How To Multiply Any Number By 11 In The Head?

Have you been looking for a fast method to multiply any number by 11 in the head? Then you’ve arrived at the right place! Jokes aside, any method that enables us to calculate fast in the head is probably also very efficient as a computer algorithm for specific applications. So, it’s probably more useful than most people would give it credit for. I’m going to demonstrate one such fast calculation algorithm involving 11 in this article. It simply takes advantage of the addition-related features of 11. That’s right. All that you’ll need for this method is mathematical addition. Sounds easy, right? Let’s find out if it indeed is!

This essay is supported by Generatebg

Multiply 2-Digit Numbers by 11

Let’s start by multiplying 2-digit numbers using 11. Let us consider 11*81. If you work this out in your calculator, you’ll get 11*81 = 891. It’s not so straightforward for everyone to calculate this in the head by using the conventional multiplication method. One could use algebra to split the problem into smaller problems and solve it like this:

But even this requires a bit of effort. The method that I’m sharing here is faster, simpler, and requires only addition. It’s perhaps more helpful if I first demonstrate it practically, and then explain how it works. So, here we go:

Let’s deconstruct what I just did in 2 steps:

1. I first wrote the two digits of the number that I am multiplying separately with a space in between.
2. I then added the two digits together and wrote the sum in the middle.

That’s it! This pretty much gives us the result of the multiplication operation. Of course, there are a couple of nuances involved, which we will cover one by one. For instance, how do we use this method to multiply 3-digit numbers? Right, we shall answer that question first.

Multiply 3-Digit Numbers by 11

To multiply 3-digit numbers using 11, the approach remains the same, except for one small change. Like before, I’ll start first by giving you a visual demonstration, and then explain what’s going on underneath. Consider the example: 11*217. Again, this multiplication operation is not straightforward using conventional methods. So, let’s give our little method a go:

Did you notice what I did there? If not, worry not; we will deconstruct what I did in 3 steps:

1. I first wrote the first and last digits of the number we are multiplying separately. But this time, I had a larger space in between; enough space for 2 numbers.
2. Then I added the first digit with the middle digit and wrote the sum down as the second digit of the product.
3. Finally, I added the middle digit with the last digit and wrote the sum down as the third digit of the product.

All this looks easy, but some questions arise. How do we scale from 3-digit numbers to bigger numbers? It is quite clear that the first and last digits of the number being multiplied are also the first and last digits (respectively) of the product. But how do we know how many numbers to fill in between them? Right, let’s answer both of those questions in one go.

---

Multiply Any Number by 11

Let us consider the example challenge that I originally proposed: 11*21241. Again, like before, I’ll first demonstrate the method, and then we can see how it answers our questions:

To answer our first question, the approach scales smoothly. We write the first and last digits of the number we are multiplying as the first and last digits of the product and leave a gap in between. To answer our second question, the gap accommodates the number of addition-combinations of digits-pairs of the original number from left to right. In our example, 21241 has 4 addition-pair combinations from left to right: 2+1, 1+2, 2+4, and 4+1. Before we celebrate having learned a cool new method, there is one final consideration we have to take into account.

Considering Carry Over

When we are adding two digits, everything works fine if the sum leads to a single digit (like 2+4 = 6). But if the sum leads to a 2-digit number (like 7+4 = 11), we need to carry the first digit in the sum over leftward. This is actually just simple addition, but it is important to make a note of this when using this method. So, I’ll just go ahead and work out a couple of examples to drive home the point:

There you have it. That’s pretty much all there is to it. You can now be proud of yourself that you can multiply any number by 11 in the head.

---

I hope you found this article interesting and useful. If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you: Why Is 3 A Special Denominator In Division? and How Many Decimal Digits Of Pi Do We Really Need?