November 14th, 2013, 09:41 PM  #1 
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  what is the square root of +1 ?
I have a question, because squaring on both sides gives the identity 1=1 . However, if you use imaginary number i which is defined by you will get Because by definition of i We will arrive at the result +1 = 1 , is there perhaps an error somewhere? 
November 14th, 2013, 10:45 PM  #2  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: what is the square root of +1 ? Quote:
That's why you have to write: Quote:
So: With complex numbers there is no attempt to define square root (or the nth root) as any particular root. Unlike real numbers, there is no clear concept of the "positive one". So, for example: And the nth root will always be n complex numbers.  
November 15th, 2013, 12:47 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
That's not right. By convention, ?(1) = i, not i. By convention, if z is a complex number with a nonzero imaginary part, ?z has an imaginary part with the same sign as the imaginary part of z. The equation (?xy) = (?x)(?y) holds for all positive values of x and y. As already shown, it doesn't hold for x = y = 1. 
November 15th, 2013, 02:19 AM  #4 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: what is the square root of +1 ?
I must admit that's news to me. To take the argument set out by the OP: This must break down somewhere. I guess you have to say that, with that convention, for complex numbers: ? 
November 15th, 2013, 03:56 AM  #5  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: what is the square root of +1 ? Quote:
Look at my first post, 1 has two roots so perhaps holds for negative values of z and w too. Look at how Wolfram Alpha calculates square roots (also known as 2nd roots) of 1: http://www.wolframalpha.com/input/?i=sqrt1 You can see two real roots +1 (real root, principal root) and 1 (real root)  
November 15th, 2013, 05:04 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 
Sometimes it works for complex numbers and sometimes it doesn't.

November 15th, 2013, 10:14 PM  #7 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: what is the square root of +1 ? So everything is right as rain again. 
November 16th, 2013, 02:21 AM  #8  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: what is the square root of +1 ? Quote:
Ok, very good. Consider now but what is Is Or is  
November 16th, 2013, 05:07 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038  The rule that (z^m)^n = z^(mn) always works if z > 0 and both m and n are real, but only sometimes works otherwise. 
November 16th, 2013, 10:59 AM  #10  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: Quote:
Again I write the same formulas but now a little differently. How to prove that holds even when both z and w are negative: Because And this time the paradox +1 = 1 is hidden.  

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