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December 24th, 2016, 11:44 PM   #21
Tau
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Quote:
Originally Posted by v8archie View Post
If I were referring to numbers on the complex plane rather than restricting myself to the real line, yes.
This is semantics, and not helpful to the discussion. De facto it is a real number and has properties that complex numbers don't have and vice versa. I answered your question. How about you comment on that i = -1/-1/-1/-1/... ?
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December 24th, 2016, 11:50 PM   #22
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You should read and understand the second half of JeffM1's post #8: "I strongly suspect that the highly confusing use of the − symbol in mathematics has led you astray".
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December 25th, 2016, 12:36 AM   #23
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Not at all, for a positive real n, -n is negative. For an imaginary number like i it doesn't mean that i is positive (i.e. greater than zero) and -i is negative, they are just conventions and we decide to run with i by default, all it means is that i + (- i) = 0.
But, here's the catch, we can't substitute i for -i willy-nilly, we have to be consistent. We either use i in all of mathematics (in place for (-1)^(1/2) ) or use -i in all of mathematics. For simplicity we chose the former.

So far so good, but nobody yet has debunked my strongest reasoning thus far
that taking the common wisdom "i is neither positive nor negative" actually results in "i is both positive and negative".
I rewrite it again:

"i is neither positive nor negative"
becomes
"i is not positive and not negative"
becomes
"i is not (not negative) and not (not positive)"
becomes
"i is (not not) negative and (not not) positive"
becomes
"i is negative and positive"

Of course I'm not suggesting writing it as +/-i as that means either +i or -i. We apparently need a new notation to denote "both positive and negative". Impossible you say? i is an impossible number to start with. Oh well.
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December 26th, 2016, 06:46 PM   #24
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Quote:
i is neither positive nor negative
Quote:
i is not positive and not negative
What? Does 'not positive' mean negative, or not positive. I think once you clearly define your difference between 'neither, nor' and 'not', you will see the flaw.
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December 26th, 2016, 07:26 PM   #25
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What? Does 'not positive' mean negative, or not positive. I think once you clearly define your difference between 'neither, nor' and 'not', you will see the flaw.
I doubt it. He implicitly extends the trichotomy to the complex numbers without defining "positive" and "negative" with respect to complex numbers. He sees that $i$ is not zero. He then concludes that $i$ must therefore either be positive or negative because of his assumed trichotomy. Of course $i$ acts neither in the way it would if it were positive nor in the way it would if it were negative. He then makes the rather astounding leap that, with $i$ failing to be positive and failing to be negative, it is simultaneously both.

The so-called logic is that because an elephant is not a bird and not a fish, it must be both a bird and a fish. Anyone who finds that compelling will not be dissuaded by pointing out that "not positive" does not mean "negative" even within the real numbers

Last edited by JeffM1; December 26th, 2016 at 07:31 PM.
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December 26th, 2016, 08:16 PM   #26
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Originally Posted by JeffM1 View Post
I doubt it. He implicitly extends the trichotomy to the complex numbers without defining "positive" and "negative" with respect to complex numbers. He sees that $i$ is not zero. He then concludes that $i$ must therefore either be positive or negative because of his assumed trichotomy. Of course $i$ acts neither in the way it would if it were positive nor in the way it would if it were negative. He then makes the rather astounding leap that, with $i$ failing to be positive and failing to be negative, it is simultaneously both.

The so-called logic is that because an elephant is not a bird and not a fish, it must be both a bird and a fish. Anyone who finds that compelling will not be dissuaded by pointing out that "not positive" does not mean "negative" even within the real numbers
It's all fun and games until someone wants to have a fallacious argument..
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December 27th, 2016, 04:23 AM   #27
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If you restrict yourself to the imaginary axis exclusively trichotomy holds. So $i > 0$ is positive , $-i<0$ is negative. Trichotomy fails when you go to ordered pairs $a + bi$

Where both $a$ , $b$ are non zero real numbers

Lol , there is no ordering of ordered pairs (in the usual sense)

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Last edited by agentredlum; December 27th, 2016 at 04:26 AM.
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December 27th, 2016, 07:11 AM   #28
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Jeff your analogy with the elephant is not comparable with the original and instead a better analogy would be "the elephant is not alive nor dead" and in science fiction those are called "dead-alive". If we see that i is like a fiction of mathematics the "both positive and negative" part is not such a stretch considering accepting i in the first place.

Last edited by Tau; December 27th, 2016 at 07:17 AM.
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December 27th, 2016, 07:57 AM   #29
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The point is that $i$ does not exist except as embedded in the conplex plane, which is a numbering system of ordered pairs. These ordered pairs do not a total ordering analogous to the total ordering of the real numbers viewed on the real number line. The real numbers, embedded in the complex plane are not subject to such a total ordering either.

On the complex plane, positive and negative have no meaning because the ordering does not exist which allows one to relate the numbers to zero.

The standard definitions of positive and negative refer to the standard ordering relation on the real number line. Since $i$ is not on that number line, the terms do not apply to that number. You are telling us that the colour blue smells more bitter than the colour black.
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December 27th, 2016, 07:58 AM   #30
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Originally Posted by agentredlum View Post
there is no ordering of ordered pairs (in the usual sense)
What about the rationals?
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