1 = -1 where is the error

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In summary, the error in the reasoning above lies in the assumption that sqrt(-1/1) is equal to sqrt(-1)/sqrt(1). This is not always true for complex numbers and results in an inconsistency when solving the equation. To avoid this, it is important to use the correct properties and consider the different choices of square roots for complex numbers.
  • #1
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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  • #2
doesnt [sqrt(a)]^2 = |a| ? maybe that's just in the reals
 
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  • #3
[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.
 
  • #4
square rooting is 1 to 2, so you need to pick a choice of square root. You've not done so consitently.
 

1. How is it possible for 1 to equal -1?

This equation, 1 = -1, is mathematically impossible and a clear error. In mathematics, the equal sign denotes that both sides of the equation are equal. Since 1 and -1 are two different values, this equation cannot be true.

2. What could have caused this error?

There are a few possible causes for this error. It could be a simple typo or mistake in writing the equation, or it could be a mistake in the calculation or manipulation of the equation. It could also be a result of using an incorrect mathematical rule or operation.

3. Can this error be corrected?

Yes, this error can be corrected. Since the equation is not mathematically possible, it can be rewritten or manipulated to represent a true statement. For example, using the identity property, we can write 1 + 0 = -1 + 2, which is a true equation.

4. Why is it important to recognize and correct this error?

Mistakes and errors in mathematics can lead to incorrect solutions and conclusions. Recognizing and correcting errors is crucial for obtaining accurate and reliable results in scientific experiments and calculations. It also helps to develop critical thinking and problem-solving skills.

5. Can this error occur in other equations or situations?

Yes, errors can occur in any mathematical equation or situation. It is important to carefully check and double-check all calculations and equations to avoid errors. It is also helpful to have a second person review your work to catch any mistakes that may have been overlooked.

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