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The Magic Cafe Forum Index » » Magical equations » » Infinite product ratio (0 Likes) Printer Friendly Version

Larry Barnowsky
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What do you get when you multiply 2 times 2x2x4x4x6x6x8x8...... and divide it by
3x3x5x5x7x7x9x9...?
Who first discovered this?


Posted: Dec 12, 2005 6:59pm
-------------------------------------------------
Hint: It was published in 1655.
stanalger
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Is it approximately equal to the square root of the square root of
2143/22?

Didn't J.W. pass this on to Simon as he was going to the fair?
Larry Barnowsky
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The answer is a more significant number but from your post I think you know what it is.
LostSoul
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OK, I'll take a stab at Pi? It seems to be pretty close using my rough math skills.

Dave
Larry Barnowsky
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Pi is correct. Who derived this first in 1655?
stanalger
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I think it
WAS LI'L JOHN.
Larry Barnowsky
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The correct answer is Pi and was derived in 1655 by John Wallis. The very astute stanalger submitted the correct answer by PM.
James F
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Very interesting! I didn't know that...Thanks for posting.
Larry Barnowsky
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There's an excellent paperback book called "A History of Pi" by Petr Beckamnn fromm 2000BC to the present.


Posted: Dec 17, 2005 11:48am
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Here's alink for the book:
amazon.com.
Heinz Weber
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Sorry, I don't get this one. Each number of the odd line (3x3x5x5x...) is bigger than the number on the same place in the even line (2x2x4x4x...). I think the division has to go to 0. Why am I wrong?
Thanks, Heinz
Larry Barnowsky
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Heinz,
Good question. As we approach infinity the individual odd numbers and correspoding even numbers get closer together in value so their quotient gets closer to 1 not zero. It starts with 2/3x2/3= 4/9 =.444 At say 100 the new product will include 100/101x100/101= .980 which is getting close to 1. This allows the product to converge to a non zero number. Also, the product should be properly written 2 times 2x2x4x4x6x6x8x8 divided by 1x3x3x5x5x7x7. In the initial post I left out the 1 which was a mistake. For the first few numbers: 2 x (2x2)/(1x3) x (4x4)/((3x5) x (6x6)/(5x7) x (8x8)/(7x9) = 2.97 and so on until be get 3.14159....
Heinz Weber
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Thank you Larry,

the missing 1 was my problem (i made an excel-sheet), now I see...

In your last message I think it should read 2/1x2/3 instead of 2/3x2/3, an corresponding 100/99x100/101....

(I think if all part-quotients were smaller than 1 the whole thing (their product!) could not reach 1. Only with infinite sums there are a couple of astonishing effects, sum of 1/n to name a well known example.)

Heinz

PS: please forgive my sometimes poor english and terminology, I never wrote math-related textes in english before
Larry Barnowsky
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Your English is very clear Heinz. It's amazing that this was pulished in 1655!
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