There's actually a really sneaky moment in here which is not totally revealed. Feynman says that he first works out 12, then 12.002, then the man with the abacus says "12.01!". He asks for more digits.
What is sneaky here is that 12.002 is a very rough approximation that Feynman is making before doing a long division. He has the formula of
But he simply hasn't had the time to do the long division yet and has just approximated 1.03 / 1728 by 1/2000. At this point he has the formula, and quite possibly even knows that the error goes like - 12 x^2 / 9 which would be something like one part in a million, but he wants to do his long division to get the extra decimals to show off.
There's actually a slightly more slick way to get to this where you start from 12 and compute:
(2/3) * guess + (1/3) * number / guess^2.
The 1/3 and 2/3 are chosen to minimize the error, and you wouldn't know this if you've never worked it out. (I only worked it out because I would occasionally be stuck without a calculator on exam problems.)
It works really well on these sorts of problems; you compute 1729.03 / 144 and aside from the leading 12 you get:
0.0071527777...
You divide this fraction by 3 to get:
0.002384259259...
which is precisely as far as Feynman got, but the reasoning is much quicker. If he'd been even faster with this he might have been able to apply guess = 12.002 to get another couple of decimals in the same time.
Slightly offtopic: The abacus is a fascinating tool; often considered an early prototype for tangible interfaces.
"Among other historical inspirations, we suggested the abacus as a compelling prototypical
example. In particular, it is key to note that when viewed from the perspective of human-
computer interaction (HCI), the abacus is not an “input device.” The abacus makes no
distinction between “input” and “output.” Instead, the abacus beads, rods, and frame serve
as manipulable physical representations of numerical values and operations. Simultaneously,
these component artifacts also serve as physical controls for directly manipulating their
underlying associations."
I think Nicholas Carr made an interesting distinction in The Shallows about the usefulness of tools in the context of short and long-term memory (interesting even if I feel ambivalent about it):
"In freeing us from the work of remembering, it’s said, the Web allows us to devote more time to creative thought. But the parallel is flawed. The pocket calculator relieved the pressure on our working memory, letting us deploy that critical short-term store for more abstract reasoning. As the experience of math students has shown, the calculator made it easier for the brain to transfer ideas from working memory to long-term memory and encode them in the conceptual schemas that are so important to building knowledge. The Web has a very different effect. It places more pressure on our working memory, not only diverting resources from our higher reasoning faculties but obstructing the consolidation of long-term memories and the development of schemas. The calculator, a powerful but highly specialized tool, turned out to be an aid to memory. The Web is a technology of forgetfulness."
That's a great quote, and articulates something that's been bothering me about the web and why I've increasingly withdrawn from it. I find the web is a great tool for fishing for stimulus to further thought, but at some point I need to get away from it to develop skills in a way that feels slower, but is actually deeper.
To use an office metaphor: it's a great water cooler, but you can't write your report at the water cooler.
> I realized something: he doesn't know numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is to learn to push the little beads up and down. You don't have to memorize 9+7=16; you just know that when you add 9, you push a ten's bead up and pull a one's bead down. So we're slower at basic arithmetic, but we know numbers.
Maybe one of "them" could post here offering some opinions about ways in which learning the abacus enhances or extends "their" thought processes in useful ways. Or are "they" too limited by their rote learning? Or are we safe in our comforting knowledge that we have nothing to learn from "them?"
Feynman would have wanted his stories shared, not hoarded. We live in an insane time when sharing the stories of a dead man could be considered an illegal act.
Note also that SYJMF (along with "What Do You Care What Other People Think?") weren't actually written by Feynman. They're transcribed and edited versions of audio recordings made by his friend Ralph Leighton.
I don't recall of Leighton compiled them himself or if there was another editor involved in the printed editions, but regardless, they're probably still alive and deserve compensation for their contribution.
You don't specify when the non-insane time you're thinking of is, but I suspect that at that time distributing copies of someone else's memoirs in appreciable volume without being granted permission was probably also illegal.
How? I can see no way this argument can be used against the metric system.
The abacus method is a fundamentally different way of doing arithmetic, that's why the abacus guy didn't know numbers. But metric and non-metric are fundamentally similar, only metric is much easier, gets out of the way, and lets you think about the quantities instead of thinking about factors.
That was sarcasm. I was surprised that he knew those factoids about feet and inches, which eventually gave him an edge in the competition. He won because of them. And that was the only benefit he ever got out of not using the metric system.
I saw the video a few months ago, and it didnt make any sense to me. I didnt imagen Feynman as a guy that would go up to a stranger just to show off his math skills. The linked story makes much more sense!
If I make an article about 224 with the only property that it's the lowest natural number currently lacking an wikipedia article, will it get deleted or will it stay wrong?
I read somewhere (perhaps a Martin Gardner's article) saying that numbers couldn't be classified as "extraordinary" (having interesting properties) and "normal" because then the first normal number could be deemed extraordinary.
I was reminded of this story while trying to explain to my 12 year old (again) the importance of knowing numbers and ways to play with them in your head. I was glad to find the story online, and I'm happy HN found it interesting enough to make the front page.
As elsewhere mentioned in the comments, if you enjoy this story you'll probably enjoy reading all of Surely You're Joking, Mr Feynman.
This is a story about using the wrong tool for the job.
The Japanese abacus is a powerful tool for addition and subtraction. As Feynman said, a skilled user can add faster than you write the numbers down. Even today, its excellent user interface still puts it ahead of a typical pocket calculator. But for cube roots ... it sucks.
I wonder how Feynman would have fared against a slide-rule salesman?
This was the one scene from the movie Infinity [1] that I really didn't like. Instead of a restaurant in Brazil, the script had Feynman challenging a shopkeeper. Unfortunately, the way is was done made him come off as arrogant and obnoxious in that scene. Other than that, though, it was a decent flick.
What is sneaky here is that 12.002 is a very rough approximation that Feynman is making before doing a long division. He has the formula of
But he simply hasn't had the time to do the long division yet and has just approximated 1.03 / 1728 by 1/2000. At this point he has the formula, and quite possibly even knows that the error goes like - 12 x^2 / 9 which would be something like one part in a million, but he wants to do his long division to get the extra decimals to show off.There's actually a slightly more slick way to get to this where you start from 12 and compute:
The 1/3 and 2/3 are chosen to minimize the error, and you wouldn't know this if you've never worked it out. (I only worked it out because I would occasionally be stuck without a calculator on exam problems.)It works really well on these sorts of problems; you compute 1729.03 / 144 and aside from the leading 12 you get:
You divide this fraction by 3 to get: which is precisely as far as Feynman got, but the reasoning is much quicker. If he'd been even faster with this he might have been able to apply guess = 12.002 to get another couple of decimals in the same time.