## A simple way to multiply

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Spider
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### A simple way to multiply

I was shown this method a week or so ago, and I just think it's brilliant.

Basically, you take your two numbers which you want to multiply, and write them side by side as two columns. The number on the right is doubled, and the number on the left is halved until you reach 1. If the number on the left is odd then you simple subtract one and then halve it.

So say we take 18 and 21, you would get a table like this:

Code: Select all

`18    21 9    42 4    84 2   168 1   336`

Once you have this table, you cross off all the rows in which the number on the left is even, so now we're left with:

Code: Select all

` 9    42 1   336`

And then you simply add up the numbers in the right hand column, so we have:

Code: Select all

`42 + 336 = 378`

Which you can easily verify on a calculator.

The great thing about this is that because you're halving the number on the left each time it doesn't take long to reach 1 (if your smallest number is 2047 or less it will take at most 10 steps), so you can very quickly multiply very large numbers and it doesn't require any difficult maths really Gelsamel
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Handy if you're not great at multiplying.

Kas
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I'm quite proficient at multiplcation, but I think I'm going to have to try this method for a while just because it is so damn nifty.
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Hix
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I was shown the following method over 20 years ago, and I find it works just as well. Basically, you take your two numbers which you want to multiply, and write them side by side as two columns. The number on the right is multiplied by 10, and the number on the left is divided by ten, until you reach a single digit number on the left. If the number on the left is not a multiple of 10, you ignore any remainder upon dividing.

Once you have this table, you cross off all digits in the left column except the one's digits. Now perform the simple multiplications in each row (they're all multiplications by a single digit), and add the results to get your final answer.

(Try it... the whole process should feel somehow familiar!)

Of course, this also works if you replace "10" with any other integer greater than 1 (also replacing "digit" with the new base-appropriate term).

rlo
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### Re: A simple way to multiply

Spider wrote:I was shown this method a week or so ago, and I just think it's brilliant.... Basically, you take your two numbers which you want to multiply, and write them side by side as two columns.

Spider, that is excellent.

holyyakker
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That is nifty, but I'm a huge fan of mental math and I could do 18 * 21 in my head faster via mental means then using the halving trick. Is there an application for it outside of how spiff it looks when you do it?

(btw, the mental way I'd tackle that is 18*20+18=36*10+18=360+18=378. About 3 or 4 seconds of mental manipulation.)
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SpitValve
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See I did 18*21 = 10*21+8*21 = 210+168 = 378 but hey rlo
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Reminds me of another multiplication trick, which I've known since I was very young but forgot about until recently.

I just tried to describe it myself, but it begs for illustrations. Luckily it's on the web elsewhere (of course):

http://www.cofc.edu/wallacegurganus/

Cool thing is I have no idea why it works.

Verysillyman
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Wow! that one's pretty cool. i was expecting the 9s multiplication thing which I think is gay.

I was pleasantly surprised.

Spider
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holyyakker wrote:That is nifty, but I'm a huge fan of mental math and I could do 18 * 21 in my head faster via mental means then using the halving trick. Is there an application for it outside of how spiff it looks when you do it?

(btw, the mental way I'd tackle that is 18*20+18=36*10+18=360+18=378. About 3 or 4 seconds of mental manipulation.)

Sadly the method is slow, and unless you really struggle to multiply single digit numbers the standard "10's" method is by far the fastest, even if you can't do mental maths it's much quicker on paper too.

I just think this is really neat, and it does simplify the calculations, at the cost of efficiency.

rlo
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Verysillyman wrote:Wow! that one's pretty cool. i was expecting the 9s multiplication thing which I think is gay. I was pleasantly surprised.

Thanks, VSM (can I call you VSM?). Any idea why it works? I guess that's a puzzle in and of itself.

Hey, you know what? I'll post that question as a puzzle over in the puzzles forum.

Verysillyman
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rlo wrote:Thanks, VSM (can I call you VSM?).

You can call me anything you like Just so long as you call me.

I don't know why things work, so long as they do.

As for the original post, it seems to me that I'd need paper to do all that working. So why not do it the normal way if I have paper handy? And a pen/pencil/sharp knife dipped in my own blood.

Peshmerga
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A simple way to multiply Zomg! D:
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thomasjmaccoll
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SpitValve wrote:See I did 18*21 = 10*21+8*21 = 210+168 = 378 but hey hmm, i think i would do 18*21 = (18*10)+(18*10)+(18*1)=180+180+18=378

which is probably the simpleton way to do it but also seems easiest to me...

(and yes, i probably would've tried to work out 18*1)
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hermaj
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thomasjmaccoll wrote:
SpitValve wrote:See I did 18*21 = 10*21+8*21 = 210+168 = 378 but hey hmm, i think i would do 18*21 = (18*10)+(18*10)+(18*1)=180+180+18=378

Yeah, I'd probably do it that way too. But this is great... even better than that thing with multiplying by 11!

rrenaud
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This repeated doubling for multiplication reminds me of doing repeated squaring for exponentiation.

See http://www.algorithmist.com/index.php/Repeated_Squaring

When combined with modular arithmetic, it makes for a practical algorithm for computing remainders of powers, eg 3^7 % 10.

3 % 10 = 3
3^2 % 10 = 9
3^4 % 10 = (3^2 % 10) * (3^2 % 10) = 9 * 9 % 10 = 1

So 3 ^ 7 % 10 = ((3^1) % 10 * (3 ^ 2) % 10 * (3^4) % 10) % 10 = (3 * 9 * 1) % 10 = 7

This is actually useful in cryptography, for things like the RSA algorithm.

http://en.wikipedia.org/wiki/RSA

Narsil
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