Quote:
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Originally Posted by vinaur
Here's an example of -1=1 (incorrect of course, but I won't tell where, yet):
1=1
1/1=1/1
-1/1=-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^2=1
-1=1
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This one is fairly easy, when you take square roots you need to assume both values that the input could have produced (i.e. sqrt(1) = -1,1). So your reasoning should have been something like:
sqrt(-1/1)=sqrt(1/-1)
sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)
(±i)/(±1)=(±1)/(±i) => i^2=±1 (if you work out all combinations)
-1=1, or -1=-1 (the latter is correct)
When people end up with something like 2=1, it doesn't mean they actually proved 2=1, it means there is a contradiction which means the they did something incorrect or their initial conditions were wrong. Proof by contradiction is very strong tool if used correctly. However I understand this is all in fun.