December 24th, 2016, 10:35 AM  #11  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,446 Thanks: 2119 Math Focus: Mainly analysis and algebra  Quote:
Logical conclusion: $i$ is negative Result: contradiction Implication: the assumption is invalid  $i$ is not positive Assumption: $i$ is negative Logical conclusion: $i$ is positive Result: contradiction Implication: the assumption is invalid  $i$ is not negative Thus $i$ is neither positive nor negative.  
December 24th, 2016, 11:29 AM  #12 
Member Joined: Jan 2014 Posts: 86 Thanks: 4 
I can rewrite your argument with the law of excluded middle as follows: Assumption: $i$ is positive Logical conclusion: $i$ is negative Result: contradiction Implication: the assumption is invalid  $i$ is not positive, thus it is negative Assumption: $i$ is negative Logical conclusion: $i$ is positive Result: contradiction Implication: the assumption is invalid  $i$ is not negative, thus it is positive Thus $i$ is both positive and negative. A simpler path to the same conclusion is if we start by your assertion "$i$ is neither positive nor negative", then becomes "$i$ is not (not negative) and not (not positive)", which becomes "$i$ is (not not) negative and (not not) positive", and finally "$i$ is negative and positive" Last edited by Tau; December 24th, 2016 at 11:32 AM. 
December 24th, 2016, 12:02 PM  #13 
Member Joined: Jan 2014 Posts: 86 Thanks: 4 
Another way to reformulate the impossible properties of $i$ is through its continued fraction: $i$ = 1/$i$ = 1/1/$i$ = 1/1/1/$i$ = 1/1/1/1/... the partial fractions alternate to infinity 1, 1, 1, 1 
December 24th, 2016, 02:11 PM  #14 
Senior Member Joined: May 2016 From: USA Posts: 524 Thanks: 232 
This is getting really silly. You have not given a definition of < and > with respect to complex numbers, let alone a coherent definition. You have not given a definition of positive and negative for complex numbers, let alone a coherent definition. And BOTH i = 1 and i =  1 are false statements. 
December 24th, 2016, 02:35 PM  #15 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,446 Thanks: 2119 Math Focus: Mainly analysis and algebra  
December 24th, 2016, 04:49 PM  #16 
Member Joined: Jan 2014 Posts: 86 Thanks: 4 
Jeff, I'm not concerned about complex numbers, why are you perplexing this? I'm investigating just the sign of i. It is a kind of 1, it can't be +1 or 1 and the common wisdom is it can't have any sign. I maintain that it has both. The signs are the usual bigger or smaller than zero definition. Archie, it is true in the case of nonzero numbers. Or to drive it home consider this statement "There has been a change in the GDP and has not been positive". Last edited by Tau; December 24th, 2016 at 05:16 PM. 
December 24th, 2016, 10:10 PM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,446 Thanks: 2119 Math Focus: Mainly analysis and algebra 
The problem is that you don't have a clear idea of what "bigger or smaller than zero" means in the complex plane. Is $i$ bigger or smaller than $1$? Not that it particularly matters how you answer. The rest of the world isn't going to follow your lead. This is utter nonsense. You can't talk about $i$ without being concerned about complex numbers. Your statement only serves to emphasise the fact that you don't know what you are talking about. Last edited by skipjack; December 24th, 2016 at 10:59 PM. 
December 24th, 2016, 11:05 PM  #18 
Member Joined: Jan 2014 Posts: 86 Thanks: 4 
$i$ is purely imaginary. Would you consider 1 + 0$i$ a complex number?

December 24th, 2016, 11:21 PM  #19 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,446 Thanks: 2119 Math Focus: Mainly analysis and algebra 
If I were referring to numbers on the complex plane rather than restricting myself to the real line, yes.

December 24th, 2016, 11:36 PM  #20 
Member Joined: Jan 2014 Posts: 86 Thanks: 4 
And to answer your question all we can say for sure is that i is not 1, but they have the same absolute value or magnitude. So though it's not i(mpossible) to know the size of i, which is just 1, its sign is elusive.


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