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cos(x) = 1?

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This seems like a homework problem, so I won''t give the answer; however, here''s a hint. The flaw is in step 3 of your proof (top set of equations). Look for a problem in the way you''ve factored e(i2pi*(x/2pi)).

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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*EDIT, saw grhodes' post, so I removed the solution.

This reminds me of this 'proof' as well:

-1 = -1

-1/1 = 1/-1

sqrt(-1/1) = sqrt(1/-1)

sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)

i/1 = 1/i

i = -i

1 = -1 //Haha!

The error in the proof above is going from line 3 to line 4. This proof plays on your assumption that a mathematical property will work for all sets of numbers. In this case it is the property that

sqrt(a/b) = sqrt(a)/sqrt(b)

Many dont realize that this is only true if and only if both a and b are nonnegative real numbers.



Qui fut tout, et qui ne fut rien
Invader's Realm

[edited by - Invader X on January 29, 2003 10:33:36 PM]

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Wow, that''s really cool. Does it have to do with the fact that one to an irrational power can have an infinity of different values (i.e. it''s not always one)? That would explain why 1^(x/2pi) doesn''t have to be one.

Here''s an example:
1^pi =
(e^(2pi*i))^pi =
e ^ (2 * (pi^2) * i)

So one to the power of pi can be e to the power of 2 pi squared i. Amazing, isn''t it? Or maybe I''m completely wrong, lol.

Firebird Entertainment

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Its not homework, just some stuff a friend sent me (which I mirrored to my own server to avoid leaching bandwidth, the original links were http://www.people.cornell.edu/pages/mfs24/pictures/cos.gif and http://www.people.cornell.edu/pages/mfs24/pictures/taylor.gif).
I''m only in calclus 2 and dunno anything about defining cosine in terms of e or taylor series (which is why I had to ask and couldn''t figure it out myself).
If you don''t believe me, of course there really isn''t any way I could prove it to you. I just asked cuz I was interested in knowing why cosine isnt equal to 1 =-)

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quote:
Original post by Extrarius
Its not homework, just some stuff a friend sent me (which I mirrored to my own server to avoid leaching bandwidth, the original links were http://www.people.cornell.edu/pages/mfs24/pictures/cos.gif and http://www.people.cornell.edu/pages/mfs24/pictures/taylor.gif).
I'm only in calclus 2 and dunno anything about defining cosine in terms of e or taylor series (which is why I had to ask and couldn't figure it out myself).
If you don't believe me, of course there really isn't any way I could prove it to you. I just asked cuz I was interested in knowing why cosine isnt equal to 1 =-)


The problem is similar to the problem with the -1 = 1 proof.

If you have a calculator that can handle powers of complex numbers (TI-82 and above can), then type in these two things:

(e^(i*2π ))*(2/2π ) //I set x = 2
e^(i*2π *(2/2π )) //Again, x = 2

Compare the results (preferably the decimal results, the TI-89 might put it into e^(..)'s, sin's, cos's, ln's etc. The TI-82 only puts it into decimal format.

Hopefully you'll be able to see the flaw, if you just get confused look over the reasoning for why the -1 = 1 proof doesn't work.

*Curses! ampersand's got messed up!


Qui fut tout, et qui ne fut rien
Invader's Realm

[edited by - Invader X on January 29, 2003 12:37:45 AM]

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quote:
Original post by Invader X
sqrt(a/b) = sqrt(a)/sqrt(b)

Many dont realize that this is only true if and only if both a and b are nonnegative real numbers.



This problem is also confused further by the fact that sqrt(1) = 1 *or* -1.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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quote:
Original post by Extrarius
Its not homework, just some stuff a friend sent me (which I mirrored to my own server to avoid leaching bandwidth, the original links were http://www.people.cornell.edu/pages/mfs24/pictures/cos.gif and http://www.people.cornell.edu/pages/mfs24/pictures/taylor.gif).
I''m only in calclus 2 and dunno anything about defining cosine in terms of e or taylor series (which is why I had to ask and couldn''t figure it out myself).
If you don''t believe me, of course there really isn''t any way I could prove it to you. I just asked cuz I was interested in knowing why cosine isnt equal to 1 =-)


Extrarius,

I have no real reason not to believe you personally, but the fact is that some people occasionally post here who are blatantly trying to take advantage of the forums to avoid doing homework, which I strongly disapprove of. I sometimes have to judge posts at face value. Unfortunately, you stated nothing other than "explain the flaw(s) in the following ''proofs'', which sounds like a flat out request for an answer to a problem that looks an awful lot like homework with no obvious relationship to game development.

I don''t mind occasional theoretical discussions of math/physics, although these forums are focused on practical game development problems. When people are interested in theoretical discussions, I prefer that they follow the same rules described for homework posts, e.g., clearly state why you''re asking such a question and show that you''re at least trying to work the theory yourself. Make sense?

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Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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