 x 15 is the square of 15, which is equal to 225. When we find the square root of the number 225, all we’re doing is figuring out what that original number was that we multiplied by itself to get 225. When dealing with numbers that aren’t perfect squares, the mechanics of finding square roots is different, but the idea is still the same. For example, what’s the square root of 343? Using a calculator, I find that the answer is approximately 18.52. If we multiply 18.52 x 18.52, we get 342.99 which is really close to 343. So, as you can see, no matter what type of number we’re dealing with, the meaning of the square root is the same.

## Square Roots in the Real World

You might be wondering why you’d ever need to calculate a square root in the first place? As with many things in math, it turns out that square roots show up all the time in the real world. For example, imagine you’re thinking about buying a house and you see in the advertisement that the lot is just over 4,000 square feet. Is that tiny? Or huge? What does it mean? Well, one way to get an idea for it is to assume that the lot is approximately square shaped (most aren’t exactly squares of course, but it’s not a bad model to use just to get an idea). The square root of the square footage of the lot then tells you how big each side of the model square lot is. Since the square root of 4,000 square feet is a little over 63 feet, we see that the house sits on a square lot that’s a little over 63 feet wide and 63 feet deep. Of course, there are tons of other examples of square roots in the real world too, but I think that gives you the picture.

Let’s now move on to the practical side of finding square roots. First, what can you do if you need to calculate a square root when you don’t have a calculator? In truth, since most phones now have calculators built into them, you probably almost always have one on you. But even still, there are times when you don’t need to waste time using it. In particular, you don’t need to bother using a calculator when you only need to know an approximate answer—and that’s exactly the case in most real world situations. When we were figuring out how big a 4,000 square foot lot is, we really didn’t need a very accurate answer and figuring out that the lot was about 60 feet on a side was good enough.

In cases like this where high precision is overkill, you can easily estimate a square root in your head by figuring out which two perfect squares the number falls between. For example, imagine you need to find the square root of 60. If you’ve memorized the multiplication table, you already know that 7^2 = 49 and 8^2 = 64. Since 60 is larger than 49 but smaller than 64, we can immediately see that the square root of 60 must be a number between 7 and 8. And we can also see that the real answer must be a little closer to 8 than 7 since 60 is closer to 64 than 49. Make sense?

You can easily estimate a square root in your head by figuring out which two perfect squares the number falls between.

## How to Calculate Square Roots Without a Calculator

But what if that rough estimate just isn’t good enough and you need a more accurate answer? Are you out of luck? No, you can use a very cool and clever 2000+ year old algorithm known as the “Babylonian method” or “Heron’s method” to improve your estimate. Here’s how it works for the square root of 60:

• Step 1 is to guess the answer…the better the guess the more quickly the accuracy of the estimate will improve. In our case, since we know the answer must be between 7 and 8, let’s guess 7.5. Since 7.5 x 7.5 = 56.25, we know this isn’t a super accurate guess, but let’s see what the algorithm can do for us.

• Step 2 is to divide the number we’re taking the square root of by our guess. So 60 / 7.5 = 8.

• Step 3 is to average this new number and our guess. In other words, add this number to our guess and divide the result by 2. That’s (7.5 + 8) / 2 = 7.75. This number is our new guess. Let’s check and see how good this guess is: 7.75 x 7.75 = 60.0625…which is really close to 60. So in only one trip through this sequence of steps we’ve improved our estimate of the square root of 60 tremendously.

• Step 4 is to decide if your new guess is accurate enough. If it is, you’re done. If not, go back to the beginning with this new guess and repeat the process until the answer is as accurate as you need it to be. Every trip through this ancient algorithm will give you a more and more accurate answer.

## Wrap Up

Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

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