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Topic: Proof that 1>1 (Read 1794 times) |
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Miguel Z
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Proof that 1>1
« on: Feb 19th, 2006, 11:07am » |
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just stumbled upon this site. i think it's great! nice work!
here's one little problem that I found in a complex analysis book. it is similar to the 0=1 problem. this one also does require a little bit of math, but it should be very easy if you know complex analysis.
first of, for those not familiar with complex analysis, note that e^(2 pi i) = 1. now for the "proof":
1 > e^(-4 pi^2)
= e^(2 pi i)(2 pi i)
= [e^(2 pi i)]^(2 pi i)
= 1^(2 pi i)
= 1
therefore, 1>1. where's the flaw?
cheers!
Miguel
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Icarus
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Re: Proof that 1>1
« Reply #1 on: Feb 19th, 2006, 11:45am » |
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There are some similar "calculations" that have come up before, but they are deeply buried now, so it is good to see the same trick brought forward again.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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oh_boy
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Re: Proof that 1>1
« Reply #2 on: Apr 6th, 2006, 8:28pm » |
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on Feb 19th, 2006, 11:07am, Miguel Z wrote:
2 pi i = 0, so what is 00? 
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Icarus
wu::riddles Moderator
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Re: Proof that 1>1
« Reply #3 on: Apr 8th, 2006, 4:34pm » |
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on Apr 6th, 2006, 8:28pm, oh_boy wrote:| 2 pi i = 0, so what is 00? |
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2 i does not equal 0. Indeed, |2i| > 6.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Michael Dagg
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Re: Proof that 1>1
« Reply #4 on: Apr 15th, 2006, 9:19pm » |
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I thought the flaw happened because we can't make a little house with the > and = characters.
See, =>, >=, >=>, =>=. I can't get the roof up.
Oh, wait . 1^1 . That's it.
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Regards,
Michael Dagg
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shadebreeze
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Re: Proof that 1>1
« Reply #5 on: Jun 16th, 2006, 6:47am » |
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Can you extend the same power properties for real numbers to complex numbers?
I would say they have to be redefined, and the
e^((2pi i)(2pi i)) = (e^(2pi i))^(2pi i)
equation might be wrong in the complex plane.
However, I am just guessing, I studied complex analysis long ago and I don't remember it very well.
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Icarus
wu::riddles Moderator
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Re: Proof that 1>1
« Reply #6 on: Jun 16th, 2006, 3:09pm » |
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You can and you can't. The problem is, in the complex plane exponentiation is multi-valued. That is, there is more than one number a for which a = bc makes sense. In the special case that the base b is positive real, we have a "preferred value" that we normally use. But even in this case it is not the only one.
The various rules of exponentiation can still be said to hold, in the sense that both sides of the equation will give a value that is valid for "bc", but sometimes the values that they settle on are not the same.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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srn437
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the dark lord rises again....
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Re: Proof that 1>1
« Reply #7 on: Sep 1st, 2007, 2:57pm » |
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(e^2(pi)(i))^2(pi)(i) does not equal e^2(pi)(i)(2)(pi)(i). It would if it didn't involve complex numbers.
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| « Last Edit: Sep 3rd, 2007, 8:41pm by srn437 » |
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