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September 1st, 2012, 03:36 AM  #1 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  How to prove that "i(ioota)" is irrational?
How to prove that "i(ioota)" is irrational by means of contradiction method?

September 1st, 2012, 03:39 AM  #2  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to prove that "i(ioota)" is irrational? Quote:
 
September 1st, 2012, 03:46 AM  #3 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  Re: How to prove that "i(ioota)" is irrational?
?1 or sqrt(1)

September 1st, 2012, 03:49 AM  #4 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to prove that "i(ioota)" is irrational?
Oh, you're talking about imaginary unit then. But, i is neither rational nor irrational. In some sense, it can be even called rational since the absolute value of i is 1, which is rational. But i is never irrational. 
September 1st, 2012, 03:53 AM  #5 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  Re: How to prove that "i(ioota)" is irrational?
but i need to prove it as irrational by using contradiction method.

September 1st, 2012, 03:55 AM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to prove that "i(ioota)" is irrational? Quote:
i is neither rational nor irrational I told you that. From where did you got this problem?  
September 1st, 2012, 04:03 AM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 462 Math Focus: Calculus/ODEs  Re: How to prove that "i(ioota)" is irrational?
I suppose you could begin be assuming i is rational: where Since we must have and , we have a contradiction. 
September 1st, 2012, 04:05 AM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to prove that "i(ioota)" is irrational? Quote:
 
September 1st, 2012, 04:08 AM  #9 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 462 Math Focus: Calculus/ODEs  Re: How to prove that "i(ioota)" is irrational?
Well, we have really only shown that i is not real, but as it's not real, then it's also not rational! 
September 1st, 2012, 04:10 AM  #10  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to prove that "i(ioota)" is irrational? Quote:
 

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