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Human Numerical Methods
Posted on October 20, 2008
Filed Under Math, Misc |
Or: I thought about calling this “Humerical Methods”, but there’s nothing funny here.
My girlfriend and I recently went on vacation to Germany. Distant relatives of mine (Wolfgang and Birgit), agreed to host us for a few days, which turned out to be a great break from cheap hostels.
One day, Wolfgang tells us that they are taking us to Öhringen, and town about 16 kilometers away.
He quickly corrects himself for the benefit of the Americans. “That should be about nine miles.”
“It’s closer to 9.6″ I said without skipping a beat.
The speed of my answer surprised him. “Are you a quick multiplier?”
“Well, not exactly. Each kilometer is a little over .6 miles, so I add 1/2 and 1/10 the original number. It’s usually really faster than multiplying or dividing.”
It should be noted that this is an approximate method. The error is around 3.5%, as the conversion factor is closer to .621371. Most of the time, approximation are good enough with respect to distances. Once you have a handle on the distance, you can make all sorts of reasonable estimations, like driving time.
Figuring it out
Last March, I went on a road trip to Canada to compete in the CS Games with my fellow Computer Science majors. Since it’s a long effing way from TCNJ to Sherbrooke in Canada, we needed something to do. My buddy Steve realized that we accidentally printed our return directions in kilometers instead of miles. He read out one of the distances on the map and asked what it was in miles/hour.
We thought for a second, and then I looked at the speedometer and read off the corresponding speed in miles per hour.
“How did you do it so fast?”
“Mental math,” I replied.
“What’s the conversion factor?” somebody asked. I saw that 100km/hr was roughly 60 miles, so I said “about .6.”
Luckily for me, all of the distances Steve read off were under 160km (the limit of my speedometer), so I was able to use the speedometer trick for a while. I figured that charade couldn’t last long, so I broke down the problem (multiplying by ~.6), and saw easy fractions I could use. I continued using this flawlessly.
Going from miles to kilometers is just as easy. Since a mile is ~1.609 kilometers, you can get an even better answer with identical math.
For a distance n, just add together n + n/2 + n/10. This approximation leaves you within .6%!
It’s a piece of cake to develop new methods along these lines. Take the expansion of the multiplicative constant, c, pick the first few digits of c, and break it up however mental arithmetic is easiest. Each digit of c that you use gives you a better approximation.
Determining the relative error is easy: it’s the unused digits of c divided by c.
You probably use this method for calculating tips already. If n is the dollar amount of the tip, it is easy to calculate 10% of the tip: x = .1 * n. The full amount of the tip is x + x/2, giving you 15%. That’s 1/10th + 1/20th, or 10% + 5%. This particular application happens to be exact.
You can even determine a generalization for any affine method, such as temperature conversions, by also adding and subtracting rounded approximations.
Feynman Vs. the Abacus
The above is child’s play! Any kid who has learned division can figure out their own tricks. The real limits of human mental math are accomplished by mental lookup tables. Special numbers are the name of the game; the more special numbers you know, the easier it is to quickly solve a problem.
The best example I have found of this is a story in Surely You’re Joking, Mr. Feynman!
The first challenge is addition. Feynman doesn’t stand a chance against the abacus, even when the numbers are written down and shown to both parties at the same time.
Next comes multiplication. Feynman still loses, but he has started to close the gap.
The next challenge is long division. It is a tie. This bothered the abacus salesman, so he challenges Feynman to cube roots. He writes the number “1729.03″ on a piece of paper and starts working the abacus. Feynman quickly figures out that the first few digits are “12.002″ while the abacus salesman struggles to even determine “12″.
As the book puts it:
How did the customer beat the abacus?
The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03 is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root’s excess is one-third of the number’s excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.
Hooray Math!
The Feynman abacus story demonstrates one of the great things about math: every little fact eventually counts for something.
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4 Responses to “Human Numerical Methods”
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You were doing math while driving to Canada? I feel unsafe.
I usually do the same thing for tips in the restaurant. Tip = price / 10 + (price / 10) / 2
It’s funny to hear that somebody else knows about CS Games! I wish I were still in school so that I could attend.
Richard Feynman is my new hero. Jake Voytko is a close second.
Loved the post.
Please proofread your dialogue! My inner English major is dying a little inside. (An inner-inner English major? Probably located in the region of one of my bellybuttons.)