The multiplication method taught in school requires more writing than necessary when multiplying by 11. It slows us down.
Consider multiplying 1234512345 by 11. Here’s the way we were taught in school:
Note that we wrote 1234512345 three times and we wrote the number 11. There’s nothing wrong with this, but when time is critical, these time-wasters can be eliminated. The technique described below does exactly the same thing, but with a different layout and a lot less writing.
A Nice Simple Example
Let’s do the above multiplication the new way. We start by writing 1234512345 and drawing some lines. The lines help clarify what we’re doing, but after gaining some experience, you won’t be drawing the lines any more.
Each line joins a digit above to a digit below. The digits below aren’t shown yet, but the lines indicate there will be 11 digits below, just like there are 10 digits above.
We start by following the right-most line and carrying the least significant digit down to the right.
From right to left, we add the two digits above to get the digit below. For example, our first addition is 4 + 5 = 9. We follow the lines down from the 4 and the 5, and write the 9 below and between the 4 and the 5.
Then, continuing right to left, we next add 3 + 4 = 7. We write the 7 below and between the 3 and the 4, using the lines as a guide.
Continuing in this fashion:
And finally, we copy the left-most digit down to the left. We can think of it as adding the 1 to an invisible zero (or not).
We have our answer: 1234512345 X 11 = 13579635795.
An Example With Carrying
The above example was nice and simple. Let’s make it a little messier. We will multiply 159357456 by 11. As we’re about to see, this multiplication will involve some carrying.
We start the same as before. We write the number and draw the lines (or if you prefer, you can simply imagine the lines and save a few more seconds).
Then we copy the right-most digit down to the right.
Now the fun begins. 5 + 6 = 11. 11 is a two digit number, so we have a carry. Write the 1 below and between the 5 and the 6, and keep the other 1 in your head.
Next is the 4 + 5, but don’t forget the 1 you’re carrying in your head. 4 + 5 + 1 = 10. 10 is a two digit number, so we have a carry. Write the 0 below and between the 4 and the 5, and keep the 1 in your head.
Next is the 7 + 4, but don’t forget the 1 you’re carrying in your head. 7 + 4 + 1 = 12. 12 is a two digit number, so we have a carry. Write the 2 below and between the 7 and the 4, and keep the 1 in your head.
Next is the 5 + 7, but don’t forget the 1 you’re carrying in your head. 5 + 7 + 1 = 13. 13 is a two digit number, so we have a carry. Write the 3 below and between the 5 and the 7, and keep the 1 in your head.
Next is the 3 + 5, but don’t forget the 1 you’re carrying in your head. 3 + 5 + 1 = 9. 9 is a one digit number, so there’s no carry this time (good thing, carrying around those 1’s in my head was starting to wear me out). Write the 9 below and between the 3 and the 5.
Continue on in this fashion, carrying as required.
We have our answer: 159357456 X 11 = 1752932016.
In our final example, we have a carry from the final (left-most) digit. We will multiply 987654321 by 11.
Now comes the fun part. We just did 9 + 8 + a carry, which gave us 18. We wrote down the 8 and we’re left with a carry floating around in our heads.
Our final step is to copy the left-most digit (the 9) down and to the left, but what about the carry? Imagining a 0 to the left of the 9 may help. We add 0 + 9 + 1 (the carry) to get 10. We write down the 0,
but what about that final carry? As you have probably guessed by now, we simply write it down to the left of our answer:
And there we have it: 987654321 X 11 = 10864197531.
Where To From Here?
Multiplying by 11 is now much more efficient. When we get used to doing it this way, we won’t have to draw the squiggly line anymore. We will just write down the starting number and the answer.
Keep your eye open for my next article. We can use a similar technique to divide by 11. You won’t want to miss it.
© Copyright 2015 by Warren Gaebel, B.A., B.C.S.. All rights reserved.