# Ask Uncle Colin: How do I multiply big numbers together?

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

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

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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