MathPages
For those who love maths and physics, here is a site that is bound to keep you hooked (especially on sleepy Friday evenings as today).
The history section is particularly impressive. Most of us know about the Hardy-Ramanujan number and how Hardy mentions 1729 as a "dull number". There is an interesting anecdote from Feynman's book where this number makes another appearance, hence proving that it isn't so dull after all.
The number 1729 also appears in Richard Feynman's collection of anecdotes (Surely You're Joking, Mr. Feynman!). In a chapter entitled "Lucky Numbers" he tells of going into a small restaurantin Brazil to eat lunch. He's the only customer, so he has four waiters standing around him. Then a Japanese man enters the restaurant,and he is selling abacuses. The man challenges the waiters to an adding contest, but they don't want to lose face, so they tell him to go challenge the customer sitting there (Feynman). They first have an addition contest, and the abacus wins easily. Then they try multiplication,and the abacus wins again, but it's a bit closer. Then they try long division, and this time it's a tie. As Feynman says, the more difficult the problem, the better he can do with pencil and paper compared with the abacus. Finally the Japanese man calls out "Raios cubicos!"... he wants to challenge Feynman to cube roots.
Feynman says the man wrote a number, "any old number", down on a piece of paper, and he still remembers the number... 1729.03. The salesman begins working furiously on his abacus, but Feynman just sits there smiling, and says "12.002...". The abacus salesman was beaten, and left the restaurant in disgust. The waiters are amazed at Feynman's calculating prowess.
He explains that he happenned to remember that there are 1728 cubic inches in a cubic foot, so the cube root of 1729 must be just slightly greater than 12. Then he just needed to account for the extra 1.03. To do this he neglected the 0.03 and used the binomial expansion
(1728 + 1)1/3 = 12(1 + 1/1728)1/3 = 12(1 + (1/3)(1/1728) + ...
so the amount by which the cube root of 1729 exceeds 12 is about4/1728 = 1/432.
You can only get two 432's out of 1000, so the firstnon-zero digit is 2, leaving a remainder of 136, and bringing downanother zero we know there are three 432's in 1360, so the nextdigit is 3, and so on. This gives 12.0023...
The history section is particularly impressive. Most of us know about the Hardy-Ramanujan number and how Hardy mentions 1729 as a "dull number". There is an interesting anecdote from Feynman's book where this number makes another appearance, hence proving that it isn't so dull after all.
The number 1729 also appears in Richard Feynman's collection of anecdotes (Surely You're Joking, Mr. Feynman!). In a chapter entitled "Lucky Numbers" he tells of going into a small restaurantin Brazil to eat lunch. He's the only customer, so he has four waiters standing around him. Then a Japanese man enters the restaurant,and he is selling abacuses. The man challenges the waiters to an adding contest, but they don't want to lose face, so they tell him to go challenge the customer sitting there (Feynman). They first have an addition contest, and the abacus wins easily. Then they try multiplication,and the abacus wins again, but it's a bit closer. Then they try long division, and this time it's a tie. As Feynman says, the more difficult the problem, the better he can do with pencil and paper compared with the abacus. Finally the Japanese man calls out "Raios cubicos!"... he wants to challenge Feynman to cube roots.
Feynman says the man wrote a number, "any old number", down on a piece of paper, and he still remembers the number... 1729.03. The salesman begins working furiously on his abacus, but Feynman just sits there smiling, and says "12.002...". The abacus salesman was beaten, and left the restaurant in disgust. The waiters are amazed at Feynman's calculating prowess.
He explains that he happenned to remember that there are 1728 cubic inches in a cubic foot, so the cube root of 1729 must be just slightly greater than 12. Then he just needed to account for the extra 1.03. To do this he neglected the 0.03 and used the binomial expansion
(1728 + 1)1/3 = 12(1 + 1/1728)1/3 = 12(1 + (1/3)(1/1728) + ...
so the amount by which the cube root of 1729 exceeds 12 is about4/1728 = 1/432.
You can only get two 432's out of 1000, so the firstnon-zero digit is 2, leaving a remainder of 136, and bringing downanother zero we know there are three 432's in 1360, so the nextdigit is 3, and so on. This gives 12.0023...
5 Comments:
Ouch, that took a while to understand.
Btw I found the equation hard to read should probably use the superscript <sup> tag.
Ok, there is an alignment mistake there..Let me correct it (the two 1/3 stands for two ends of the "=", they should not be together)
Try something like
(1728 + 1)<sup>1/3</sup> = 12(1 + 1/1728)<sup>1/3</sup>
or go for an image of the equation. Both are perfectly respectable. Better than trying to manually do the layouting. :)
Wokay - done!
Impressive
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