Multiplying Big Numbers in Your Head

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by Richard Crews
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In response to a storm of inquiries* I hereby reveal my best discovery for multiplying two 2-digit numbers in your head.

First step: it's relatively easy to square a 2-digit number, right? Like 40 squared is 1600 and 70 squared is 4900, but also 35 squared is 30 squared (900) plus 5 squared (25) plus two times 30 x 5 (2 x 150 = 300) so 35 squared is 900 + 25 + 300 which = 1225.

So squaring any 2-digit number in your head isn't that hard if you practice it a little.

Now, suppose the starting numbers are different--we're not just squaring some number. If both the starting numbers are odd (or both are even) they "surround" some number, that is they are the same distance from it. Like 35 and 45 surround 40--they are each 5 away from it.

But 39 x 41 comes out to one less than 40-squared.
And 38 x 42 comes out to 4 less than 40-squared.
And 37 x 43 comes out to 9 less than 40-squared..
And 36 x 44 comes out to 16 less.
And so on--it's always the square of the distance from the number surrounded.

So to multiply 35 x 45 you square 40, the number they surround (which is easy, it's 1600) and subtract 5-square which is 25 (which is also easy: 1600 - 25 = 1575).

Voila! 35 x 45 is 1575.

It gets a little harder if the number that is surrounded is not so easy to square. Like take 29 x 41. They surround 35 (each is 6 away from 35). So 35 squared is 1225 (as above) and then minus 6-squared (which is 36)--so 1225 minus 36 is 1189.

Et voila! 29 x 41 is 1189.

Some people might not find it easy to subtract 36 from 1225 in your head, but since 36 is just 11 more than 25, all you really need to do is subtract 11 from 1200.

It gets a little harder (it might take 20 or 30 seconds) when the two starting numbers are not close to one another, and when one is odd and the other is even. But it's doable, in fact, there are some other tricks you can use.

When it comes to multiplying larger numbers (like 4123 x 7809), you just do them as smaller sets (a problem like this might take 2 or 3 minutes)--
4100 x 7800 (in other words, 41 x 78 and then followed by 4 zeroes, that's 31980000), plus
23 x 7800 (that's 23 x 78 followed by 2 zeroes, 179400), plus
4100 x 9 (with 2 zeroes, 36900), and plus
23 x 9 (with no zeroes, 207).

So 4123 x 7809 = 31980000 + 179400 + 36900 + 207 = 32196507

The hard part is remembering the intermediate results as you go along so that you can add them up when you get to the end of the problem. That's where my mnemonic system comes in. Ask me about it if you care.

I like to check my results as I go along using check-sums or "casting out nines." That's an interesting process too.

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* OK, so no one actually asked me--so sue me.