How to Square Numbers Quickly

Quick, what’s 25 squared? If that’s too easy for you, how about something tougher like 37 squared? Or 111 squared? Want to learn an easy way to calculate these squares and the square of any other number? Keep on reading to find out!

Jason Marshall, PhD
5-minute read
Episode #160
Math on chalkboard

Using Algebra to Square Numbers

Although I said earlier that we’re going to take a break from algebra for a bit, it turns out that there’s actually a little algebra in the mental math trick we’re going to learn today—goes to show you that algebra is everywhere in the world!

What we’re about to do may seem a little strange at first, but stick with me for a minute and I promise we’ll wind up in a good place. As you know, our goal for today is to learn how to square numbers in our heads. Let’s imagine that the number we’re going to square can be written as the sum of two numbers. For example, if we’re squaring 25, we know that 25 can be written as 25 = 20 + 5.

Instead of using actual numbers, let’s express this idea algebraically by saying that the number we’re trying to square can be written as the sum of two numbers—as in a + b. So, in the example of 25 = 20 + 5, a = 20 and b= 5. Now, what happens if we square our number? In particular, what happens if we square our number written as the sum of a + b? You don’t need to know how to actually do this multiplication to use the mental math trick we’re about to uncover, you just need to know that:

(a + b)^2 = (a + b) x (a + b) = a^2 + 2ab + b^2

OK, great…but what does this have to do with squaring numbers in your head?

How to Square Numbers…Quickly

We can use the fact that (a + b)^2 = a^2 + 2ab + b^2 to turn the problem of finding the square of one big number into several smaller and simpler problems. Let’s see how this works with the example of calculating 25^2. Since we know that 25^2 = (20 + 5)^2 (so a = 20 and b = 5), we can use the algebraic formula we found to simplify the problem. Namely, we know that the square of 25 must be equal to the square of 20 (which is 20^2 = 400) plus twice the product of 20 and 5 (which is 2 x 20 x 5 = 200) plus the square of 5 (which is 5^2 = 25). In other words, 25^2 = 400 + 200 + 25 = 625.

Is that really easier that simply calculating 25^2 directly? I’d say our method is better here, but maybe just marginally. The real advantage comes when you’re trying to square larger or more complex numbers (meaning numbers that aren’t simple powers of 5 and 10. For example, what’s 52^2? Dunno! But we can easily find out using the fact that 52 = 50 + 2, and therefore 52^2 = 50^2 + 2 * 50 * 2 + 2^2 = 2,500 + 200 + 4 = 2,704. Once you get the hang of it—which means you definitely need to practice a bit—you’ll find that this method for calculating squares is much, muchfaster.

Wrap Up

Okay, that’s all the math we have time for today. If you want to learn some more mental math techniques, check out these earlier Math Dude articles:

Also, be sure to check out my mental math audiobook called The Math Dude’s 5 Tips to Mastering Mental MathAnd for even more math goodness, be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra.

Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via FacebookTwitter, or 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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About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

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