Stacking multiplications 5

We have learned how to multiply by stacking the factors and calculating, digit by digit. Does this work with any number? Any number at all? 3,494,862 times 3,289? Yes, in fact it does.

This method, or algorithm, to calculate a product step by step works for all multiplications. Sometimes there are many steps, but it always works. Let’s have a look at some examples. 287 times 642. Now we have three digits in both factors but the algorithm is the same.

Stack the numbers on top of each other. We’ll calculate slowly, step by step. Remember that you can always pause the video to think a bit longer, or go back to a previous step. If you want to, you can pause and try for yourself first. Let’s calculate, from the right!

7 times 2 is 14. The 1 becomes the memory digit, and the 4 ends up here. 7 times 4 is 28. Plus the memory digit, 1, this is 29. We get a new memory digit which is 2.

And we write the 9 here. 7 times 6 is 42. Plus the memory digit, 2, this is 44, which we write here. The next digit in the lower row is 8. 8 times 2 is 16.

The 1 becomes the memory digit, and the 6 goes here. 8 times 4 is 32. Plus the memory digit 1, this is 33. Then we get a new memory digit which is 3, and we write the other 3 here. 8 times 6 is 48.

Plus the memory digit, 3, this is 51, which we write here. We are not completely done yet, because we need to multiply a bit more before we use addition. This 2 also needs to be taken into account. 2 times 2 is 4. Then we get no memory digit, and we write 4 here.

2 times 4 is 8. We write 8 here. And finally: 2 times 6 is 12. This is the last multiplication, so we write 12 here. And we add everything up.

The product is 184 254! This is a large and advanced multiplication, but we have learned an algorithm to split it into many small, simpler, calculations. One single large calculation is replaced by many small ones. But what about this one? A multiplication with three different factors.

41 times 28 times 12. Can we calculate it? Yes, but not in one single stack. Instead, we can make two stacks, one after the other. First, 41 times 28.

We stack it like this and calculate the result: 1148. And this we multiply by 12. So, 1148 times 12. And after all the calculations we get the final product. 41 times 28 times 12 equals 13,776.

But what about that multiplication in the beginning? We need to complete a lot of steps and it looks rather complicated. But it’s the same idea as before: We split one large and difficult calculation into many small, simpler calculations. And the algorithm works, for all factors.