Dear Uncle Colin,

How do you multiply big numbers like $2158 \times 1812$? I try to do it using the column method or the grid, but I always make mistakes.

-- A Desperately Desired Error Reduction

I’ve been playing with something midway between the grid and the column recently, as it happens. I’ll show you it with 56 × 84 first, and then go on to do your example.

Here’s the working:

$5$   $6$ $8$ 4:0 + 4:8 $4$       2:0 + 2:4

As you can see, each digit in one number is multiplied by each digit in the other, but split across a colon. The second row is offset by one column (and each following row would be offset by one more).

All that’s left to do is to add up each column - everything between the colons - to get 4:6:10:4. 10 isn’t a digit, so you carry the 1 onto the 6 to get 4704, which is the correct answer.

Your example would look like this:

$2$   $1$   $5$   $8$ $1$ 0:2 + 0:1 + 0:5 + 0:8 $8$       1:6 + 0:8 + 4:0 + 6:4 $1$             0:2 + 0:1 + 0:5 + 0:8 $2$                   0:4 + 0:2 + 1:0 + 1:6

Adding the columns gives 3:7:19:19:12:9:6. This needs a bit of care with the carries, so I’d do it one carry at a time from the right, giving 3:7:19:20:2:9:6, then 3:7:21:0:2:9:6 and finally 3:9:1:0:2:9:6; the correct answer is 3,910,296.

This requires a few fewer steps than the traditional column method, and a lot less writing out zeros than the grid method. It’s roughly equivalent to the Napier’s Bones method, which is a nightmare to typeset.

Hope that helps!

-- Uncle Colin