Two Plus Two Older Archives Ok so I just proved 1 = -1. Someone help me find my error.
 FAQ Members List Calendar Search Today's Posts Mark Forums Read

#1
10-13-2005, 11:00 PM
 DrPublo Member Join Date: Jan 2004 Location: Princeton, NJ Posts: 38
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.

(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
10-13-2005, 11:23 PM
 Guest Posts: n/a
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
10-13-2005, 11:28 PM
 DrPublo Member Join Date: Jan 2004 Location: Princeton, NJ Posts: 38
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
10-13-2005, 11:30 PM
 benkahuna Junior Member Join Date: Oct 2004 Posts: 4
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
10-13-2005, 11:37 PM
 DrPublo Member Join Date: Jan 2004 Location: Princeton, NJ Posts: 38
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
10-13-2005, 11:38 PM
 Guest Posts: n/a
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).
#7
10-13-2005, 11:41 PM
 Guest Posts: n/a
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).
#8
10-13-2005, 11:49 PM
 DrPublo Member Join Date: Jan 2004 Location: Princeton, NJ Posts: 38
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
#9
10-13-2005, 11:52 PM
 benkahuna Junior Member Join Date: Oct 2004 Posts: 4
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.
#10
10-13-2005, 11:56 PM
 gumpzilla Senior Member Join Date: Feb 2005 Posts: 1,401
Re: Ok so I just proved 1 = -1. Someone help me find my error.

You're picking the two different branches, since both i and 1 / i when squared yield -1, but themselves differ by a factor of -1, and this is where the problem comes in. So the answer to your question is that you are being inconsistent in how you define sqrt(-1), and if you're consistent in that regard then the problem goes away. Difficulties of this kind are extremely common when working with complex variables, which is part of what makes them confusing.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Two Plus Two     Two Plus Two Internet Magazine     About the Forums     MOD DISCUSSION     ISOP General Poker Discussion     Texas Hold'em     Beginners Questions     Books and Publications     Televised Poker     News, Views, and Gossip     Brick and Mortar     Home Poker     Poker Beats, Brags, and Variance     Poker Theory Limit Texas Hold'em     Mid- and High-Stakes Hold'em     Medium Stakes Hold'em     Small Stakes Hold'em     Micro-Limits     Mid-High Stakes Shorthanded     Small Stakes Shorthanded PL/NL Texas Hold'em     Mid-, High-Stakes Pot- and No-Limit Hold'em     Medium-Stakes Pot-, No-Limit Hold'em     Small Stakes Pot-, No-Limit Hold'em Tournament Poker     Multi-table Tournaments     One-table Tournaments Other Poker     Omaha/8     Omaha High     Stud     Other Poker Games General Gambling     Probability     Psychology     Sports Betting     Other Gambling Games     Rake Back     Computer Technical Help Internet Gambling     Internet Gambling     Internet Bonuses     Software 2+2 Communities     Other Other Topics Other Topics     Sporting Events     Politics     Science, Math, and Philosophy     The Stock Market

All times are GMT -4. The time now is 08:26 PM.