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September 19th, 2011, 06:03 AM   #1
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find the mistake ?

We know that $\displaystyle i^{2} = -1$ and $\displaystyle i = \sqrt{-1}$

$\displaystyle i = \sqrt{-1} = \frac{\sqrt{-1}}{1} = \frac{\sqrt{-1}}{\sqrt{1}} = \sqrt{\frac{-1}{1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \frac{1}{\sqrt{-1}} = \frac{1}{i}$

if the last expression $\displaystyle i = \frac{1}{i}$ both sides multiply with $\displaystyle i$ and find $\displaystyle i^{2} = 1$

$\displaystyle i^{2}=1$ ?????

Last edited by skipjack; August 15th, 2019 at 04:54 AM.
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September 19th, 2011, 06:07 AM   #2
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Re: find the mistake ?

It's the middle "equality", when you say that the quotient of the radicals is the radical of quotients.
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September 19th, 2011, 04:02 PM   #3
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Re: find the mistake ?

√(-1) has two possibilities, i or -i, depending on the context.

Last edited by skipjack; August 15th, 2019 at 05:02 AM.
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September 20th, 2011, 02:53 AM   #4
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That's confusing. There are two square roots of -1, namely i and -i, but (-1) refers to i, not -i.

The rule that (a/b) = a / b does not apply when a is positive and b is negative.
Use √(a/b) = - √a / √b instead.

Similarly, the rule that (a) (b) = (ab) is replaced with (a) (b) = - (ab) if a and b are both negative.

Last edited by skipjack; August 15th, 2019 at 05:01 AM.
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