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September 19th, 2011, 06:03 AM  #1 
Member Joined: Aug 2011 From: student Posts: 55 Thanks: 0  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. 
September 19th, 2011, 06:07 AM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5  Re: find the mistake ?
It's the middle "equality", when you say that the quotient of the radicals is the radical of quotients.

September 19th, 2011, 04:02 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,849 Thanks: 742  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. 
September 20th, 2011, 02:53 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,103 Thanks: 2321 
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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