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December 24th, 2016, 10:35 AM   #11
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Originally Posted by Tau View Post
So, if i is positive then it is negative and if i is negative then it is positive. So i is both positive and negative, a contradiction. How do you get around this?
Assumption: $i$ is positive
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.
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December 24th, 2016, 11:29 AM   #12
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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.
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December 24th, 2016, 12:02 PM   #13
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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
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December 24th, 2016, 02:11 PM   #14
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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.
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December 24th, 2016, 02:35 PM   #15
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I can rewrite your argument with the law of excluded middle as follows
You are making stuff up. It is not true to say that "not positive" is equal to "negative".
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December 24th, 2016, 04:49 PM   #16
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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 non-zero 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.
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December 24th, 2016, 10:10 PM   #17
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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.

Quote:
Originally Posted by Tau View Post
I'm not concerned about complex numbers
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.
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December 24th, 2016, 11:05 PM   #18
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$i$ is purely imaginary. Would you consider 1 + 0$i$ a complex number?
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December 24th, 2016, 11:21 PM   #19
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If I were referring to numbers on the complex plane rather than restricting myself to the real line, yes.
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December 24th, 2016, 11:36 PM   #20
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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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