26 ms·

Feynman vs. the Abacus (1985)

54 points by marcodiego 5y ago

ypeterholmes 5y ago
They said "When you grow up, you won't always have an abacus in your pocket!" And yet here we are.
- retrac 5y ago
  And if anyone saw that coming it was Feynman. This isn't quite prophetic for 1959 but it's close: > I don't know how to do this on a small scale in a practical way, but I do know that computing machines are very large; they fill rooms. Why can't we make them very small, make them of little wires, little elements – and by little, I mean little. For instance, the wires should be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across.
  - mrtnmcc 5y ago
    Quite a coincidence... Robert Noyce invented the first monolithic integrated circuit chip at Fairchild Semiconductor in 1959.
- MaxBarraclough 5y ago
  The you won't always have a calculator justification was always a poor one, even ignoring that we all carry computers these days. You don't study mathematics just to improve your mental arithmetic. If mental arithmetic were the point, you'd just practice mental arithmetic for your whole mathematical education, rather than progressing to more advanced topics.
r3trohack3r 5y ago
It feels like, in every one of Feynman’s arithmetic stories I’ve read, he always says how lucky he was that someone used a specific number. At some point it stops being luck.
- BurningFrog 5y ago
  Or, when you enough special cases, you're almost always lucky to hit one of them. Which is another way of saying what you're saying, I think.
  - wolfgang42 5y ago
    Feynman actually touches on this in the same chapter, a few pages before the anecdote quoted in the article: > I had a lot of fun trying to do arithmetic fast, by tricks, with Hans [Bethe] [...] He was nearly always able to get the answer to any problem within a percent. It was easy for him—every number was near something he knew.
- marcodiego 5y ago
  “In the fields of observation chance favors only the prepared mind.”
- mhh__ 5y ago
  Murray Gell-Mann did say once that he was annoyed by Feynman's tendency to invent anecdotes about himself. He wasn't lucky in the sense that he was a brilliant scientist but he also had a huge ego which he needed to validate publicly.
  - svachalek 5y ago
    It may be annoying, but I'll grant he earned it. The world is full of gigantic egos with little of substance to say.
  - cpp_frog 5y ago
    If I remember correctly, according to Leonard Mlodinow's Feynman's Rainbow, Gell-Mann wrote that in Feynman's obituary for the Physics Review. It "raised quite a few eyebrows".
    - mjreacher 5y ago
      He says the same in this interview too https://www.youtube.com/watch?v=rnMsgxIIQEE&t=109s https://www.youtube.com/watch?v=rnMsgxIIQEE&t=109s
FreeRadical 5y ago
1729 was also Ramanujan’s taxi number https://mathworld.wolfram.com/Hardy-RamanujanNumber.html https://mathworld.wolfram.com/Hardy-RamanujanNumber.html
- paulpauper 5y ago
  it's because 12^3 differs from 1729 by just 1, making the error very small when using series
- dhosek 5y ago
  This is how I remember that 1728 = 12³. I remember getting puzzled looks from classmates in my elliptic curves class first when I was able to provide 12³ without a moment's delay and then again when I explained why I knew that.
wolfi1 5y ago
I was always wondering what method the Abacus guy used for calculating the cubic root
- pavpanchekha 5y ago
  Suppose you have a number: x = a0 10^1 + a1 10^0 + a2 10^-1 + ... If you write down x * x * x, you get something like: x^3 = a0^3 10^3 + 3 a0^2 a1 10^2 + (3 a0 a1^2 + 3 a0^2 a2) 10^1 + ... Now equate this term by term to your goal, which is: x^3 = 1 10^3 + 7 10^2 + 2 10^1 + ... From the first term, you get: 1 = a0^3 So a0 = 1, giving us an answer of 10 so far. Plug that in to the second term and you get: 7 = 3 a1 which gives a1 = 2---you always round down. The answer is 12 so far. We have a carry of 1, which we need to add to the next one. That gives us: 12 = 3 * 1 * 2^2 + 3 * 1^2 * a2 = 12 + 3 a2 Which leaves a2 = 0. So the answer is 12.0 so far. As you go further, there are more and more terms hence the "scaling" phenomenon you see. Every time you are solving a polynomial equation in one variable, where the solution is an integer 0 through 9; my guess is that on an abacus you do binary search instead of root finding. On paper this sounds easy, but on the abacus it sounds impossible—each of those adds and multiplies is its own crazy sequence of steps.
smoldesu 5y ago
Low-precision mathematics saves lives.
- Koshkin 5y ago
  (Allowing wider margins of error and larger tolerances means better reliability and stability.)
klyrs 5y ago
I'll have to pull this one out next time I hear somebody complaining about the imperial system of measures being useless
- ithkuil 5y ago
  nah, that's clearly an argument in favour of full blown duodecimal system. 1729 decimal is 1001 duodecimal, Every kid raised in a duodecimal system that had to learn how to convert into decimal in order to interact with us barbarians knows this fact.
- Koshkin 5y ago
  I love the chaos of the imperial system. It has the taste of life itself.
  - mensetmanusman 5y ago
    The clock is base 12 because it has more rational fractions, those Babylonians!
    - Koshkin 5y ago
      Their number system was base 60, actually (one prime factor more).
uwagar 5y ago
feynman is always smart in his autobiography.
- mandmandam 5y ago
  If you're gonna try and convince people Feynman wasn't smart, try a bible group.
dang 5y ago
One past thread: Feynman vs. The Abacus - https://news.ycombinator.com/item?id=5849665 https://news.ycombinator.com/item?id=5849665 - June 2013 (33 comments)
beloch 5y ago
"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." There are people lauding and panning Feynman in this thread, but this story illustrates a very important lesson that I've seen a lot of very smart people fail to learn. Whenever you're doing a calculation, especially if you're using a computer or calculator, make an approximate estimate of what the result should be. Don't just assume that whatever method you use will produce a correct answer. A man with an abacus (or computer) is probably going to be faster than someone like Feynman most of the time, but he's also going to make mistakes. Big ones. He might not make them frequently, but mistakes always happen sooner or later. The trick is to catch them when they do happen. That's hard to do if you have absolutely no sense of what the answer should be. If, however, you start from a rough estimate using anything your brain can come up with, as Feynman does in his anecdote, you're a lot less likely to produce a badly wrong answer without realizing it. If this contest had gone on long enough, Feynman would have been beaten badly in several rounds if he messed up a calculation and had to start over. The thing is, Feynman would know when this happened because the outcome wouldn't agree with his initial estimate. The man with the abacus would, eventually, have produced answers off by several orders of magnitude without realizing. Rough estimates are important.
- Stratoscope 5y ago
  Back in the slide rule era, we had to learn this lesson. The slide rule would give you the (rather approximate) digits, but you had to know where to put the decimal point.
pmoriarty 5y ago
Flash Anzan at the All-Japan National Soroban Championship 2012: [1] Background on how students train to do this using the Soroban (the Japanese Abacus): [2] Another demo: [3] Two nine-year-old girls play shiritori[4] ("a Japanese word game in which the players are required to say a word which begins with the final kana of the previous word") while adding 30 three-digit numbers flashed in 20 seconds: [5] [1] - https://www.youtube.com/watch?v=7ktpme4xcoQ https://www.youtube.com/watch?v=7ktpme4xcoQ [2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y https://www.youtube.com/watch?v=Px_hvzYS3_Y [3] - https://www.youtube.com/watch?v=JawF0cv50Lk https://www.youtube.com/watch?v=JawF0cv50Lk [4] - https://en.wikipedia.org/wiki/Shiritori https://en.wikipedia.org/wiki/Shiritori [5] - https://www.youtube.com/watch?v=_vGMsVirYKs https://www.youtube.com/watch?v=_vGMsVirYKs

54 points by marcodiego 5y ago

ypeterholmes 5y ago

They said "When you grow up, you won't always have an abacus in your pocket!" And yet here we are.

retrac 5y ago
And if anyone saw that coming it was Feynman. This isn't quite prophetic for 1959 but it's close: > I don't know how to do this on a small scale in a practical way, but I do know that computing machines are very large; they fill rooms. Why can't we make them very small, make them of little wires, little elements – and by little, I mean little. For instance, the wires should be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across.
- mrtnmcc 5y ago
  Quite a coincidence... Robert Noyce invented the first monolithic integrated circuit chip at Fairchild Semiconductor in 1959.
MaxBarraclough 5y ago
The you won't always have a calculator justification was always a poor one, even ignoring that we all carry computers these days. You don't study mathematics just to improve your mental arithmetic. If mental arithmetic were the point, you'd just practice mental arithmetic for your whole mathematical education, rather than progressing to more advanced topics.

r3trohack3r 5y ago

It feels like, in every one of Feynman’s arithmetic stories I’ve read, he always says how lucky he was that someone used a specific number. At some point it stops being luck.

BurningFrog 5y ago
Or, when you enough special cases, you're almost always lucky to hit one of them. Which is another way of saying what you're saying, I think.
- wolfgang42 5y ago
  Feynman actually touches on this in the same chapter, a few pages before the anecdote quoted in the article: > I had a lot of fun trying to do arithmetic fast, by tricks, with Hans [Bethe] [...] He was nearly always able to get the answer to any problem within a percent. It was easy for him—every number was near something he knew.
marcodiego 5y ago
“In the fields of observation chance favors only the prepared mind.”
mhh__ 5y ago
Murray Gell-Mann did say once that he was annoyed by Feynman's tendency to invent anecdotes about himself. He wasn't lucky in the sense that he was a brilliant scientist but he also had a huge ego which he needed to validate publicly.
- svachalek 5y ago
  It may be annoying, but I'll grant he earned it. The world is full of gigantic egos with little of substance to say.
- cpp_frog 5y ago
  If I remember correctly, according to Leonard Mlodinow's Feynman's Rainbow, Gell-Mann wrote that in Feynman's obituary for the Physics Review. It "raised quite a few eyebrows".
  - mjreacher 5y ago
    He says the same in this interview too https://www.youtube.com/watch?v=rnMsgxIIQEE&t=109s https://www.youtube.com/watch?v=rnMsgxIIQEE&t=109s

FreeRadical 5y ago

1729 was also Ramanujan’s taxi number https://mathworld.wolfram.com/Hardy-RamanujanNumber.html https://mathworld.wolfram.com/Hardy-RamanujanNumber.html

paulpauper 5y ago
it's because 12^3 differs from 1729 by just 1, making the error very small when using series
dhosek 5y ago
This is how I remember that 1728 = 12³. I remember getting puzzled looks from classmates in my elliptic curves class first when I was able to provide 12³ without a moment's delay and then again when I explained why I knew that.

wolfi1 5y ago

I was always wondering what method the Abacus guy used for calculating the cubic root

pavpanchekha 5y ago
Suppose you have a number: x = a0 10^1 + a1 10^0 + a2 10^-1 + ... If you write down x * x * x, you get something like: x^3 = a0^3 10^3 + 3 a0^2 a1 10^2 + (3 a0 a1^2 + 3 a0^2 a2) 10^1 + ... Now equate this term by term to your goal, which is: x^3 = 1 10^3 + 7 10^2 + 2 10^1 + ... From the first term, you get: 1 = a0^3 So a0 = 1, giving us an answer of 10 so far. Plug that in to the second term and you get: 7 = 3 a1 which gives a1 = 2---you always round down. The answer is 12 so far. We have a carry of 1, which we need to add to the next one. That gives us: 12 = 3 * 1 * 2^2 + 3 * 1^2 * a2 = 12 + 3 a2 Which leaves a2 = 0. So the answer is 12.0 so far. As you go further, there are more and more terms hence the "scaling" phenomenon you see. Every time you are solving a polynomial equation in one variable, where the solution is an integer 0 through 9; my guess is that on an abacus you do binary search instead of root finding. On paper this sounds easy, but on the abacus it sounds impossible—each of those adds and multiplies is its own crazy sequence of steps.

smoldesu 5y ago

Low-precision mathematics saves lives.

Koshkin 5y ago
(Allowing wider margins of error and larger tolerances means better reliability and stability.)

klyrs 5y ago

I'll have to pull this one out next time I hear somebody complaining about the imperial system of measures being useless

ithkuil 5y ago
nah, that's clearly an argument in favour of full blown duodecimal system. 1729 decimal is 1001 duodecimal, Every kid raised in a duodecimal system that had to learn how to convert into decimal in order to interact with us barbarians knows this fact.
Koshkin 5y ago
I love the chaos of the imperial system. It has the taste of life itself.
- mensetmanusman 5y ago
  The clock is base 12 because it has more rational fractions, those Babylonians!
  - Koshkin 5y ago
    Their number system was base 60, actually (one prime factor more).

uwagar 5y ago

feynman is always smart in his autobiography.

mandmandam 5y ago
If you're gonna try and convince people Feynman wasn't smart, try a bible group.

dang 5y ago

One past thread: Feynman vs. The Abacus - https://news.ycombinator.com/item?id=5849665 https://news.ycombinator.com/item?id=5849665 - June 2013 (33 comments)

beloch 5y ago

"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." There are people lauding and panning Feynman in this thread, but this story illustrates a very important lesson that I've seen a lot of very smart people fail to learn. Whenever you're doing a calculation, especially if you're using a computer or calculator, make an approximate estimate of what the result should be. Don't just assume that whatever method you use will produce a correct answer. A man with an abacus (or computer) is probably going to be faster than someone like Feynman most of the time, but he's also going to make mistakes. Big ones. He might not make them frequently, but mistakes always happen sooner or later. The trick is to catch them when they do happen. That's hard to do if you have absolutely no sense of what the answer should be. If, however, you start from a rough estimate using anything your brain can come up with, as Feynman does in his anecdote, you're a lot less likely to produce a badly wrong answer without realizing it. If this contest had gone on long enough, Feynman would have been beaten badly in several rounds if he messed up a calculation and had to start over. The thing is, Feynman would know when this happened because the outcome wouldn't agree with his initial estimate. The man with the abacus would, eventually, have produced answers off by several orders of magnitude without realizing. Rough estimates are important.

Stratoscope 5y ago
Back in the slide rule era, we had to learn this lesson. The slide rule would give you the (rather approximate) digits, but you had to know where to put the decimal point.

pmoriarty 5y ago

Flash Anzan at the All-Japan National Soroban Championship 2012: [1] Background on how students train to do this using the Soroban (the Japanese Abacus): [2] Another demo: [3] Two nine-year-old girls play shiritori[4] ("a Japanese word game in which the players are required to say a word which begins with the final kana of the previous word") while adding 30 three-digit numbers flashed in 20 seconds: [5] [1] - https://www.youtube.com/watch?v=7ktpme4xcoQ https://www.youtube.com/watch?v=7ktpme4xcoQ [2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y https://www.youtube.com/watch?v=Px_hvzYS3_Y [3] - https://www.youtube.com/watch?v=JawF0cv50Lk https://www.youtube.com/watch?v=JawF0cv50Lk [4] - https://en.wikipedia.org/wiki/Shiritori https://en.wikipedia.org/wiki/Shiritori [5] - https://www.youtube.com/watch?v=_vGMsVirYKs https://www.youtube.com/watch?v=_vGMsVirYKs