Monday, February 20, 2006

The last post along these lines demonstrated a 'trick' I learned for squaring integers near 50 in your head. I can't remember where I learned it, but now after a websearch for any other reference to that method, I am suspecting one of the Feynmann biographies I've read (but not "Genius") mentioned it.

This one, I thought of myself. Inspired by the other method, I idly thought "hey, I bet you can do something obvious and useful with numbers near 100!" I now present to you:

First, represent the multiplication of the two integers near 100 as (100 +/- x)*(100 +/- y); -9 <= (x|y) <= +9. To multiply these numbers together, you either have to do long hand multiplication, memorize ~361 combinations (I'm betting I only have about half that many multiplications memorized already, just from [0..12]*[0..12]), use my method, or possibly by addition in log space via memorized log tables (there aren't many of these people around anymore). (or, I suppose, or another method I've never heard of, if you want to be pedantic) We don't bother considering the case where you're using a calculator because that would make these whole optimizations moot, anyway. An abacus is also considered a calculator here, as well as a slide rule. Head only.

The double or triple digit multiplication gets reduced to a memorization, an addition/subtraction, and memorizing the tables of single digit multiplications (e.g. the stuff you learned in elementary school).

(100 +/- x)*(100 +/- y) = 10000 + 100 * (+/-x +/-y) +/- x*y

A beauty of this quick mental method is that the x & y values become the equivalent of vectors in the same reference frame and you can use your intuitive spacial relations skills to 'see' quicker-to-calculate cancellations (like x=-5,y=+5). As with the other method I mentioned, because you're adding successive (and smaller magnitude) terms to get the answer, you can quickly "ballpark" magnitudes and bark out a rough answer immediately. While everybody is reeling from your answer, looking up into their heads to try and figure out if you're right, you can slip in the successive approximations until you are at the final answer.

For example,
Other Person: real quick, what's 94 * 107?
You (thoughts): Both of those are near 100. 10000ish!
You (speaking): 10000ish. [elapsed time: 0.5 seconds]
Y(t): 94 is -6 from 100, and 107 is +7 from 100. Addition leaves +1. 100.
Y(s): 10100 or so. [elapsed time: another second]
OP: [arches eyebrow]
Y(t): -6*7 = -42. 10100-42. 10058.
Y(s): 10058 exactly. [total elapsed time: 2.5 seconds]
OP: [unrolling eyes] Fudge off it's 10058.
OP: [looks around for calculator]
OP: [multiplies numbers]
OP: Oh.

And there you have it.

P.S. Oh, and in case you're wondering, no, I don't think this works on the ladies at a party. Please don't try it. It's probably only marginally more successful than a grown man doing magic tricks as a method of trying to pick up women, and buddy you're fooling yourself if you think that's working. This trick should only be used to impress other nerds.

Burton MacKenzie
http://www.burtonmackenzie.com

Leesa said...

I can divide most numbers by seven. And I am sure you know how to do that.

Derek said...

Thank God I own a calculator!

eldsbergy said...

I too have a trick. It almost is like long multiplication. I'll call it "short multiplication".

Basically to solve your example I would multiple the 94 by 100 and get 9400, multiply the remaining 7 by 90 and get 630, multiply the remaining 4 by 7 and get 28. Then add 9400+630+28 which equals 10058.

This process works for any numbers. Like your other topic "HOWTO: What are the small multiples of 9?" example; 9 x 5. I would multiply 10 by 5 and get 50, and the remaining -1 by 5 and get -5. Add them 50+(-5)=45.

I don't know if this is any faster but it works for me.

cheers

burton mackenzie said...

That's cool! When I have a second I'm going to have to sit down and work that out on paper. (the general method, not that specific example :-)

burton mackenzie said...

leesa - i hadn't read any of your stories. I came across your post about blog popularity while i was searching on a peripheral topic. However, i'd love to hear how you divide numbers by seven!