Why Does the Equation 1 = -1 Seem Correct?

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Discussion Overview

The discussion revolves around the reasoning that leads to the conclusion that 1 = -1, specifically examining the manipulation of square roots involving real and imaginary numbers. The scope includes mathematical reasoning and conceptual clarification regarding the properties of square roots in different number systems.

Discussion Character

  • Mathematical reasoning, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant presents a series of steps leading to the conclusion that 1 = -1, questioning the validity of the manipulation involving square roots of negative numbers.
  • Another participant suggests that the equation [sqrt(a)]^2 = |a| may only apply in the realm of real numbers, implying a limitation in the reasoning presented.
  • A third participant points out that dividing a real number by an imaginary number requires careful handling, such as using the conjugate, to avoid inconsistencies in the reasoning.
  • Another participant notes that the inconsistency arises from not consistently choosing a square root, highlighting the need for clarity in the choice of roots when dealing with square roots.

Areas of Agreement / Disagreement

Participants express differing views on the manipulation of square roots and the properties of numbers in different systems, indicating that multiple competing views remain without a consensus on the reasoning's validity.

Contextual Notes

Limitations include the dependence on the properties of square roots in real versus complex numbers and the need for consistent choices in mathematical operations involving roots.

sparkster
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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
 
Last edited:
[tex]\frac{\sqrt{1}}{\sqrt{-1}}[/tex] 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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