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Ok so I just proved 1 = -1. Someone help me find my error.
Hi guys. First post in this forum.
Working on a problem set recently, a few friends and I accidentally discovered a proof of -1=1, and for the life of us we can't find out what we did wrong. And it's not like we're math slouches either; we're all graduate students in physical/theoretical chemistry. From what I understand posting TeX doesn't work on 2+2, so you'll have to follow my algebra. Start with the identity (E-V)^(1/2) = (E-V)^(1/2) Now multiply each side by -1, except on the RHS substitute i^2 for -1 (where i of course is the imaginary number). (-1)(E-V)^(1/2) = (i^2)(E-V)^(1/2) Now divide through by i (-1/i)(E-V)^(1/2) = i*(E-V)^(1/2) But since i is just the square root of -1, we can subsume it into the square root of E-V (-1)[(E-V)/-1]^(1/2) = [(-1)(E-V)]^(1/2) and then rearrange the interior of the square root to find (-1)(V-E)^(1/2) = (V-E)^(1/2) or -1 = 1. No dividing by zero in this proof either. Where did I make a mistake? The Doc |
#2
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
I suspect the error occurs when you take the square root of -1. Normally the square root of a positive number can be positive or negative. eg sqrt(9) = +3 or -3
When you take the square root of -1 you are saying there is only one possible answer (i) I might be out to lunch, but I suspect that is where the problem lies. |
#3
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
No question that -1 has both "positive" and "negative" square roots, as i^2 = (-i)^2 = -1. But I think taking the same root on both sides of the equation should maintain equality? For example, it would be odd to remark that 9 = 9 but then upon taking the square root of both sides of the equation conclude 3 = -3.
The Doc |
#4
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
You have performed an invalid operation between steps 3 and 4.
The 2 terms in step 3 equal each other. The 2 in step 4 do not. Maybe you can't use -1/i instead of i algebraicly as you have. |
#5
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
[ QUOTE ]
You have performed an invalid operation between steps 3 and 4. The 2 terms in step 3 equal each other. The 2 in step 4 do not. [/ QUOTE ] Yes, I know that. The question is what principle of mathematics have I violated. [ QUOTE ] Maybe you can't use -1/i instead of i algebraicly as you have. [/ QUOTE ] -1/i = i. Try multiplying through by i. The Doc |
#6
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
[ QUOTE ]
[ QUOTE ] Maybe you can't use -1/i instead of i algebraicly as you have. [/ QUOTE ] -1/i = i. Try multiplying through by i. The Doc [/ QUOTE ] I know they're equivalent, thanks. In other news 4/2=2. I'm sure an elementary search of a place like wikipedia will tell you why you can't deal with i as you have. I've told you what I know and I'm not going to search for the answer for you. I might know, but I just don't deal with a imaginary numbers very often in daily life. I was thinking what the other poster was thinking about there being positive and negative square roots being the issue, but since no operations were performed taking a sqaure root of a number, I don't think it applies here. It's not like that lame proof that 1+1=1. |
#7
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
I'm not sure if this helps, but I agree with the previous poster that that particular step was incorrect.
sqrt(-1/1) = sqrt(-1)/sqrt(1), but sqrt(1/-1) is - sqrt(1)/sqrt(-1) not sqrt(1)/sqrt(-1). |
#8
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
(-1)(-1) = 1, so sqrt((-1)(-1)) = 1, but sqrt(-1)sqrt(-1) = i^2 = -1 (not 1).
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#9
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
[ QUOTE ]
(-1)(-1) = 1, so sqrt((-1)(-1)) = 1, but sqrt(-1)sqrt(-1) = i^2 = -1 (not 1). [/ QUOTE ] You just blew my mind. What is the technical reason this proof is incorrect? The Doc |
#10
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Re: Ok so I just proved 1 = -1. Someone help me find my error.
I'm not sure I can give you a technical reason, but certain mathematical laws or priciples that work for real numbers don't necessarily work for complex numbers.
sqrt(a/b) = sqrt(a) / sqrt(b) is only true for real numbers. I have a hard enough time even understanding what a complex number actually is. [img]/images/graemlins/smile.gif[/img] |
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