Why Does the Equation 1 = -1 Seem Correct?

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The discussion centers on the flawed reasoning that leads to the incorrect conclusion that 1 equals -1 through manipulation of square roots. The error arises from improperly applying the property of square roots to complex numbers, particularly in the step involving the division of square roots of negative and positive values. It is highlighted that the equation sqrt(-1/1) = sqrt(-1)/sqrt(1) does not hold true in the context of complex numbers. The inconsistency is resolved by noting that when dealing with square roots, especially in polar form, one must consistently choose the correct branch of the square root. This illustrates the importance of understanding the properties of complex numbers in mathematical operations.
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I've tried and tried and I cannot find the error in the reasoning below. It's probably something simple and I'll feel like an idiot when someone explains it.
i = i
sqrt(-1) = sqrt(-1)
sqrt(1/-1) = sqrt (-1/1)
sqrt(1)/sqrt(-1) = sqrt(-1)/sqrt(1)
sqrt(1) * sqrt(1) = sqrt(-1) * sqrt(-1)
[sqrt(1)]^2 = [sqrt(-1)]^2
1 = -1

Does it have something to do with sqrt(-1/1) = sqrt(-1)/sqrt(1)? Does complex numbers not obey this property?
 
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doesnt [sqrt(a)]^2 = |a| ? maybe that's just in the reals
 
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\frac{\sqrt{1}}{\sqrt{-1}} pretty much sums up what's wrong with the train of thought. This is a good reason why when you divide a real number by an imaginary number that you must first multiply by the conjugate on the numerator and denominator. I can't nail down a good reason other than that.

I should note that this inconsitency does not exist if you do the division with sqrt(1) and sqrt(-1) in polar form.
 
square rooting is 1 to 2, so you need to pick a choice of square root. You've not done so consitently.
 

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