How to Square Two-Digit Numbers in Your Head
Review: What are Perfect Squares?
We’ve talked about squaring numbers and perfect squares many times before, so everybody who’s been following along for a while should be up to speed. But just in case you’re a little fuzzy, the quick and dirty summary is that squaring a number is simply the process of multiplying that number by itself. And the result of doing that is a number that’s called a perfect square. So the square of 5 is just 5 x 5 and that’s equal to the perfect square 25.
Is There a Way to Square 2-Digit Numbers Quickly?
While it’s easy to calculate the squares of single-digit numbers like 5 in your head (since those squares are part of the basic multiplication table we learned about many moons ago), it’s not so easy to multiply two-digit numbers in your head. Or…actually…is it? What do you think: If I was to ask you to quickly find the square of a number like 32, could you do it? In truth, probably not—but that’s just because you don’t know the trick that my friend showed me. So it’s time to let you in on this mental math secret.
How to Square 2-Digit Numbers Ending with 5
Let’s start by talking about the special case of squaring a two-digit number that ends with 5. For example, what’s the square of 35? Well, it turns out that the result of squaring any 2-digit number that ends with 5 starts with the number you get by multiplying the first digit of the number you’re squaring with the next highest digit and ends with the number 25. Which means that the answer to 35 x 35 must begin with the number 3 x 4 = 12 (since 3 is the first digit in 35 and 4 is the next number higher than 3) and ends with the number 25. So, as you can check for yourself by hand (just to make sure it works!), the answer to 35 x 35 is 1,225.
How about the square of 75? Well, the answer must begin with 7 x 8 = 56 and end with 25. So it’s 5,625, right? As you can check by hand or with a calculator, it is! And as you can check with the rest of the two-digit numbers ending with 5, this trick always works—mentally squaring two-digit numbers that end with 5 is a cinch. But what if the number doesn’t end with 5?
How to Square Any 2-Digit Number in Your Head
Squaring any two-digit number in your head, let’s say 32 x 32, is a bit more difficult. We don’t have time to go into all the details about why this works right now, but the first step is to figure out the distance (more accurately the absolute value) from the number you’re squaring to the nearest multiple of ten. In our example, the nearest multiple of 10 to 32 is 30, and the distance between 32 and 30 is 2. If you were instead squaring 77, the nearest multiple of 10 is 80, and the distance between 80 and 77 is 3. Now that we’ve figured out this distance, all that we have to do to find the answer to the problem is multiply the number we get when we subtract this distance from the original number by the number we get when we add this distance to the original number, and then add the square of the distance to the result.
I know that sounds like a mouthful, but it’s really not so bad. In our example, the method says that 32 x 32 must be equal to 30 (that’s the original number minus the distance of 2) times 34 (that’s the original number plus the distance of 2) plus 4 (that’s the square of the distance of 2). In other words, 32 x 32 = (30 x 34) + 4. Wait, that actually looks morecomplicated! How exactly is it better? Because as long as you use the fact that 30 = 3 x 10 to make the multiplication problem easy (as in 30 x 34 = 3 x 10 x 34 = 3 x 340 = 1,020), this is now an easy problem to solve! Practice at it a bit, and you’ll see that the beauty of this method is that it turns a single problem that’s hard to solve in your head into multiple easy problems.
To make sure you understand how this all works, let’s do a couple of practice problems. I’ll start by showing you one more worked example, and then I’ll leave you to try the rest for yourself.
- 77 x 77 = 5,929 The distance from 80 to 77 is 3, so 77 x 77 = (74 x 80) + 9 = (8 x 10 x 74) + 9 = (8 x 740) + 9 = 5,920 + 9 = 5,929
- 21 x 21 = _____
- 54 x 54 = _____
- 98 x 98 = _____
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