posted
Ye-p, it's due to the complex areas where it breaks down. Also, log 1 != 0 when you consider complex values.
Posts: 15770 | Registered: Dec 2001
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posted
its been awhile since i saw this, so forgive me for being sort of vague, but i saw a really neat investigation of what happens when you try to differentiate the function
f(x)= x^x
which if i remember correctly got made complex (f(x)=x^z). Anyway, the function requires some really crazy differentiation which i understood a few years ago but since i've been avoiding math for some time now i don't remember well enough to explain.
anyway, the point is that the derivative of the function at 0 ends up being either 0 OR 1, depending on how you look at it.
maybe someone who has taken some math classes more recently than i will recognize this and be able to explain it better...
[edit] i remember now, it's because 0 raised to any number is 0, but any number raised to the 0 power is 1, which creates a contradiction at 0 for the original function
posted
Wait a second. If a set of factors equal zero when multiplied together, that doesn't mean that any one of them could be zero if you just divide both sides by the others. It only means that at least ONE of the factors is zero.
Still doesn't fully answer your paradox, but still ... Math makes no sense. When I think about how many ridiculous things astronomers believe about black holes and the like just because they can make it happen with abstract mathematics ...
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posted
And yet a lot of those ridiculous things have turned out to be right: Hawking radiation, for instance.
Posts: 15770 | Registered: Dec 2001
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... But then the manager discovers that he's made a mistake, and the room only costs 25$, and gives five dollar to the porter to return to the guests. The porter, not being very good at math nor a very honest person, considers it too much trouble to divide five $ on three persons, and gives back one $ to each of the guests and keeps two for his own pockets. The guests have now paid nine $ each. Three times nine is 27. The porter has two. That makes 29.
quote:... But then the manager discovers that he's made a mistake, and the room only costs 25$, and gives five dollar to the porter to return to the guests. The porter, not being very good at math nor a very honest person, considers it too much trouble to divide five $ on three persons, and gives back one $ to each of the guests and keeps two for his own pockets. The guests have now paid nine $ each. Three times nine is 27. The porter has two. That makes 29.
Hey, where did the last dollar go?!
Whoa. That's kinda trippy. I mean, despite having an American ejukashun, I know $25 + $3 = $28 -- not $27 -- but I'm still following the logic of the paragraph above.
A few more of these, mixed with weed and airplane model glue, and I'm well on my way to unravelling the secrets of the universe...
Posts: 3293 | Registered: Jul 2002
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posted
$30 paid. $25 stays with the hotel. $5 goes with the clerk. clerk gives $3 to men, keeps $2. hotel has $25, men have $3, clerk has $2. $25 + $3 + $2 = $30. All the money is accounted for.
Posts: 15770 | Registered: Dec 2001
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posted
Well, that's all well and good, fugu13, but it does not really answer the question. The fun thing is to figure out exactly where and how the one buck disappears in the alternative way of counting that the problem provides. It isn't exactly high mathematics but that makes it perfect for those who, like me, aren't mathematicians but enjoy logical problems nevertheless.
posted
The problem occurs with adding the $2. The $2 should be subtracted, just as the $3 was (though in a roundabout way, by phrasing it as an addition of what the three men had each paid), resulting in the price remaining with the hotel of $25, just as is expected.
Posts: 15770 | Registered: Dec 2001
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How fond I am of that proof. Actually, the one I saw was slightly different; it was in Zero: The Biography of a Dangerous Idea, and it started out by letting a and b equal one. Ultimately it proved that 1=0. Then, as I recall, the book went on to show that Winston Churchill was a carrot.
By keeping the two dollars, the desk clerk made the price of the room 27 dollars. This fits with the part of the riddle where you multiply three times nine to get the twenty seven dollars.
But the problem is in the next step. You can't add the two dollars to the 27 again, because it's already included in what the clients paid--that two dollars came out of the 27 dollars.
Rather, you have to add the three dollars that was given back to the customers--which don't even figure in to the original riddle, but were obviously part of the original transaction.
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