March 3rd, 2014, 08:09 PM  #1 
Senior Member Joined: Oct 2013 From: Far far away Posts: 431 Thanks: 18  i am imaginary Is i < or > or = 1? 1) Since i is not on the real number line this question is meaningless ON THE OTHER HAND 2) Since and 1 < 1 it follows that i < 1 What is the right answer? 
March 4th, 2014, 11:54 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,026 Thanks: 2257 
If i is not a real number, none of the choices mentioned in the question applies. The second suggestion is incorrect, as what would follow is i² < 1, which doesn't imply i < 1. 
March 4th, 2014, 12:06 PM  #3  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: i am imaginary Quote:
i cannot be positive because positive number times positive number is positive number. i is neither negative nor positive. But what are +i and i ? Is +i positive and i negative?  
March 4th, 2014, 01:03 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,026 Thanks: 2257 
Both +i (which equals i) and i are imaginary, not real. Hence neither should be called positive or negative. However, the imaginary part of +i is 1 (which is positive) and the imaginary part of i is 1 (which is negative). 
March 4th, 2014, 01:29 PM  #5 
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: i am imaginary
How is this possible? Does i have two values: Because 
March 5th, 2014, 01:49 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 21,026 Thanks: 2257 
No, just as 3² = (3)² = 9 doesn't mean that 3 has two values. Like any nonzero number, 1 has two square roots, i and i. A positive real has one positive and one negative square root. Any other nonzero number has one square root with a positive imaginary part and one square root with a negative imaginary part.

March 5th, 2014, 02:30 AM  #7  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Re: Quote:
3 has two values +3 and 3. Isn't it only a general agreement that 3=+3 ? Why we could not instead use a convention that 3=3 ? If 1 has two square roots, i and i it is the same as meaning that i has two values: +i and i, which we can plot on the complex plane on the opposite sides of unit circle. Last edited by TwoTwo; March 28th, 2014 at 11:33 PM.  
March 5th, 2014, 03:14 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 21,026 Thanks: 2257 
With such a convention, adding 3 = 3 to 3 = 3 would give 6 = 0, etc., which doesn't seem useful. In the case of +i and i, you have acknowledged these correspond to opposite sides of a unit circle, and this wording acknowledges that the opposite sides are not the same side, so it would be confusing to hold that +i and i are the same or equal.


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