Error in proof that $1 = -1$

Question

I have created a proof that$ 1 = -1$ but I know that this is impossible. Could someone help me find the flaw in this proof...

$i = \sqrt{-1}$

Given

$i^2 = -1$

Given

$i^4 = 1$

Given $i^8 = 1$

Given --------------------------All Common Knowledge Above

$i^4= i^8$

Take sqrt of both sides...

$i^2=i^4$

Take sqrt of both sides...

$i= i^2$

$i=-1$

$i^2= -1$ (sub for $i$)

$-1 \times -1$ (sub for $i^2$) = $1$

$1=-1$

Thankyou all for helping me. I looked at the other questions and this question is not a duplicate. However, we all have one common error; we forgot +- when taking the square root of i^4 = i^2

We have fairly good intuition about positive numbers. Negative, not so much. Nonreal, less still. Applying so-called rules of algebra beyond our zone of comfort has to be done very carefully. — André Nicolas, Jul 18 '14 at 06:23
@AsafKaragila I don't think it's a duplicate of the one you linked to, which is about getting $\sqrt{ab} = \sqrt{a} \sqrt b$ wrong. I'm sure this is a duplicate of something, though. — , Jul 18 '14 at 08:14

syusim · Accepted Answer · 2014-07-18T06:50:14.473

6

What people seem to be calling you out for without explanation is that if $a^2 = b^2$, then we can have that $a = \pm b$. We can't know which of $b$ or $-b$ we started with, though.

So when you say that $i^4 = i^8$, then good. You're on the right track. But your next step needs to be that $i^2 = \pm i^4$. You then have no contradiction because one of $i^4$ and $-i^4$ is certainly equal to $i^2$.

I hope this helps you. Always try things like these, even if others call you out for being silly. Before being amazing you have to be a little silly.

edited Jul 18 '14 at 06:50

answered Jul 18 '14 at 06:38

syusim

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+1 for "Before being amazing you have to be a little silly." – Vincent Jul 18 '14 at 08:08
Ah, thank you so much. The irony is that none of my highschool math teachers knew why it is wrong either. Do you know who else made this famous mistake? Einstein. E = +-MC^2 which demonstrates the existence of negative energy. – user1939991 Jul 18 '14 at 14:15

5xum · Answer 2 · 2014-07-18T08:18:00.357

2

This is a typical example of the misuse of the "$\sqrt .$" function. You said that $\sqrt{i^4}=i^2$ which is not true since $i^4=1$ and $\sqrt{1}=1$, not $i^2=-1$.

edited Jul 18 '14 at 08:18

answered Jul 18 '14 at 06:23

5xum

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Can anyonw show me what I did wrong instead of just downvoting me? Does anyone think that $\sqrt{i^4}=i^2$? – 5xum Jul 18 '14 at 06:26
I did not downvote, but $\sqrt{i^4} = i^2$ is certainly possible since $i^2 \cdot i^2 = i^4$. The "true" answer would be that $\sqrt{i^4} = \pm i^2$, as the others have pointed out. – naslundx Jul 18 '14 at 08:01
@naslundx That is completely false. The square root function is a function mapping positive reals to positive reals. For example, $\sqrt 4 = 2$. The point is that from $x^2=y$, if $y>0$, we can only conclude that $x=\sqrt y$ OR $x=-\sqrt y$. However, $\sqrt y$ is still a well defined number that is not one of two numbers. – 5xum Jul 18 '14 at 08:04
There ought to be $i^2$ in the last phrase, I believe. – Jul 18 '14 at 08:15
@T.Bongers Thank you. Fixed. – 5xum Jul 18 '14 at 08:17

Error in proof that $1 = -1$

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