"I'll Calcu Later!"

April 30, 2009 1:48 PM

I've seen people compute complex arithmetic off the top of their head. How do they do it?

I remember an infomercial from the 90s where this guy was selling a system for calculating large sums quickly off the top off his head (they'd ask him to multiply two large numbers like 42506 and 397 and he'd pop off with an answer in less than a couple seconds). Pretty sure I've seen it on talk shows too.

I never had the $100+ to shell out for the program as a kid at the time, but I've always been fascinated with techniques on how to calculate faster.

What sources or techniques are available out there so that I may teach myself these heuristics?

(And I'm in no way asking for Daniel Tammet-level savant, just merely some tricks for the already-math-sharp/geek layman.)

posted by Christ, what an asshole to sports, hobbies, & recreation (13 comments total)

Here's a video showing how to do it.
posted by nitsuj at 1:55 PM on April 30

I have a book called Mathemagics which is full of that kind of stuff.
posted by imaswinger at 2:14 PM on April 30

One simple move that a lot of people use for the sort of big multiplication problem you mention there is to round and then subtract/add. For instance, in the problem you have there, 397 is pretty close to 400, so it's quicker to just think "42506 X 4" to get 170024, then tack two zeroes on to get 17002400, then get 42506 x 3 = 127518, then subtract that to get 16874882.

I can recall seeing a guy in high school who could square three-digit numbers very quickly using a similar approach. Say you wanted to square 476. That's close to 500. And if you know a little algebra, you know that:

x^2 = (x + y)(x - y) + y^2

or in our case,

476^2 = (476 +24)(476-24) + (24)^2
= 500 * 452 + 576
= 226000 + 576
= 226576

The first side is fairly easy, in that you're just dividing the one number in half and adding three zeroes. You'd have to remember that 24^2 was 576, but remembering 100 squares for two-digits numbers isn't terribly hard. At least some of this is just good working memory and being able to keep a few numbers in your head reliably.

When I was 14 or so, I could do these in a couple seconds. I could also find four- and five-digit square routes pretty quickly in other ways, too. I would try to explain this to people, but they usually couldn't do any of the steps, so it seemed inexplicable to them. It was more a matter of practice than anything, and I sort of got tired of it. I started to think that if I couldn't write something algebraically, it wasn't really math. So I did that stuff instead.
posted by el_lupino at 2:20 PM on April 30 [2 favorites]

I'm not sure if you'd include "determine the day of the week for any given date" in the tricks you're looking for, but if you would, you want the Doomsday algorithm.
posted by DevilsAdvocate at 2:34 PM on April 30

I had a physics professor once who would write down trig functions to three decimal places off the top of his head. He ocassionally fudged the last digit, but was generally right. Darned if I know how he did it.

The kinds of tricks el_lupino mentions are good for mental multiplication, which is useful and something it's reasonable to expect to be able to get better at. I also second imaswinger's recommendation of Mathemagics; I once owned a copy, and while it wasn't my cup of tea (didn't feel like spending the time and effort practicing mental math), it seemed to be full of effective tricks.
posted by musicinmybrain at 2:40 PM on April 30

You'll want to watch this video of Arthur Benjamin at TeD.
posted by monkeymadness at 2:58 PM on April 30 [2 favorites]

Developing your ability to estimate comes before anything else. This allows you to roughly check your answer.

All of the above suggestions are applicable, but also start doing without a calculator, then practice. Naturally, doing the hyperbolic cosine of some odd number isn't going to go well, but for your standard arithmetic, practice makes perfect.
posted by adipocere at 4:42 PM on April 30

My Indian high school physics teacher credited Vedic math for his calculating abilities. Tutorial
posted by djb at 7:00 PM on April 30 [1 favorite]

See also the Trachtenberg system
posted by djb at 7:01 PM on April 30

Once upon a time, in a high-school long, long away, I had a teacher introduce me to Chisanbop (which she naturally redneckified to "chisombop", despite the appearance of the actual word both written and spoken throughout the source material...) The demo video she showed us was astonishing and intimidating. The astonishment faded.
posted by EnsignLunchmeat at 8:30 PM on April 30

It's basically tricks & more tricks. Here's one you know. To multiply by 10, add a zero at the end. Stupid and easy, right? Well there are more tricks: hundreds, probably thousands. You just need to memorize and practice them and eventually you have enough to cover all problems you'll encounter. One concrete example is you probably know your times table up to 12 or so, these guys usually memories them all the way up to 99. What these guys to is impressive and inspiring but not magical.
posted by chairface at 10:43 AM on May 1

Yeah, I think I've seen the Kevin Trudeau-alike infomercial, too. Not sure if it was him of not, I just associate it with him for some reason (maybe the speed reading infomercial).

That's not to say that this isn't possible, but like any skill, requires lots of practice. The Secrets of Mental Math by Arthur Benjamin is a nice place to start. It starts out pretty easily, and gets harder quickly. There are whole communities online devoted to mental math, like the yahoo! group for Mental Calculation.
posted by Barry B. Palindromer at 10:55 AM on May 1

Yup -- tricks. In Surely You're Joking, Mr. Feynman! there's a chapter called "Lucky Numbers" where Feynman talks about some of his (link goes to full text, but there are no anchors so you'll have to search for the chapter title).
posted by amery at 1:12 AM on May 2